To reach your long-term goal of earning a college degree, you may need to make some short-term financial sacrifices along the way. With an average 4–6 year degree completion rate, your college education is an investment of your time, intellect, and finances. Attending college takes money, and the responsibility of paying tuition falls on you and/or your family. Speak openly with your family about how you plan to pay for your education now and in the future. When conversing with your family, make a clear determination of who will be paying which expenses. Also, remember that costs are not limited to tuition and housing (Education Financial Council, 2009), but also include:
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Fees:
Fees include costs such as activity fees, parking, etc. Please check the Bursar Services Web site to obtain the most recent price of fees.
Student Accounts – www.uta.edu/uta/tuition.php
Student Money Management Center –
https://www.uta.edu/business-affairs/smmc/
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Books and Supplies:
Books are surprisingly expensive, particularly if they are accompanied by a CD or workbook. In 2006–2007, the national average cost of books was $1000 per year. Supply costs can include pens, pencils, USB drives, notebooks, paper, etc.
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Equipment, housing materials, and miscellaneous items:
Equipment includes desktop and laptop computers, calculators, printers, etc. Housing materials include microwaves, refrigerators, toasters, pillows, comforters, etc. Miscellaneous expenses include clothes, cell phones, movies, iPods, etc.
Activity 9-1
Computing College Expenses
The first step toward managing your money is to create a realistic budget, followed by minimizing expenses and borrowing wisely. As a college student, you will have some expenses that occur intermittently (2–3 times per year) such as tuition, fees, and books. Using the College Expenses Budget worksheet below, compute your college expenses by semester. After you have computed your semester costs, you will need to compare it to the total resources from grants, loans, and scholarships.
College Expenses Budget
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Expenses for One Semester
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Semester Total
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Tuition and Fees
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Books
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Parking
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Total Expenses
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Resources to Pay for College Expenses
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Scholarships
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Grants
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Student Loan(s)
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Veterans Benefits
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Money from Parents/Relatives
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Money from Employment
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Other
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Total Resources
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Education Financial Council. (2009). Make your college plan. Making College More Affordable. Retrieved from www.efc.org/cs/root/resources/resources#plan
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Subtract the Total Expenses from the Total Resources. If this is a negative number, you will need to find additional funds to cover college expenses. The latter part of the chapter will discuss means to secure additional funds.
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If the difference between Total Expenses and Total Funds is a positive number, divide this number by 4 (the number of months in the semester) to determine the surplus amount of money you will have to help cover monthly living expenses.
Total College Expenses $ _______________
Minus Total Resources $ _______________ = ___________________
Positive Net Resources $ _______________ ÷ 4 = ___________________
College expenses and financial aid resources only make up half of your total costs. Next, you will need to estimate your monthly living expenses to get a better idea of how your money is being spent each month.
Living Expenses Budget
After covering the costs of tuition, housing, books, and supplies, you will still need money to live on while in school. While you may not have as much control over your income and college expenses as you would like, you can control most of your living expenses just by the choices you make. You will need to map out your “cash flow”: how much money is coming in (income) and where it is being spent (expenses) over a 12-month period.
Regardless of the method used, the process of creating a budget is basically the same: Record monthly income, track and document monthly expenses, and, finally, make adjustments to balance the cash flow. With a budget, you can make informed decisions about spending and saving your money.
Thought Question
You may have a general idea of where most of your money goes each month – housing, food, utilities, tuition – but can you account for every dollar?
The first rule of budgeting is to track and document your monthly expenses. Recording your monthly expenses can be a daunting task, so it is always helpful to use an accurate budgeting tool to track expenditures. Take a look below and you will notice that expenses are broken down into three broad categories (Federal Trade Commission, n.d.):
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Fixed Expenses:
Expenses that are typically the same each month, such as rent or mortgage, car payments, childcare, credit card payments, and savings. Fixed expenses are generally the most difficult to change.
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Variable Expenses:
Expenses that occur each month, but the amounts you spend on them vary from month to month. Examples include food, clothing, and utilities. Variable expenses generally have the greatest degree of flexibility.
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Periodic Expenses:
Expenses that occur annually, semi-annually, quarterly, or seasonally. Such expenses include car registration, car or home maintenance, gifts, taxes, and insurance.
Complete the following activity to identify your monthly expenditures.
ACTIVITY 9-2:
Tracking Your Weekly and Monthly Expenditures
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Gather information about how your money was spent during the past month from your checkbook register, receipts, credit card bills, online statements, and any other financial records you have. This will help you get the most accurate information.
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If you do not have complete financial records for the past month, begin recording all expenditures for the next several weeks.
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Transfer these daily totals to the Record of Weekly Expenditures worksheet.
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Complete the worksheet after you have at least 30 days expenditure information.
The grand total in your “Record of Weekly Living Expenses” shows you how much money you spend or should set aside each month to cover your expenses. Once you have documented all of your expenses, you will be ready to compute your monthly income and compare it to your monthly expenses to determine, what, if any, changes are necessary to balance your budget.
Thought Question
Do you spend more than you earn each month?
 Money. Retrieved from
http://www.afsaef.org
Compare Your Income to Expenditures
To determine your monthly income, you will want to include your “take-home pay” (earnings from your paycheck after taxes are deducted) and any other forms of income. Subtract your total monthly expenditures from your monthly income. If the number is positive, you have a surplus income and are living within your means. If the number is negative, your expenses are exceeding your income and you will need to cut back on spending.
Total Income $ ________________ – Total Expenses = ______
Balance Your Income and Expenses
If your expenses exceed your income, you have three options:
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Increase your monthly income
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Decrease your monthly expenses
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Combine increasing income and decreasing expenses
As a college student, you may not be able to eliminate a cash shortage by earning a higher income. However, you can look for ways to decrease your monthly expenses without depriving yourself of things you really need.
Decrease Fixed and Variable Expenses
There are fewer opportunities to decrease fixed expenditures than variable expenditures; however, some fixed expenses such as rent, car payments, and cell phone bills may be reduced through comparison shopping. While these steps may reduce your expenses, they can take time to complete and may result in additional charges.
Variable expenses, on the other hand, may be the easiest and quickest solution. Review your variable expenses with the intent of reducing the totals in some categories, rather than eliminating entire expense categories. For example, if you need to cut $200 from your budget and you usually spend $100 a month on cable television, would you be willing to decrease that by $50, thereby saving $50? Keep looking for places in which you can make similar, small adjustments until you have reduced your expenses by $200 (National Foundation for Credit Counseling, n.d.).
Use these pointers as you review your monthly expenses and continue to evaluate ways to cut back on any unnecessary expenses.
Tips on How to Trim Unnecessary Expenses
Shopping
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Don’t shop when you are hungry or bored you may spend more.
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Take advantage of sales, but do not buy items you do not need or items of poor quality.
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Do not buy on impulse. Think things over and make sure you really need the item.
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Compare unit prices to find the best deals. For example, it may be a better deal to buy an item in a larger quantity.
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Buy generic or store brands when you can. They are usually less expensive.
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Compare prices on items both within the store and at different stores. Try discount or thrift stores.
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Use coupons and rebates for additional items you need to buy.
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Make a list of what you need. Only buy what is on it.
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Keep in mind that nonfood items may be more expensive at supermarkets.
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Rent tools and equipment you will not use often instead of buying them.
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Beware of sales gimmicks. Don’t lose sight of what you need.
Food
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Plan meals so you can shop with a list.
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Buy fruits and vegetables that are in season.
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Pack your lunches and snacks instead of buying them.
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Use cheaper cuts of meat than a recipe calls for or substitute a less expensive type of meat.
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Be careful when buying prepared foods. They may be faster, but they are usually more expensive.
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Eat out less.
Transportation
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Use public transportation when possible. Ask if reduced fares are available. For example, you may be able to buy a pass for several trips instead of paying for each trip separately.
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Have regular maintenance done on your vehicle to avoid costly repairs.
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Compare fares on different airlines before purchasing tickets.
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Consider buying a used vehicle that is in good shape instead of a new one to keep monthly payment lower.
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Use the lowest octane gas recommended by your manufacturer.
Entertainment
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Attend free events offered at UTA and in the community. Check the UTA Calendar of Events and the Arlington Chamber of Commerce for more information.
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Rent a movie or go to a matinee instead of paying full price.
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Visit the UTA Central Library or Arlington Public Library. They often sponsor events, and you can also check out books and other materials for free.
Phone Services
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Comparison shop for the best cell phone plan.
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Make calls when rates are lowest or free.
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Get rid of services you do not use that often.
Housing
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In warm weather, raise the thermostats of air conditioners when no one is there or at night when it is cooler.
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In colder weather, lower the heat when no one is home.
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Lower your water heater to 120 degrees.
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Look into energy-saving devices, such as energy-saving light bulbs and temperature control devices.
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Learn to make repairs yourself to save money.
National Foundation for Credit Counseling (NFCC). (n.d.). Consumer Tools. Retrieved from www.nfcc.org/Financial Education/ConsumerTools.cfm
The budget allocation chart below shows the average percentage that families spend on certain budget categories. You will choose to spend money your own way, to meet your own goals as a student. The amount you spend in each category will vary by person and income level. The less you earn, the greater the percentage you will spend on basic expenditures.
Figure 9-3. Model Budget Allocation
Thought Question
Have you established a system for paying your bills each month?
Managing Cash Flow
Once you have enough money to cover all expenses, you may still find it difficult to pay your bills on time if due dates do not coincide with paydays. However, there are several tools available including the ones listed below. These methods help you to manage the flow of income and expenses in different ways (National Foundation for Credit Counseling, n.d.).\
Envelope Method
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This tool is useful if you pay your bills with cash each month.
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Make an envelope for each expense category (rent, utilities, food, etc.).
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Label the envelope with the purpose, amount needed, and due date.
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When you receive income, divide it into amounts to cover the expenses listed on the envelope.
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Pay bills as soon as you receive them so you will not be tempted to spend the money on something else.
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If you prefer using a check to pay larger expenses, such as rent or car payments, you can write the check and place it in the envelope until the payment is due.
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Any excess income can be applied toward next month’s expenses, other expenditures, or placed into a savings/emergency fund.
Calendar Method
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This method utilizes a monthly calendar.
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Record the income receipt dates and expense due dates on the calendar.
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It is helpful to use two different color pens when recording this information.
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This method gives you a visual representation of when you get paid versus when the bills are due.
