41.12: Human Osmoregulatory and Excretory Systems - Kidney Function and Physiology
- Outline the process by which kidneys filter blood, reabsorb nutrients and water, and produce urine
Blood Filtration and Nutrient and Water Reabsorption
Kidneys filter blood in a three-step process. First, the nephrons filter blood that runs through the capillary network in the glomerulus. Almost all solutes, except for proteins, are filtered out into the glomerulus by a process called glomerular filtration. Second, the renal tubules collect the filtrate. Most of the solutes are reabsorbed in the PCT by a process called tubular reabsorption. In the loop of Henle, the filtrate continues to exchange solutes and water with the renal medulla and the peritubular capillary network.
Finally, some substances, such as electrolytes and drugs, are removed from blood through the peritubular capillary network into the distal convoluted tubule or collecting duct. Urine is a collection of substances that have not been reabsorbed during glomerular filtration or tubular reabsorbtion.
Glomerular Filtration
The formation of urine occurs through three steps: glomerular filtration, tubular reabsorption, and tubular secretion. The process of glomerular filtration filters out most of the solutes due to the high blood pressure and specialized membranes in the afferent arteriole. The blood pressure in the glomerulus is maintained independent of factors that affect systemic blood pressure. The “leaky” connections between the endothelial cells of the glomerular capillary network allow solutes to pass through easily. All solutes in the glomerular capillaries, including sodium ions and negatively and positively charged ions, pass through by passive diffusion; the only exception is macromolecules such as proteins. There is no energy requirement at this stage of the filtration process. Glomerular filtration rate (GFR) is the volume of glomerular filtrate formed per minute by the kidneys. GFR is regulated by multiple mechanisms and is an important indicator of kidney function.
Tubular Reabsorption and Secretion
Tubular reabsorption occurs in the PCT part of the renal tubule. Almost all nutrients are reabsorbed; this occurs either by passive or active transport. Reabsorption of water and key electrolytes are regulated and influenced by hormones. Sodium (Na + ) is the most abundant ion; most of it is reabsorbed by active transport and then transported to the peritubular capillaries. Because Na + is actively transported out of the tubule, water follows to even out the osmotic pressure. Water is also independently reabsorbed into the peritubular capillaries due to the presence of aquaporins, or water channels, in the PCT. This occurs due to the low blood pressure and high osmotic pressure in the peritubular capillaries. Every solute, however, has a transport maximum; the excess solute is not reabsorbed. Kidneys’ osmolarity of body fluids is maintained at 300 milliosmole (mOsm).
In the loop of Henle, the permeability of the membrane changes. The descending limb is permeable to water, not solutes; the opposite is true for the ascending limb. Additionally, the loop of Henle invades the renal medulla, which is naturally high in salt concentration. It tends to absorb water from the renal tubule and concentrate the filtrate. The osmotic gradient increases as it moves deeper into the medulla. Because two sides of the loop of Henle perform opposing functions, it acts as a countercurrent multiplier. The vasa recta around the loop of Henle acts as the countercurrent exchanger.
Additional solutes and wastes are secreted into the kidney tubules during tubular secretion, which is the opposite process to tubular reabsorption. The collecting ducts collect filtrate coming from the nephrons and fuse in the medullary papillae. From here, the papillae deliver the filtrate, now called urine, into the minor calyces that eventually connect to the ureters through the renal pelvis.
Contributions and Attributions
- OpenStax College, Biology. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44809/latest...ol11448/latest . License : CC BY: Attribution
- Human Physiology/The Urinary System. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Human_P...tem%