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Any excess income can be applied to next month’s expenses, other expen- ditures, or placed into a savings/emergency fund.
Personal Financial Management Software
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This method helps you to organize income and expenses electronically, produce reports and graphs that compare data over any time period, and categorize spending to assist with tax preparation.
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There are many personal finance software products available on the market. You can find the products compared online.
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Before purchasing, try to sample several software programs to find one that meets your needs. For example, does the software allow you to access online banking, pay bills online, split bills and expenses when recording data, and meet other criteria important to you?
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Evaluate whether you have the time and expertise to use the features of the software program.
Monitor Your Progress
A workable budget can take anywhere from six months to a year to develop. Each week, record and document your income and expenses for that month. Then, evaluate the findings. If you consistently overspend in a budget category (such as eating out), you need to change the projected amount for that line. A purpose of a budget is to help you recognize what you can and cannot afford. If you find that you never have enough money at the end of the month, you may need to consider making some bigger changes.
Refer to the budget allocation guidelines in Figure 9-3, comparing what you spend in these categories with the suggested percentages. For example, if you are spending 35% on your car, could you consider trading your existing one for a less expensive, used, and more fuel-efficient model? If not, it may be time to consider increasing your income with an additional job, more hours on a cur- rent job, or a salary increase.
It is important to make a commitment to your established budget without too much focus on the temporary adjustments. If followed consistently, your budget will become an effective working tool that will help you take control of your finances (American Financial Services Educational Foundation, n.d.).
A realistic budget will help you:
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Live within your income
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Spend your money wisely
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Reach your financial goals
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Prepare for financial emergencies
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Develop intelligent money management habits
Student Loans
As you continue to pay for college, try to rely on sources that do not require repayment: scholarships, personal savings, summer work income, and prepaid or 529 savings plans. Like many other students, you may not be able to cover all expenses through these resources alone and must depend upon other avenues such as the federal government and institutional and state aid.
Students receive over $83 million yearly in federal student aid. To qualify for federal student aid, you must be a U.S. citizen or eligible non-citizen with a valid Social Security number. In addition, recipients must maintain satisfactory academic progress once in school (U.S. Department of Education, 2008). Federal student aid can come in the following forms:
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Grants:
Money that does not have to be repaid. Many students qualify based on demonstrated financial need.
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Work-Study:
Students work while simultaneously paying for college expenses.
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Loans
: Money is borrowed and must be repaid with interest.
As you plan for college, it is important to maximize sources of aid that do not require repayment.
Table 9-2. Maximize Sources of Aid You Do Not Have to Repay
Maximize the Sources on the Top of the chart
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DO NOT HAVE TO REPAY
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Scholarships and Grants
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Savings and Work-Study Earnings
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CHEAPEST LOANS
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EXPENSIVE LOANS
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Private Educational, Home Equity Loans
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Credit Cards
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U.S. Department of Education. (2008).  Washington, D.C.: U.S. Department of Education.
If you would like to get an estimate of your federal student aid eligibility, you can access FAFSA4caster. This free online tool will list the types of aid for which you are eligible, along with estimated grant award amounts and scenario-based award packages (U.S. Department of Education, 2009).
FAFSA4caster*
www.fafsa4caster.ed.gov/F4CApp/index/index.jsf
* A tool from Federal Student Aid—www.FederalStudentAid.ed.gov
While it would be ideal to cover all expenses through scholarships, grants, and work-study, you may have to rely on private or federal loans to cover all expenses. It is wise to closely compare private and federal loans before making a decision on which to use. Federal loans have lower fixed interest rates, reasonable repayment options, no repayment penalties, and no credit checks (excluding PLUS loans) (U.S. Department of Education, 2007).
Private loans, in contrast, are considerably more expensive than federal loans because they have higher, variable interest rates that increase with your repayment amount. Be careful: many private lenders aggressively market themselves through TV, radio, and on-campus solicitations.
Table 9-3. Federal Student Loans vs. Private Loans
Federal Student Loans
(Loans from the government or guaranteed by the government)
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Private Student Loans
(Non-federal loans from a bank, credit union, or other financial institution)
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You will not have to start repaying your federal student loans until you graduate, leave school, or change enrollment status to less than full-time.
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Many private student loans require payments while you are still in school.
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The interest rate on Stafford Loans is fixed, currently at 4.29 percent for subsidized loans for undergraduate students and 5.84 percent for unsubsidized loans for undergraduate and graduate students, and almost always lower than on a private loan—and much lower than on a credit card.
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Private loans can have variable interest rates greater than 18 percent.
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Students with greater financial need might qualify for a subsidized loan. The government pays the interest on subsidized loans while a borrower is enrolled in school at least half-time during certain periods.
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Private student loans are not subsidized. No one pays the interest on the loan but you.
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You do not need to pass a credit check to get a federal student loan (except for PLUS Loans). Federal student loans help you establish a good credit record.
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Private student loans require an established credit record. The cost of a private student loan depends on your credit score, which you may not yet have as a student.
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You do not need a co-signer to get a federal student loan.
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You may need a co-signer to get the best possible deal.
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Free help is available at 1-800-4-FED-AID.
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You need to find out if there is free help.
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Some interest is tax deductible.
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Interest may not be tax deductible.
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Loans can be consolidated into the Direct or FFEL Consolidation programs which have favorable repayment plans and other benefits. See www.loanconsolidation.ed.gov for more information.
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Private student loans cannot be consolidated into a federal loan consolidation program. They can only be consolidated into a private bank loan, if available.
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U.S. Department of Education. (2008).  Washington, D.C.: U.S. Department of Education.
Applying for Loans
To apply for a federal student loan, you will need to complete the Free Application for Federal Student Aid (FAFSA). The FAFSA determines if you are eligible to receive federal grants, work- study, and state and institutional aid. You will need to complete the FAFSA on or after January 1st of the year you expect to attend school. State and institutional aid awards are granted on a first-come, first-serve basis, so apply as early as possible (U.S. Department of Education, 2008).
Have the following documents (if applicable) available at the time of submission:
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Your Social Security Number
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Your driver’s license (if you have one)
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Your W-2 forms and other records of income
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Your (and your spouse’s, if you are married) Federal Income Tax Return
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Your parents’ Federal Income Tax Return (if you are a dependent)
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Your untaxed business records
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Your alien registration or permanent resident card (if you are not a U.S. citizen)
Even if you do not plan to take any federal student loans, it is always a good idea to complete the FAFSA anyway. In some cases, scholarship applications may require that students show financial need based on their FAFSA. And, you never know—the best-laid plans may change, and you may decide to take the federal aid mid-year.
Free Application for Federal Student Aid (FAFSA)
www.FAFSA.ed.gov
A Note on Interest Rates
Like credit cards and private loans, educational loans have interest rates. Interest is “a percentage of the original loan amount (the principle) that is added to the total repayment amount” (U.S. Department of Education, 2008). Essentially, it is a charge for borrowing money. Not all federal loans are created equal, and some are attached with larger interest rates.
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Unsubsidized loans:
Interest, which is your responsibility to pay, accrues from the time you receive the loan. You can repay the interest while you are attending school or you can allow it to accrue and be added to the principle balance, making it far more expensive.
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Subsidized Loans:
The federal government pays the interest on the loan while you are still attending college.


15.4: Aromaticity and the Hückel 4n + 2 Rule
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Objectives
After completing this section, you should be able to
- define aromaticity in terms of the Hückel 4 n + 2 rule.
- use the Hückel 4 n + 2 rule to determine whether or not a given polyunsaturated cyclic hydrocarbon should exhibit aromatic properties.
- describe the difference in properties between an aromatic hydrocarbon, such as benzene, and a non-aromatic polyunsaturated cyclic hydrocarbon, such as cyclobutadiene or cyclooctatetraene.
- draw molecular orbital diagrams for aromatic species, such as benzene, the cyclopentadienyl anion and pyridine, and compare these diagrams with those obtained for non-aromatic species, such as cyclobutadiene and the cyclopentadienyl cation.
Study Notes
The following mnemonic device will help you establish the approximate energy levels for the molecular orbitals of various organic ring systems.
Whatever the size of the ring, place one point of the ring down to the bottom. The corners of the ring, where the carbons are located, will roughly approximate the location and pattern of the molecular orbital energy levels. Cut the ring exactly in half. The energy levels in the top half will be anti-bonding ( Ψ * ) orbitals and those in the bottom will be bonding ( Ψ ) orbitals. If the carbons fall directly in the centre of the ring (e.g., four-membered rings) the energy levels there are non-bonding.
cyclopropenyl ring (three-membered ring)
cyclobutadienyl ring (four-membered ring)
cyclopentadienyl ring (five-membered ring)
In 1931, German chemist and physicist Erich Hückel proposed a theory to help determine if a planar ring molecule would have aromatic properties. His rule states that if a cyclic, planar molecule has \(4n+2\) \(π\) electrons, it is considered aromatic . This rule would come to be known as Hückel's Rule .
Four Criteria for Aromaticity
When deciding if a compound is aromatic, go through the following checklist. If the compound does not meet all the following criteria, it is likely not aromatic.
- The molecule is cyclic (a ring of atoms)
- The molecule is planar (all atoms in the molecule lie in the same plane)
- The molecule is fully conjugated (p orbitals at every atom in the ring)
- The molecule has 4n+2 π electrons (n=0 or any positive integer)
Counting π Electrons?
Perhaps the toughest part of Hückel's Rule is figuring out which electrons in the compound are actually π electrons. Once this is figured out, the rule is quite straightforward. Pi electrons lie in the p orbitals and sp 2 hybridized atoms have 1 of these p orbitals each. When looking at a C=C double bond we know that there is one sigma bond and one pi bond. The pi bond is formed by the overlap of 1 p orbital from each carbon in the double bond. Each p orbital contains one electron so when the two p orbitals overlap the two electrons become a pi bond and are thus called pi electrons. Because aromaticity deals directly with double bonds and conjugation it is simpler to just count the number pi electrons in a compound. Each double bond ( π bond) always contributes 2 π electrons. Benzene has 3 double bonds, so it has 6 π electrons.
Why 4n+2 π Electrons?
According to Hückel's Molecular Orbital Theory , a compound is particularly stable if all of its bonding molecular orbitals are filled with paired electrons. This is true of aromatic compounds, meaning they are quite stable. With aromatic compounds, 2 electrons fill the lowest energy molecular orbital, and 4 electrons fill each subsequent energy level (the number of subsequent energy levels is denoted by n), leaving all bonding orbitals filled and no anti-bonding orbitals occupied. This gives a total of 4n+2 \(\pi\) electrons. You can see how this works with the molecular orbital diagram for the aromatic compound, benzene, below. Benzene has 6 \(\pi\) electrons. Its first 2 \(\pi\) electrons fill the lowest energy orbital, and it has 4 \(\pi\) electrons remaining. These 4 fill in the orbitals of the succeeding energy level. Notice how all of its bonding orbitals are filled, but none of the anti-bonding orbitals have any electrons.
To apply the 4n+2 rule, first count the number of π electrons in the molecule. Then, set this number equal to 4n+2 and solve for n. If is 0 or any positive integer (1, 2, 3,...), the rule has been met. For example, benzene has six \(\pi\) electrons:
\[\begin{align} 4n + 2 &= 6 \\ 4n &= 4 \\ n &= 1 \end{align} \nonumber \]
For benzene, we find that \(n=1\), which is a positive integer, so the rule is met.
With Hückel's theory we can assume that if a molecule meets the other criteria for aromaticity and also has 2, 6, 10, 14, 18 ect. pi electrons it will most likely be aromatic.
Antiaromaticity
More than 100 years ago, chemists recognized the possible existence of other conjugated cyclic polyalkenes, which at least superficially would be expected to have properties like benzene. The most interesting of these are cyclobutadiene, whose shape and alignment of p orbitals suggested it should have substantial electron-delocalization energies.
Cyclobutadiene
However, cyclobutadiene was found to be an extremely unstable molecules. In fact, cyclobutadiene is even more reactive than most alkenes. The synthesis of cyclobutadiene eluded chemists for almost 100 years. As more work was done, it became increasingly clear that the molecule, when formed in reactions, was immediately converted to something else. Finally, cyclobutadiene was captured in an essentially rigid matrix of argon at 8K. On warming to even 35K , it dimerizes through a Diels Alder reaction to yield a tricyclicdiene.
Due to the square ring, cyclobutadiene was expected to have some degree of destabilization associated with ring strain. However, estimations of the strain energies, though substantial, did not account for cyclobutadiene's high degree of instability. Also, why was it not being stabilized through cyclic conjugation in the same way as benzene. The answer can be seen in the molecular orbitals.
We considering the molecular orbitals diagram of the analogous 1,3-butadiene, the four 2p atomic orbitals combine to form four pi molecular orbitals of increasing energy. Two bonding pi orbitals and two antibonding pi* orbitals. The 4 pi electrons of 1,3-butadiene completely fill the bonding molecular orbitals giving is the additional stability associated with conjugated double bonds.
However, when placed into a ring the molecular orbitals undergo a significant change. The four p orbitals of cyclobutadiene combine to form the following 4 molecular orbitals:
- a bonding molecular orbital which is lower in energy than the atomic orbitals
- two degenerate non-bonding molecular orbitals which are of equivalent energy to the atomic orbitals
- one antibonding molecular orbital which is higher in energy
The non-bonding orbitals represent that there is no direct interaction between adjacent atoms. When adding cyclobutadiene's 4 pi electrons to the molecular orbital diagram, the bonding orbital is filled and both non-bonding orbitals are singly occupied. If cyclobutadiene's double bonds were delocalized, all the pi electrons would be in low energy bonding orbitals. However, only two of the pi electrons are in bonding orbitals; the other two are non-bonding. In addition, the molecular orbital diagram shows that two of the electrons are unpaired, a situation called a triplet state, which usually makes organic molecules very reactive. For cyclobutadiene cyclic conjugation has made the molecule less stable. This unexpected instability in 4n π-electron cyclic conjugated compounds is termed antiaromaticity and the compounds are called antiaromatic.
Antiaromaticity gives cyclobutadiene some interesting structural features. Cyclobutadiene's single and double bonds have different bond lengths, 158 pm and 135 pm respectively which give it a rectangular shape. If the double bonds were conjugated, there would only be one average bond length, like benzene, and shape would be square. In fact, cyclobutadiene's double bonds are not conjugated but locked into position.
Just like 4n + 2 rule with aromaticity, compounds which are flat, cyclic, have a p orbital at every member of the ring, and 4n pi electrons should be antiaromatic. Another potentially antiaromatic molecule is 1,3,5,7-cyclooctatetraene.
Cyclooctatetraene was first synthesized in 1911 by a German chemist, R. Willstatter (Nobel Prize 1915), who reported an extraordinary thirteen-step synthesis of cyclooctatetraene from a rare alkaloid called pseudopelletierine isolated from the bark of pomegranate trees. However, during the Second World War, the German chemist W. Reppe found that cyclooctatetraene can be made in reasonable yields by the tetramerization of ethyne under the influence of a nickel cyanide catalyst:
If cyclooctatetraene was flat it would have all the requirements to be antiaromatic. It is cyclic, has a p orbital in every member of the ring, and has 8 pi electrons. When looking at the molecular orbital diagram of cyclooctatetraene we see it shares certain characteristics in common with cyclobutadiene. Two of the molecular orbitals are degenerate non-bonding orbitals. When the molecular orbitals are fill with cyclooctatetraene's 8 pi electrons the last two electrons are unpaired in the two nonbonding orbitals creating a triplet state. Based off the molecular orbital diagram cyclooctatetraene should be antiaromatic.
However, cyclooctatetraene is easily prepared and relatively stable. Also, it undergoes addition reactions typical of alkenes. In reality, cyclooctatetraene is not flat but has a tub-like shape. By assuming this shape, the p orbtials between double bond are out of alignment for overlap. Because the double bonds are not conjugated, cyclooctatetraene escapes the destabilizing effects of antiaromaticity. This makes its its pi bonds react like 'normal' alkenes. Because cyclooctatetraene is not flat nor conjugated it is properly defined as non-aromatic. In general, if an antiaromatic compound has the ability to form a non-planar shape it will do so to avoid destabilization by becoming non-aromatic.
Determining if a Compound is Aromatic, Antiaromatic, or Nonaromatic
To make this determination it is important to first as if the compound has the possibility of cyclic conjugation. This requires that the compound be cyclic and have a p orbital at every atom in the ring. If the compound does not have both of these criteria it cannot be aromatic or antiaromatic and must therefore be nonaromatic. Next the number of pi electrons in the ring is determined to see if it follows the count of aromaticity (4n + 2) or antiaromaticity (4n). Before making the final determination, it is vital to know the actual geometry of the molecule. Despite the number of pi electrons, a compound cannot be aromatic or antiaromatic if its geometry is not planar to allow for p orbital overlap.
Example \(\PageIndex{1}\)
A Common Misconception
A very common misconception is that hybridization can be used to predict the geometry, or that hybridization somehow involves an energy cost associated with 'promoting' electrons into the hybrid orbitals. This is entirely wrong. Hybridization is always determined by geometry. You can only assign hybridization states to an atom if you already know its geometry, based on some experimental or theoretical evidence. The geometry of the oxygen in furan is trigonal planar and therefore the hybridization must be sp 2 .
The specific rule is that if you have an sp 2 conjugated system, the lone pair will be involved if it makes the system more stable. In this case, conferring Hückel 4n+2 aromaticity. For furan with two lone pairs on the oxygen atom, if we count electrons from the carbon atoms, we have 4 (one per carbon). So adding two electrons from one of the lone pairs will give 6 = 4(1)+2, so Hückel rule is applicable and furan is aromatic.
Exercise \(\PageIndex{1}\)
Using the criteria for aromaticity, determine if the following molecules are aromatic:
- Answer
-
a) Aromatic - only 1 of S's lone pairs counts as π electrons, so there are 6 π electrons, n=1
b) Not aromatic - not fully conjugated, top C is sp 3 hybridized
c) Not aromatic - top C is sp 2 hybridized, but there are 4 π electrons, n=1/2
d) Aromatic - N is using its 1 p orbital for the electrons in the double bond, so its lone pair of electrons are not π electrons, there are 6 π electrons, n=1
e) Aromatic - there are 6 π electrons, n=1
f) Not aromatic - all atoms are sp 2 hybridized, but only 1 of S's lone pairs counts as π electrons, so there 8 π electrons, n=1.5
g) Not aromatic - there are 4 π electrons, n=1/2
h) Aromatic - only 1 of N's lone pairs counts as π electrons, so there are 6 π electrons, n=1
i) Not aromatic - not fully conjugated, top C is sp 3 hybridized
j) Aromatic - O is using its 1 p orbital for the elections in the double bond, so its lone pair of electrons are not π electrons, there are 6 π electrons, n=1
Exercise \(\PageIndex{2}\)
To be aromatic, a molecule must be planar conjugated, and obey the 4n+2 rule. The following is the following molecule aromatic?
- Answer
-
No, it is not. It does not obey the 4n+2 rule. Also it is not planar.
References
- Vollhardt, Peter, and Neil E. Schore. Organic Chemistry: Structure and Function . 5th ed. New York: W. H. Freeman & Company, 2007.
- Berson, Jerome. Chemical Creativity: Ideas from the Work of Woodward, Hückel, Meerwein, and Others . New York: Wiley-VCH, 1999.
- Badger, G.M. Aromatic Character and Aromaticity . London, England: Cambridge University Press, 1969.
- Lewis, David and David Peters. Facts and Theories of Aromaticity . London, England: Macmillan Press, 1975.


15.19A: Anatomy of the Eye
- Explain how eyes have evolved to benefit organisms
Anatomy of the Eye
The retina, a thin layer of cells located on the inner surface of the back of the eye, consists of photoreceptive cells, which are responsible for the transduction of light into nervous impulses. However, light does not enter the retina unaltered; it must first pass through other layers that process it so that it can be interpreted by the retina.
The cornea, the front transparent layer of the eye, along with the crystalline lens, refract (bend) light to focus the image on the retina. After passing through the cornea, light passes through the aqueous humour, which connects the cornea to the lens. This clear gelatinous mass also provides the corneal epithelium with nutrients and helps maintain the convex shape of the cornea. The iris, which is visible as the colored part of the eye, is a circular muscular ring lying between the lens and the aqueous humour that regulates the amount of light entering the eye. Light passes through the center of the iris, the pupil, which actively adjusts its size to maintain a constant level of light entering the eye. In conditions of high ambient light, the iris contracts, reducing the size of the pupil. In conditions of low light, the iris relaxes and the pupil enlarges.
The main function of the lens is to focus light on the retina and fovea centralis. The lens is a transparent, convex structure located behind the cornea. On the other side of the lens is the vitreous humour, which lets light through without refraction, maintains the shape of the eye, and suspends the delicate lens. The lens focuses and re-focuses light as the eye rests on near and far objects in the visual field. The lens is operated by muscles that stretch it flat or allow it to thicken, changing the focal length of light coming through to focus it sharply on the retina. With age comes the loss of the flexibility of the lens; a form of farsightedness called presbyopia results. Presbyopia occurs because the image focuses behind the retina. It is a deficit similar to a different type of farsightedness, hyperopia, caused by an eyeball that is too short. For both defects, images in the distance are clear, but images nearby are blurry. Myopia (nearsightedness) occurs when an eyeball is elongated and the image focus falls in front of the retina. In this case, images in the distance are blurry, but images nearby are clear.
There are two types of photoreceptors in the retina: rods and cones. Both are named for their general appearance. Rods, strongly photosensitive, are located in the outer edges of the retina. They detect dim light and are used primarily for peripheral and nighttime vision. Cones, weakly photosensitive, are located near the center of the retina. They respond to bright light; their primary role is in daytime, color vision.
The fovea is the region in the center back of the eye that is responsible for acute (central) vision. The fovea has a high density of cones. When you bring your gaze to an object to examine it intently in bright light, the eyes orient so that the object’s image falls on the fovea. However, when looking at a star in the night sky or other object in dim light, the object can be better viewed by the peripheral vision because it is the rods at the edges of the retina, rather than the cones at the center, that operate better in low light. In humans, cones far outnumber rods in the fovea.
Key Points
- The cornea and the lens bend light to focus the image on the retina; the iris and pupil regulate the amount of light entering the eye.
- The aqueous humour maintains the convex shape of the cornea; the vitreous humour supports the lens and maintains the shape of the entire eye.
- Presbyopia occurs because the image focuses behind the retina; it is similar to hyperopia (farsightedness), which is caused by an eyeball that is too short.
- Myopia (nearsightedness) occurs when an eyeball is elongated; images in the distance appear blurry, but images nearby are clear.
- Rods are used for peripheral and nighttime vision; cones are used for daytime and color vision.
- The fovea is responsible for acute vision because it has a high density of cones.
Key Terms
- rod : a rod-shaped cell located in the outer retina of the eye that is extremely sensitive to light
- retina : the thin layer of cells at the back of the eyeball where light is converted into neural signals sent to the brain
- cone : cell located near the center of the retina that is weakly photosensitive and is responsible for color vision in relatively bright light


2.5: Properties of Multiplication
- understand and appreciate the commutative and associative properties of multiplication
- understand why 1 is the multiplicative identity
The Commutative Property of Multiplication
Commutative Property of
Multiplication
The product of two whole numbers is the same regardless of the order of the factors.
Multiply the two whole numbers.
Solution
\(6 \cdot 7 = 42\)
\(7 \cdot 6 = 42\)
The numbers 6 and 7 can be multiplied in any order. Regardless of the order they are multiplied, the product is 42.
Practice Set A
Use the commutative property of multiplication to find the products in two ways.
- Answer
-
\(15 \cdot 6 = 90\) and \(6 \cdot 15 = 90\)
Practice Set A
- Answer
-
\(432 \cdot 428 = 184,896\) and \(428 \cdot 432 = 184,896\)
The Associative Property of Multiplication
Associative Property of Multiplication
If three whole numbers are multiplied, the product will be the same if the first two are multiplied first and then that product is multiplied by the third, or if the second two are multiplied first and that product is multiplied by the first. Note that the order of the factors is maintained.
It is a common mathematical practice to use parentheses to show which pair of numbers is to be combined first.
Multiply the whole numbers.
Solution
\((8 \cdot 3) \cdot 14 = 24 \cdot 14 = 336\)
\(8 \cdot (3 \cdot 14) = 8 \cdot 42 = 336\)
Practice Set B
Use the associative property of multiplication to find the products in two ways.
- Answer
-
168
Practice Set B
- Answer
-
165,564
The Multiplicative Identity
The whole number 1 is called the multiplicative identity , since any whole number multiplied by 1 is not changed.
Multiply the whole numbers.
Solution
\(12 \cdot 1 = 12\)
\(1 \cdot 12 = 12\)
Practice Set C
Multiply the whole numbers.
- Answer
-
843
Exercises
For the following problems, multiply the numbers.
Exercise \(\PageIndex{1}\)
- Answer
-
234
Exercise \(\PageIndex{2}\)
Exercise \(\PageIndex{3}\)
- Answer
-
4,032
Exercise \(\PageIndex{4}\)
Exercise \(\PageIndex{5}\)
- Answer
-
326,000
Exercise \(\PageIndex{6}\)
Exercise \(\PageIndex{7}\)
- Answer
-
252
Exercise \(\PageIndex{8}\)
Exercise \(\PageIndex{9}\)
- Answer
-
21,340
Exercise \(\PageIndex{10}\)
Exercise \(\PageIndex{11}\)
- Answer
-
8,316
Exercise \(\PageIndex{12}\)
For the following 4 problems, show that the quantities yield the same products by performing the multiplications.
Exercise \(\PageIndex{13}\)
\((4 \cdot 8) \cdot 2\) and \(4 \cdot (8 \cdot 2)\)
- Answer
-
\(32 \cdot 2 = 64 = 4 \cdot 16\)
Exercise \(\PageIndex{14}\)
\((100 \cdot 62) \cdot 4\) and \(100 \cdot (62 \cdot 4)\)
Exercise \(\PageIndex{15}\)
\(23 \cdot (11 \cdot 106)\) and \((23 \cdot 11) \cdot 106\)
- Answer
-
\(23 \cdot 1,166 = 26,818 = 253 \cdot 106\)
Exercise \(\PageIndex{16}\)
\(1 \cdot (5 \cdot 2)\) and \((1 \cdot 5) \cdot 2\)
Exercise \(\PageIndex{17}\)
The fact that
\((\text{a first number } \cdot \text{ a second number}) \cdot \text{a third number} = \text{a first number} \cdot (\text{a second number } \cdot \text{ a third number})\)
an example of the property of multiplication.
- Answer
-
associative
Exercise \(\PageIndex{18}\)
The fact that \(1 \cdot \text{ any number} = \text{that particular number}\) is an example of the property of multiplication.
Exercise \(\PageIndex{19}\)
Use the numbers 7 and 9 to illustrate the commutative property of multiplication.
- Answer
-
\(7 \cdot 9 = 63 = 9 \cdot 7\)
Exercise \(\PageIndex{20}\)
Use the numbers 6, 4, and 7 to illustrate the associative property of multiplication.
Exercises for Review
Exercise \(\PageIndex{21}\)
In the number 84,526,098,441, how many millions are there?
- Answer
-
6
Exercise \(\PageIndex{22}\)
Replace the letter m with the whole number that makes the addition true.
\[\begin{array} {r} {85} \\ {\underline{+\ \ \ m}} \\ {97} \end{array}\nonumber\]
Exercise \(\PageIndex{23}\)
Use the numbers 4 and 15 to illustrate the commutative property of addition.
- Answer
-
\(4 + 15 = 19\)
\(15 + 4 = 19\)
Exercise \(\PageIndex{24}\)
Find the product. \(8,000,000 \times 1,000\)
Exercise \(\PageIndex{25}\)
Specify which of the digits 2, 3, 4, 5, 6, 8,10 are divisors of the number 2.
- Answer
-
2, 3, 4, 6


9.3: Employee Motivation
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Learning Objectives
- Explain how employees are motivated according to Maslow’s hierarchy of needs.
- Explain how ERG theory addresses the limitations of Maslow’s hierarchy.
- Describe the difference between factors contributing to employee motivation and how these differ from factors contributing to dissatisfaction.
- Describe the needs for achievement, power, and affiliation, and how these needs affect work behavior.
The earliest answer to motivation involved understanding individual needs. Specifically, early researchers thought that employees try hard and demonstrate goal-driven behavior to satisfy needs. For example, an employee who is always walking around the office talking to people may have a need for companionship and his behavior may be a way of satisfying this need. There are four major theories in the need-based category: Maslow’s hierarchy of needs, ERG theory, Herzberg’s dual factor theory, and McClelland’s acquired needs theory.
Maslow’s Hierarchy of Needs
Abraham Maslow is among the most prominent psychologists of the 20th century and the hierarchy of needs, accompanied by the pyramid representing how human needs are ranked, is an image familiar to most business students and managers. Maslow’s theory is based on a simple premise: Human beings have needs that are hierarchically ranked (Maslow, 1943; Maslow, 1954). There are some needs that are basic to all human beings, and in their absence, nothing else matters. As we satisfy these basic needs, we start looking to satisfy higher-order needs. Once a lower-level need is satisfied, it no longer serves as a motivator.
The most basic of Maslow’s needs are physiological needs . Physiological needs refer to the need for air, food, and water. Imagine being very hungry. At that point, all your behavior may be directed at finding food. Once you eat, though, the search for food ceases and the promise of food no longer serves as a motivator. Once physiological needs are satisfied, people tend to become concerned about safety . Are they safe from danger, pain, or an uncertain future? One level up, social needs refer to the need to bond with other human beings, to be loved, and to form lasting attachments. In fact, having no attachments can negatively affect health and well-being (Baumeister & Leary, 1995). The satisfaction of social needs makes esteem needs more salient. Esteem needs refer to the desire to be respected by one’s peers, feeling important, and being appreciated. Finally, at the highest level of the hierarchy, the need for self-actualization refers to “becoming all you are capable of becoming.” This need manifests itself by acquiring new skills, taking on new challenges, and behaving in a way that will lead to the satisfaction of one’s life goals.
Maslow’s hierarchy is a systematic way of thinking about the different needs employees may have at any given point and explains different reactions they may have to similar treatment. An employee who is trying to satisfy her esteem needs may feel gratified when her supervisor praises her. However, another employee who is trying to satisfy his social needs may resent being praised by upper management in front of peers if the praise sets him apart from the rest of the group.
So, how can organizations satisfy their employees’ various needs? By leveraging the various facets of the planning-organizing-leading-controlling (P-O-L-C) functions. In the long run, physiological needs may be satisfied by the person’s paycheck, but it is important to remember that pay may satisfy other needs such as safety and esteem as well. Providing generous benefits, including health insurance and company-sponsored retirement plans, as well as offering a measure of job security, will help satisfy safety needs. Social needs may be satisfied by having a friendly environment, providing a workplace conducive to collaboration and communication with others. Company picnics and other social get-togethers may also be helpful if the majority of employees are motivated primarily by social needs (but may cause resentment if they are not and if they have to sacrifice a Sunday afternoon for a company picnic). Providing promotion opportunities at work, recognizing a person’s accomplishments verbally or through more formal reward systems, job titles that communicate to the employee that one has achieved high status within the organization are among the ways of satisfying esteem needs. Finally, self-actualization needs may be satisfied by providing development and growth opportunities on or off the job, as well as by assigning interesting and challenging work. By making the effort to satisfy the different needs each employee may have at a given time, organizations may ensure a more highly motivated workforce.
ERG Theory
ERG theory of Clayton Alderfer is a modification of Maslow’s hierarchy of needs (Alderfer, 1969). Instead of the five needs that are hierarchically organized, Alderfer proposed that basic human needs may be grouped under three categories, namely, Existence , Relatedness , and Growth (see the following figure). Existence need corresponds to Maslow’s physiological and safety needs, relatedness corresponds to social needs, and growth need refers to Maslow’s esteem and self actualization.
Source : Based on Alderfer, C. P. (1969). An empirical test of a new theory of human needs. Organizational Behavior and Human Performance, 4 , 142–175.
ERG theory’s main contribution to the literature is its relaxation of Maslow’s assumptions. For example, ERG theory does not rank needs in any particular order and explicitly recognizes that more than one need may operate at a given time. Moreover, the theory has a “frustration-regression” hypothesis, suggesting that individuals who are frustrated in their attempts to satisfy one need may regress to another one. For example, someone who is frustrated by the lack of growth opportunities in his job and slow progress toward career goals may regress to relatedness needs and start spending more time socializing with one’s coworkers. The implication of this theory is that we need to recognize the multiple needs that may be driving an individual at a given point to understand his behavior and to motivate him.
Two-Factor Theory
Frederick Herzberg approached the question of motivation in a different way. By asking individuals what satisfies them on the job and what dissatisfies them, Herzberg came to the conclusion that aspects of the work environment that satisfy employees are very different from aspects that dissatisfy them (Herzberg, et. al., 1959; Herzberg, 1965). Herzberg labeled factors causing dissatisfaction of workers as “hygiene” factors because these factors were part of the context in which the job was performed, as opposed to the job itself. Hygiene factors included company policies, supervision, working conditions, salary, safety, and security on the job. To illustrate, imagine that you are working in an unpleasant work environment. Your office is too hot in the summer and too cold in the winter. You are being harassed and mistreated. You would certainly be miserable in such a work environment. However, if these problems were solved (your office temperature is just right and you are not harassed at all), would you be motivated? Most likely, you would take the situation for granted. In fact, many factors in our work environment are things that we miss when they are absent, but take for granted if they are present.
In contrast, motivators are factors that are intrinsic to the job, such as achievement, recognition, interesting work, increased responsibilities, advancement, and growth opportunities. According to Herzberg’s research, motivators are the conditions that truly encourage employees to try harder.
Herzberg’s research, which is summarized in the figure above, has received its share of criticism (Cummings & Elsalmi, 1968; House & Wigdor, 1967). One criticism relates to the classification of the factors as hygiene or motivator. For example, pay is viewed as a hygiene factor. However, pay is not necessarily a contextual factor and may have symbolic value by showing employees that they are being recognized for their contributions as well as communicating to them that they are advancing within the company. Similarly, quality of supervision or relationships employees form with their supervisors may determine whether they are assigned interesting work, whether they are recognized for their potential, and whether they take on more responsibilities. Despite its limitations, the two-factor theory can be a valuable aid to managers because it points out that improving the environment in which the job is performed goes only so far in motivating employees.
Acquired Needs Theory
Among the need-based approaches to motivation, Douglas McClelland’s acquired needs theory is the one that has received the greatest amount of support. According to this theory, individuals acquire three types of needs as a result of their life experiences. These needs are need for achievement, need for affiliation, and need for power. All individuals possess a combination of these needs.
Those who have high need for achievement have a strong need to be successful. A worker who derives great satisfaction from meeting deadlines, coming up with brilliant ideas, and planning his or her next career move may be high in need for achievement. Individuals high on need for achievement are well suited to positions such as sales where there are explicit goals, feedback is immediately available, and their effort often leads to success (Harrell & Stahl, 1981; Trevis & Certo, 2005; Turban & Keon, 1993). Because of their success in lower-level jobs, those in high need for achievement are often promoted to higher-level positions (McClelland & Boyatzis, 1982). However, a high need for achievement has important disadvantages in management. Management involves getting work done by motivating others. When a salesperson is promoted to be a sales manager, the job description changes from actively selling to recruiting, motivating, and training salespeople. Those who are high in need for achievement may view managerial activities such as coaching, communicating, and meeting with subordinates as a waste of time. Moreover, they enjoy doing things themselves and may find it difficult to delegate authority. They may become overbearing or micromanaging bosses, expecting everyone to be as dedicated to work as they are, and expecting subordinates to do things exactly the way they are used to doing (McClelland & Burnham, 1976).
Individuals who have a high need for affiliation want to be liked and accepted by others. When given a choice, they prefer to interact with others and be with friends (Wong & Csikszentmihalyi, 1991). Their emphasis on harmonious interpersonal relationships may be an advantage in jobs and occupations requiring frequent interpersonal interaction, such as social worker or teacher. In managerial positions, a high need for affiliation may again serve as a disadvantage because these individuals tend to be overly concerned about how they are perceived by others. Thus, they may find it difficult to perform some aspects of a manager’s job such as giving employees critical feedback or disciplining poor performers.
Finally, those with high need for power want to influence others and control their environment. Need for power may be destructive of one’s relationships if it takes the form of seeking and using power for one’s own good and prestige. However, when it manifests itself in more altruistic forms, such as changing the way things are done so that the work environment is more positive or negotiating more resources for one’s department, it tends to lead to positive outcomes. In fact, need for power is viewed as important for effectiveness in managerial and leadership positions (Mcclelland & Burnham, 1976; Spangler & House, 1991; Spreier, 2006).
McClelland’s theory of acquired needs has important implications for motivating employees. While someone who has high need for achievement may respond to goals, those with high need for affiliation may be motivated to gain the approval of their peers and supervisors, whereas those who have high need for power may value gaining influence over the supervisor or acquiring a position that has decision-making authority. And, when it comes to succeeding in managerial positions, individuals who are aware of the drawbacks of their need orientation can take steps to overcome these drawbacks.
Key Takeaway
Need-based theories describe motivated behavior as individual efforts to meet needs. According to this perspective, the manager’s job is to identify what people need and then to make sure that the work environment becomes a means of satisfying these needs. Maslow’s hierarchy categorizes human needs into physiological, safety, social, esteem, and self-actualization needs. ERG theory is a modification of Maslow’s hierarchy, where the five needs are collapsed into three categories (existence, relatedness, and growth). The two-factor theory differentiates between factors that make people dissatisfied on the job (hygiene factors) and factors that truly motivate employees. Finally, acquired-needs theory argues that individuals possess stable and dominant motives to achieve, acquire power, or affiliate with others. Each of these theories explains characteristics of a work environment that motivate employees.
Exercises
- Many managers assume that if an employee is not performing well, the reason must be lack of motivation. What is the problem with this assumption?
- Review Maslow’s hierarchy of needs. Do you agree with the particular ranking of employee needs?
- Review the hygiene and motivators in the two-factor theory. Are there any hygiene factors that you would consider to be motivators and vice versa?
- A friend of yours is competitive, requires frequent and immediate feedback, and enjoys accomplishing things. She has recently been promoted to a managerial position and seeks your advice. What would you tell her?
- Which motivation theory have you found to be most useful in explaining why people behave in a certain way? Why?
References
Alderfer, C. P. (1969). An empirical test of a new theory of human needs. Organizational Behavior and Human Performance, 4 , 142–175.
Baumeister, R. F., & Leary, M. R. (1995). The need to belong: Desire for interpersonal attachments as a fundamental human motivation. Psychological Bulletin, 117 , 497–529.
Cummings, L. L., & Elsalmi, A. M. (1968). Empirical research on the bases and correlates of managerial motivation. Psychological Bulletin, 70 , 127–144.
Harrell, A. M., & Stahl, M. J. (1981). A behavioral decision theory approach for measuring McClelland’s trichotomy of needs. Journal of Applied Psychology, 66 , 242–247.
Herzberg, F. (1965). The motivation to work among Finnish supervisors. Personnel Psychology , 18 , 393–402.
Herzberg, F., Mausner, B., & Snyderman, B. (1959). The motivation to work . New York: Wiley.
House, R. J., & Wigdor, L. A. (1967). Herzberg’s dual-factor theory of job satisfaction and motivation: A review of the evidence and a criticism. Personnel Psychology, 20 , 369–389.
Maslow, A. H. (1943). A theory of human motivation. Psychological Review, 50 , 370–396.
Maslow, A. H. (1954). Motivation and personality . New York: Harper.
McClelland, D. C., & Boyatzis, R. E. (1982). Leadership motive pattern and long-term success in management. Journal of Applied Psychology, 67 , 737–743.
McClelland, D. C., & Burnham, D. H. (1976). Power is the great motivator. Harvard Business Review, 25 , 159–166.
Spangler, W. D., & House, R. J. (1991). Presidential effectiveness and the leadership motive profile. Journal of Personality and Social Psychology, 60 , 439–455.
Spreier, S. W. (2006). Leadership run amok. Harvard Business Review, 84 , 72–82.
Trevis, C. S., & Certo, S. C. (2005). Spotlight on entrepreneurship. Business Horizons, 48 , 271–274.
Turban, D. B., & Keon, T. L. (1993). Organizational attractiveness: An interactionist perspective. Journal of Applied Psychology , 78 , 184–193.
Wong, M. M., & Csikszentmihalyi, M. (1991). Affiliation motivation and daily experience: Some issues on gender differences. Journal of Personality and Social Psychology, 60 , 154–164.


1.3.9: Inverse Functions
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Learning Objectives
- Determine if a function is 1-1.
- Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
- Find or evaluate the inverse of a function.
- Verify inverse functions.
- Use the graph of a one-to-one function to graph its inverse function on the same axes.
Are You Ready?
Try these questions prior to beginning this section to help determine if you are set up for success:
- Simplify: \(\dfrac{2+\dfrac{1}{x}}{\dfrac{1}{x}-8}\nonumber\)
- What is the domain of \(f(x)=\sqrt{2x-5}\). State your answer in interval notation.
- Answer
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- \(\quad \dfrac{2x+1}{1-8x}\), If you missed this problem or feel you could use more practice, review [ 2.14: Adding and Subtracting Rational Expressions ]
- \(\quad \left[\dfrac{5}{2},\infty\right)\), If you missed this problem or feel you could use more practice, review [ 3.3: Functions and Function Notation ]
A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.
If some physical machines can run in two directions, we might ask whether some of the function “machines” we have been studying can also run backwards. Figure \(\PageIndex{1}\) provides a visual representation of this question. In this section, we will consider the reverse nature of functions.
Figure \(\PageIndex{1}\): Can a function “machine” operate in reverse?
This idea of operating a function in reverse is known as an inverse operation. It is important when doing so that we know if the inverse is itself a function also. Let's take a moment to review what a function is and how we may determine if an inverse of a function will itself be a function.
Recall: A function of (\x\) is a relation in which each input, (\x\), value in the domain is assigned to exactly one output, value in the range.
One-to-One Functions
Some functions have a given output value that corresponds to two or more input values. However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions .
To visualize this concept, let’s look again at some simple relations in Figures \(\PageIndex{2}\) below. The function in part (a) shows a relationship that is not a one-to-one function because inputs \(q\) and \(r\) both give output \(n\). The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.
One-to-One Function
A one-to-one function is a function in which each output value corresponds to exactly one input value.
Example \(\PageIndex{1}\)
Is the area of a circle a function of its radius? If yes, is the function one-to-one?
Solution
A circle of radius \(r\) has a unique area measure given by \(A={\pi}r^2\), so for any input, \(r\), there is only one output, \(A\). The area is a function of radius\(r\).
If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure \(A\) is given by the formula \(A={\pi}r^2\). Because areas and radii are positive numbers, there is exactly one solution:\(\sqrt{\frac{A}{\pi}}\). So the area of a circle is a one-to-one function of the circle’s radius.
You Try \(\PageIndex{1}\)
- Is a balance in a bank account a function of the bank account number?
- Is a bank account number a function of the balance?
- Is a balance a one-to-one function of the bank account number?
- Answer
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- yes, because each bank account has a single balance at any given time;
- no, because several bank account numbers may have the same balance;
- no, because the same output may correspond to more than one input.
Horizontal Line Test
An easy way to determine if a function is a one-to-one function is to use the horizontal line test . Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, as seen in Figure \(\PageIndex{3}\), then the graph does not represent a one-to-one function.
Example \(\PageIndex{2}\)
Consider the relations shown below in Figure \(\PageIndex{4}\). Are any of these one-to-one functions?
Solution
The graphs in (a) and (b) are functions since they satisfy the vertical line test. The graph in (a) is not a one-to-one function since a horizontal line will intersect the graph more than once. The graph in (b) is a one-to-one function since there is not any horizontal line that will intersect the graph more than once. The graph is part (c) is not a function so it not a one-to-one function.
Inverse Functions
Suppose a fashion designer will be traveling to Milan for a fashion show and wants to know what the temperature will be. She finds the weather forecast online, as shown in Figure \(\PageIndex{5}\). She is not familiar with the Celsius scale and her internet connection went down right after she found the forecast. She decided to convert the temperature from degrees Celsius to degrees Fahrenheit because she remembers the formula
\[C=\dfrac{5}{9}(F−32)\nonumber\]
75 degrees Fahrenheit is a comfortable temperature for his. So, she substitutes 75 for \(F\) into the formula
\[\dfrac{5}{9}(75−32)\approx24^{\circ}\nonumber\]
and finds that 75 degrees Fahrenheit is about 24 degrees Celsius.
She could do this for other temperatures but quickly realized that the most efficient thing to do is solve the equation for F. Then she can just plug in the Celsius degrees into the equation and evaluate to get the temperatures in Fahrenheit degrees. The equation, \[\begin{align*} F&=\dfrac{9}{5}C+32\end{align*} \nonumber\] is known as an inverse function .
Inverse Function
If \(f(x)=y\), then a function \(g\) is an inverse of \(f\) if \(g(y)=x\).
The inverse function of \(f(x)\) is typically notated as \(f^{-1} (x)\), which is read “\(f\) inverse of \(x\)”. , so equivalently, if \(f(x)=y\) then \(f^{-1} (y)=x\).
The notation \(f^{-1} (x)\) may only be used when the inverse of \(f\) is itself a a function. There are some functions that do not have inverse functions.
Caution!
The -1 used in the notation \(f^{−1} (x)\) is simply a function notation where the name of the function is "\(f\) inverse", and does not designate an exponent or power of -1.
Keep in mind that \[f^{−1}(x)\neq\dfrac{1}{f(x)}\nonumber\]
Example \(\PageIndex{3}\)
Given that \(h^{-1}(6)=2\), what are the corresponding input and output values of the original function \(h\)?
Solution
\(h(2)=6\)
Example \(\PageIndex{4}\)
A function \(g(x)\) is given as a graph below. Find \(g(3)\) and \(g^{-1} (3)\)
Solution
To evaluate \(g(3)\), we find 3 on the horizontal axis and find the corresponding output value on the vertical axis. The point (3, 1) tells us that \(g(3)=1\)
To evaluate \(g^{-1} (3)\), recall that by definition \(g^{-1} (3)\) means \(g(x) = 3\). By looking for the output value 3 on the vertical axis we find the point (5, 3) on the graph, which means \(g(5) = 3\), so by definition \(g^{-1} (3) =5\).
Example \(\PageIndex{5}\)
A function \(f(t)\) is given as a table below, showing distance in miles that a car has traveled in \(t\) minutes. Find and interpret \(f^{-1} (70)\)
| \(t\) (minutes) | 30 | 50 | 70 | 90 |
| \(f(t)\) (miles) | 20 | 40 | 60 | 70 |
Solution
The inverse function takes an output of \(f\) and returns an input for \(f\). So in the expression \(f^{-1} (70)\), the 70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function \(f\), 90 minutes, so \(f^{-1} (70)=90\). Interpreting this, it means that to drive 70 miles, it took 90 minutes.
Alternatively, recall the definition of the inverse was that if \(f(a)=b\) then \(f^{-1} (b)=a\). By this definition, if you are given \(f^{-1} (70)=a\) then you are looking for a value a so that \(f(a)=70\). In this case, we are looking for a \(t\) so that \(f(t)=70\), which is when \(t = 90\).
Notice that original function and the inverse function undo each other. If \(f(a)=b\), then \(f^{-1} (b)=a\), returning us to the original input. More simply put, if you compose a function and its inverse function together, you get the original input as your answer. Composition may be used to verify this.
Determining if Two Functions are Inverses of Each Other
How To Determine if Two Functions are Inverses of Each Other:
For two one-to-one functions, \(f\) and \(g\),
- Determine whether \(f(g(x))=x\) and \(g(f(x))=x\).
- If both statements in part (1) are true, then the functions are inverses of each other (\(g=f^{-1}\) and \(f=g^{-1}\)). If one or both statements in part (1) are false, then the functions are not inverses of each other (\(g{\neq}f^{-1}\) and \(f{\neq}g^{-1}\)).
Example \(\PageIndex{6}\)
If \(f(x)=\dfrac{1}{x+2}\) and \(g(x)=\dfrac{1}{x}−2\), is \(g=f^{-1}\)?
Solution
\[\begin{align*} g(f(x))&=\dfrac{1}{(\dfrac{1}{x+2})}−2 \\[5pt] &=x+2−2 \\[5pt] &=x \end{align*} \nonumber\]
so
\[g=f^{-1} \text{ and } f=g^{-1} \nonumber\]
This is enough to answer yes to the question, but we can also verify the other formula.
\[\begin{align*} f(g(x))&=\dfrac{1}{(\dfrac{1}{x}-2)+2} \\[5pt] &= \dfrac{1}{\dfrac{1}{x}} \\[5pt] &=x \end{align*} \nonumber\]
You Try \(\PageIndex{2}\)
If \(f(x)=x^3−4\) and \(g(x)=\sqrt[3]{x+4}\), is \(g=f^{-1}\)?
- Answer
-
Yes
Example \(\PageIndex{7}\)
If \(f(x)=x^3\) (the basic cubic function) and \(g(x)=\frac{1}{3}x\), is \(g=f^{-1}\)?
Solution
\[f(g(x))=\dfrac{x^3}{27}{\neq}x\nonumber\]
No, the functions are not inverses.
Note that the correct inverse to the basic cubic function is, of course, the cube root \(\sqrt[3]{x}=x^{\frac{1}{3}}\), that is, the one-third is an exponent, not a multiplier.
You Try \(\PageIndex{3}\)
If \(f(x)=(x−1)^3\) and \(g(x)=\sqrt[3]{x}+1\), is \(g=f^{-1}\)?
- Answer
-
Yes
Domain and Range of Inverse Functions
We have seen that the outputs of the function \(f\) are the inputs to \(f^{-1}\), so the range of \(f\) is also the domain of \(f^{-1}\). Likewise, because the inputs to \(f\) are the outputs of \(f^{-1}\), the domain of \(f\) is the range of \(f^{-1}\). We can visualize the situation as in Figure \(\PageIndex{6}\).
Figure \(\PageIndex{6}\): Domain and range of a function and its inverse.
This is important to remember when finding inverse functions. For example, the inverse of \(f(x)=\sqrt{x}\) is \(f^{-1}(x)=x^2, x\geq{0}\), because a square “undoes” a square root; but the square is only the inverse of the square root on the domain \(\left[0,\infty\right)\), since that is the range of \(f(x)=\sqrt{x}\).
Finding Inverses of Functions Represented by Formulas
Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula— for example, \(y\) as a function of \(x\)— we can often find the inverse function by solving to obtain \(x\) as a function of \(y\).
How To Find an Inverse Function:
- Make sure \(f\) is a one-to-one function.
- Solve for \(x\)
- Interchange \(x\) and \(y\) to write the inverse as a function of \(x\). Do remember to include any restrictions for the domain if it is not clearly implied in the equation you found.
Example \(\PageIndex{8}\)
Returning to our designer traveling to Milan, find a formula for the inverse function that gives Fahrenheit temperature given a Celsius temperature.
Solution
A quick Google search would find the inverse function, but alternatively, the designer might look back at how she solved for the Fahrenheit temperature for a specific Celsius value, and repeat the process in general
\[\begin{array}{l} {C=\dfrac{5}{9} (F-32)} \\ {C\cdot \dfrac{9}{5} =F-32} \\ {F=\dfrac{9}{5} C+32} \end{array}\nonumber \]
By solving in general, we have uncovered the inverse function. If
\[C=h(F)=\dfrac{5}{9} (F-32)\nonumber \]
Then
\[F=h^{-1} (C)=\dfrac{9}{5} C+32\nonumber \]
In this case, we introduced a notation for a function h to represent the conversion since the input and output variables are descriptive and writing \(C^{-1}\) could get confusing.
Example \(\PageIndex{9}\)
Find the inverse of the function \(f(x)=2+\sqrt{x−4}\).
Solution
\[ \begin{align*} y&=2+\sqrt{x-4} \\ y-2&=\sqrt{x-4} \\ (y-2)^2&=x-4 \\ x&=(y-2)^2+4 \end{align*} \nonumber\]
interchanging \(x\) and \(y\),
\(y=(x−2)^2+4 \nonumber\)
Notice that the range of \(f\) is \(\left[2,\infty\right)\), so this means that the domain of the inverse function \(f^{-1}\) is also \(\left[2,\infty\right)\). The formula we found for \(f^{-1}(x)\) above looks like it would be valid for all real \(x\). However, \(f^{-1}\) itself must have an inverse (namely, \(f\) ) so we have to state the restricted domain of \(f^{-1}\) to \(\left[2,\infty\right)\) in order to make \(f^{-1}\) a one-to-one function.
So, \(f^{-1}(x)=(x−2)^2+4\) where \(x\geq{2}\).
You Try \(\PageIndex{4}\)
What is the inverse of the function \(f(x)=2-\sqrt{x}\)?
- Answer
-
\(f^{-1}(x)=(2−x)^2\) where \(x\leq{2}\)
Inverse Functions and Their Graphs
Now that we know how to find the inverse of a function, let's explore the graphical relationship of functions and their inverses. Let's look at the quadratic function \(f(x)=x^2\) restricted to the domain \(\left[0,\infty\right)\). This is a one-to-one function, and it's graph is shown in Figure \(\PageIndex{7}\).
Figure \(\PageIndex{7}\): Quadratic function with domain restricted to \([0, \infty)\).
The inverse of this function is \(f^{-1}(x)=\sqrt{x}\). The graph of both \(f\) and \(f^{-1}\) are shown in Figure \(\PageIndex{8}\) on the same set of axes.
Notice that the graph of \(f^{-1}(x)\) is the graph of \(f(x)\) reflected about the diagonal line \(y=x\), which we will call the identity line, shown in Figure \(\PageIndex{8}\) .
.
Figure \(\PageIndex{8}\): Square and square-root functions on the non-negative domain
This relationship will be observed for all one-to-one functions and their inverses, because it is a result of the function and its inverse swapping inputs and outputs. This is useful to know for checking for errors when finding an inverse of a one-to-one function since the inverse equation found should visually be a reflection over the line (y=x\).
Example \(\PageIndex{10}\)
Given the graph of \(f(x)\) in Figure \(\PageIndex{9}\), sketch a graph of \(f^{-1}(x)\).
This is a one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain of \((0,\infty)\) and range of \((−\infty,\infty)\), so the inverse will have a domain of \((−\infty,\infty)\) and range of \((0,\infty)\).
If we reflect this graph over the line \(y=x\), the point \((1,0)\) reflects to \((0,1)\), the point \((4,2)\) reflects to \((2,4)\), and so on. Sketching the inverse on the same axes as the original graph gives Figure \(\PageIndex{10}\).
Is there any function that is equal to its own inverse?
Yes. If \(f=f^{-1}\), then \(f(f(x))=x\), and we can think of several functions that have this property. The identity function
does, and so does the basic reciprocal function, because
\[\dfrac{1}{\dfrac{1}{x}}=x\nonumber\]
Any function \(f(x)=c−x\), where \(c\) is a constant, is also equal to its own inverse.
Contributors and Attributions
-
Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .


17.1: Introduction to Dysrhythmias
By the end of this section, you should be able to:
- 17.1.1 Describe specific cardiac dysrhythmias and their significance to cardiac function.
- 17.1.2 Discuss the principles of managing dysrhythmias to improve cardiac functioning.
Dysrhythmias
Dysrhythmia s, also known as arrhythmias , occur when there is abnormal automaticity (spontaneous impulse generation) or abnormal conduction that results in a problem with the heart rate and/or rhythm. Medications address dysrhythmias by manipulating the electrolytes and other receptors involved in the cardiac conduction system to either treat the manifestations of the underlying arrhythmia or restore a normal heart rate, suppress the dysrhythmia, and thus restore normal sinus rhythm in a process known as cardioversion . When cardioversion is accomplished via drug administration, may be referred to as chemical cardioversion.
Dysrhythmias can be classified in multiple ways. One way is according to the ventricular contraction rate, or heart rate. A heart rate less than 60 beats per minute is considered bradycardia . Bradycardia can occur due to the SA node incorrectly pacing too slowly or to the AV node conducting incorrectly, leading to AV block. Bradycardia can lead to problems because it decreases cardiac output, which is affected by both stroke volume and heart rate. A heart rate greater than 100 beats per minute is called tachycardia . Tachycardia can further be categorized as supraventricular tachycardia (the stimulus physically originates above the ventricle but not from the sinus node) or ventricular tachycardia (the origin of the stimulus is physically located in the ventricle). Tachycardia is problematic because at very high heart rates, the ventricles do not have enough time to fill with an adequate amount of blood in between contractions. Thus, the stroke volume and cardiac output are both decreased. Tachycardia also increases the workload of the heart and can lead to a type of heart failure called tachycardia-induced cardiomyopathy.
Sinus Bradycardia
In sinus bradycardia, although the heart rate is less than 60 beats per minute, the rhythm is regular, and the SA node acts as the pacemaker for the heart. Aside from the slow heart rate, there is no other problem with conduction. On an ECG, every P wave is followed by a QRS complex, which is followed by a T wave.
Sinus Tachycardia
In sinus tachycardia , although the heart rate exceeds 100 beats per minute, the rhythm is regular, and the SA node acts as the pacemaker for the heart. Aside from the rapid heart rate, there is no other problem with conduction. Every P wave is followed by a QRS complex, which is followed by a T wave. Sinus tachycardia is often caused by exercise, pain, stress, anxiety, or dehydration.
Atrial Fibrillation
Atrial fibrillation is described as an irregularly irregular rhythm in which multiple areas of the atria generate spontaneous impulses such that the atria quiver at 400–600 beats per minute. Only some of the impulses are conducted through the AV node, resulting in a variable ventricular contraction rate or heart rate. When the heart rate is high, it is called atrial fibrillation with rapid ventricular response . Because the atria are quivering because so many impulses are being formed, the individual’s ECG will show no distinguishable P waves; however, the baseline will appear wavy due to the presence of “fibrillatory” waves (Figure 17.2).
There are two overall strategies for treating atrial fibrillation: rhythm control and rate control. The aim of rhythm control is to cardiovert and then maintain normal sinus rhythm. Rate control is a strategy that allows the client’s rhythm to remain in atrial fibrillation but slows the rate of conduction through the AV node, thereby slowing the ventricular rate/heart rate.
Atrial Flutter
Atrial flutter is a dysrhythmia closely related to atrial fibrillation. The atria quiver at rates of 250–400 beats per minute. As with atrial fibrillation, abnormal impulses are generated in the atria. Commonly, there is 2:1 conduction through the AV node, meaning that for every two impulses from the atria, one conducts through the AV node and leads to ventricular contraction. This process commonly leads to a ventricular rate of approximately 150 beats per minute (assuming an average atrial rate of 300 beats per minute). It is noteworthy that the conduction is not always 2:1; it may be 3:1, 4:1, or 5:1. As in atrial fibrillation, there are no typical P waves on the ECG of a client with atrial flutter. Instead, there are “flutter waves” in a characteristic sawtooth pattern between QRS complexes.
Premature Ventricular Contraction
A premature ventricular contraction (PVC) represents the ventricle contracting earlier than it should during the cardiac cycle due to a spontaneous impulse from the Purkinje fibers (rather than from an impulse carried from the SA node).
Ventricular Tachycardia
When multiple PVCs occur consecutively, it is known as ventricular tachycardia . Sustained ventricular tachycardia (lasting more than 30 seconds) can be serious and cause hemodynamic compromise necessitating advanced cardiac life support. Ventricular tachycardia can occur due to abnormalities of electrolytes such as magnesium or potassium or to myocardial ischemia (oxygen deprivation in the heart muscle). It can devolve into ventricular fibrillation, described next. On an ECG, ventricular tachycardia has multiple wide QRS complexes (Figure 17.2). Torsade de pointes is a dangerous ventricular tachycardia associated with medications that prolong the QT interval (the time it takes the heart to contract and recover).
Ventricular Fibrillation
Ventricular fibrillation is a medical emergency that necessitates advanced cardiac life support (ACLS). This is another ventricular arrhythmia, meaning it originates from spontaneous impulses from the Purkinje fibers (rather than from an impulse carried from the SA node), causing the ventricles to quiver erratically rather than pump, which, in turn, causes cardiac arrest. On an ECG, ventricular fibrillation appears as multiple and varied wide complexes, without any pattern or discernable P waves, QRS complexes, or T waves (Figure 17.2).
Special Considerations
Older Adults
The risk of being diagnosed with dysrhythmia skyrockets after age 60. Not only does the risk for dysrhythmias increase, but older adults have a higher risk for concomitant disease states that interfere with the action of antidysrhythmic drugs. For example, beta-adrenergic blockers, quinidine, and procainamide can exacerbate the postural hypotension that is particularly prevalent in older adults, leading to falls. Clients with heart failure (who are usually older adults) can experience exacerbation of their underlying condition when treated with flecainide or sotalol. Furthermore, older clients are more likely to be on multiple medications, increasing the likelihood of experiencing drug interactions. Age-related decreases in metabolic processing and excretion of drugs can increase antidysrhythmic plasma levels, putting these individuals at higher risk for adverse effects. Thus, antidysrhythmic drugs require extra vigilance to prevent harm in the older adult population.
(Source: Curtis et al., 2018)
Special Considerations
Pediatrics
Pediatric clients may experience arrhythmias that must be treated with antidysrhythmic therapy. Some of this use is off-label. Off-label prescription drug use means that the drug may not be specifically approved for a particular client or diagnosis, but health care providers may choose to use the drug anyway because the potential benefit outweighs the potential risks. In these circumstances, drug dosing and administration may not be standardized. The nurse should consult institutional protocols to avoid medication errors, especially overdoses.
(Source: Oeffl et al., 2023)


14: Gabriel García Márquez (1927-2014)
Thumbnail: Gabriel García Márquez Winner of the Nobel Prize in Literature in 1982. (CC BY-SA 2.0; Jose Lara via Wikipedia )
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Gabriel García Márquez (1927-2014) A Very Old Man with Enormous Wings Colombian Contemporary Literature Gabriel García Márquez and Jorge Luis Borges are the best known Latin American writers in history. Márquez won the Nobel Prize for Literature in 1982, arguably mostly for his best known novel, One Hundred Years of Solitude (1967). Later novels that were also critically acclaimed include Love in the Time of Cholera (1985) and The General in his Labyrinth (1989). Márquez was born in Colombia and raised by his maternal grandparents. His grandfather, a liberal war veteran, was an enthusiastic storyteller, and his grandmother was fond of telling fantastic stories that included ghosts, premonitions, omen, and other fantastical and gothic elements. Márquez was particularly influenced by her matter of fact treatment of the supernatural, which he later imitated in his writings through the use of "magical realism," the incorporation of supernatural or "magical" elements into fiction that is predominately realistic. In 1948, Márquez began his writing career as a journalist in Colombia while studying law at the National University of Colombia. In the mid-fifties, he moved to Caracas, Venezuela, and continued working as a columnist there. He later worked as a correspondent in Europe and travelled widely in the Southern United States (he was a devoted Faulkner fan) before settling in Mexico City. He later moved his family to Spain, where they were living when he became famous. A committed leftist, Márquez eventually was critical of U.S. imperialism and for many years denied entrance to the U.S. until the ban was removed in 1991 by President Bill Clinton. In later life, Márquez moved to Cuba, where he supported the Castro regime. In the short story "A Very Old Man with Enormous Wings" (1955), which Márquez claimed was written for children, includes elements typical of Márquez's writing, particularly magical realism and folk elements. The elements of magical realism are characterized by the matter of fact inclusion of supernatural intrusions into everyday life. The folk characteristics here are directly traceable to Márquez's upbringing in rural Colombia, where he regularly heard folktales from the region as part of his early education. Consider while reading:
Written by Anita Turlington
Thumbnail: Gabriel García Márquez Winner of the Nobel Prize in Literature in 1982. (CC BY-SA 2.0; Jose Lara via Wikipedia )


Art History - Renaissance through Post Modern
- Front Matter
- 1: Introduction
- 2: International Style
- 3: Renaissance
- 4: The Early Renaissance (1400s)
- 5: The High Renaissance (late 1400s through mid-1500s)
- 6: The Late Renaissance/Mannerism
- 7: Renaissance in the North and Spain
- 8: The Baroque Period in Italy and Spain
- 9: The Baroque in France and England in the 1600s
- 10: The Baroque Period in Northern Europe
- 11: Europe During the Rococo Period
- 12: Europe in the Second Half of the 1700s
- 13: Europe and North America in the Romantic Era (1789-1848)
- 14: The Age of Positivism
- 15: Europe in the Late 1800s
- 16: Europe and North America during the Modernist Revolution of 1904-1914
- 17: Art Between the Wars (WWI and WWII)
- 18: Postwar Europe and North America (after 1945)
- 19: The Postmodern World
- Back Matter


10.5: Ribosomal RNA Processing in Eukaryotic Nuclei
In most eukaryotes, a large rRNA gene in most eukaryotes transcribes a 45S precursor transcript containing (from shortest to longest) 5.8S rRNA, 18SrRNA and 28S rRNA. The ‘S’ stands for Svedberg, the biochemist who developed the sedimentation velocity ultracentrifugation technique to separate molecules like RNA by size. The higher the S value, the larger the molecule and therefore the faster it moves through the viscous sugar gradient during centrifugation. RNA Polymerase I transcribes 45S precursor rRNAs (preRNAs) from multiple large transcription units in the genome (shown below).
The 45S pre-rRNA is processed by cleavage. The many copies (200-400!) of the 45S gene in eukaryotic cells might be expected, since making proteins (and therefore ribosomes) will be an all-consuming cellular activity. In humans, 45S genes (45S rDNA) are distributed among five acrocentric chromosomes (those that have a centromere very near one end of the chromosome). The 45S rDNA in chromosomes is packed in the nucleolus inside nuclei.
Because these genes are present in so many copies and organized into a specific region of chromatin, it is possible to visualize 45S transcription in progress in electron micrographs such as the ones below.
The term lampbrush came from the shape of the 45S genes in the process of transcription; the RNAs extending from the DNA template look like an old-fashioned brush used to clean the chimney of a kerosene lamp.
Multiple gene copies encode 5S rRNAs. However, unlike the 45S rRNA genes, 5S rRNA gene may be spread among many chromosomes (seven in Neurospora crassa , the bread mold). Or in the case of humans, 5S RNA gene copies are distributed along chromosome 1. The 5S rRNA genes are transcribed by RNA polymerase III with minimal posttranscriptional processing. As already noted, the promoters of the 5S genes are within the transcribed part of the genes, rather than upstream of their 5S transcription units.


6.18: Factors in Analyzing Trading Area
- Identify the main factors to consider when analyzing a retail trading area
A trading area is the geographic area from where you draw your customers. There are three types of trading areas that are important to note:
- Primary Trading Area. Where the store is exactly located.
- Secondary Trading Area. The shopping center or area where the store is located.
- Fringe Trading Area. The city or town where the store is located.
You can picture these three areas as follows:
Your store is in the center of the primary area. Most of your customers will come from this smaller area as well. As you expand out, fewer and fewer customers will be coming to you. This is especially true if, for example, you are a small convenience store or drugstore. If, on the other hand, you are planning to open a Trader Joe’s, Marshall’s or T.J. Maxx, you may get your draw from a wider area. So, if you are planning to open a small, local retail store, you need to make sure that the area immediately surrounding your location has:
- Enough consumers. Are there enough people living in the area, with access to your retail space to create sales for your product or service?
- Enough money to spend. Spending potential index- compares the local average expenditure by product to the national average amount spent.
- A need for the product or service you are bringing to the area. Is there a market for your product? Here is where the census review, GIS services and other analysis processes will help determine if your market exists!
- Competition in the trading are. Using the internet as a tool, review sites like Yelp, use the map function and note who is located in your geographic area. It is also possible to use directories published by trade associations, the local chamber of commerce, municipal and city government or Chain Store Guide (published by CSG Information Services) to see who the competition might be.
So, again, analysis is the crucial component. Even with analysis, there is a possibility that a business may not work. As an example, a yoga teacher opened a studio in a small town where no other studios existed. It seemed like a perfect opportunity to tap an untapped market right? Well, the problem was, there were not enough people in the market for yoga classes. So what happened? The studio was open for a few years, marketing heavily to the local area, but ended up shutting their doors when they couldn’t create enough sales to be profitable.
Contributors and Attributions
- Factors in Analyzing Trading Area. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution


5.2: The Contents of International Law
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One distinguishes broadly between domestic, regional and (public and private) international law. Domestic law stems from domestic lawmakers and regulates the life of the citizens of a particular state. Regional law, such as European Union law or the law of regional human rights mechanisms, stems from regional intergovernmental institutions and addresses the governments and individuals of a particular geographical region or legal regime. Public international law is the subject of this chapter and addresses – in most general terms – relations involving states, intergovernmental organisations and non-state actors, which include today individuals, non-governmental organisations (NGOs) and private corporations. Private international law concerns conflicts of laws that may arise in cases where the domestic laws of different states could apply, for example in cases of cross-border e-commerce, marriages or liabilities.
Within public international law, a distinction is traditionally drawn between the law of peace and the law of war (humanitarian law). The law of peace regulates peaceful relations and includes such subject matters as international treaty law, the law of diplomatic and consular relations, international organisation law, the law of state responsibility, the law of the sea, the environment and outer space or international economic law.
International humanitarian law (IHL) is the law of armed conflicts ( jus in bellum – the law applicable in war) and regulates the conduct of international and non-international hostilities. In times of war, the use of force, including the killing of human beings, is not prohibited. The legal regulation of armed conflicts goes back to the mid-nineteenth century and comprises a large body of customary rules and a series of important conventions and additional protocols to these conventions adopted primarily in The Hague and Geneva.
International humanitarian law regulates, among other things, the methods and means of warfare and the protection of certain categories of persons – for example, the sick and wounded, prisoners of war and civilians. More specific treaties prohibit the use of certain types of weapons (such as chemical or biological weapons, mines or cluster munitions) or the protection of cultural property during armed conflict. Much of the development and codification of this body of law is the merit of the International Committee of the Red Cross, founded in 1863 by Henry Dunant, which is a private humanitarian institution based in Geneva and forms part of the International Red Cross and Red Crescent Movement.
At the transitional points between the law of peace and the law of armed conflict lies the legal regulation of the resort to force ( jus ad bellum – the law to engage in war) which concerns the conditions that need to be met to use force legally as, for example, in instances of self-defence (Article 51, UN Charter). More recently, scholars also speak of the regulation of the transition to peace after the end of armed conflicts ( jus post-bellum – the law after war) which includes questions over how to end armed conflicts, transitional justice and post-war reconstruction.
The strict distinction between the law of peace and the law of armed conflict has been somewhat blurred with the rise of international human rights law and international criminal law. Human rights law builds on and develops fundamental principles of humanitarian law for the protection of individuals. On the other hand, human rights have considerably influenced the refinement of humanitarian rules for the protection of combatants and civilians. International criminal law has seen a rapid development after the end of the Cold War first with the establishment of the international criminal tribunals for the former Yugoslavia and Rwanda and then with the establishment of the International Criminal Court in 2002.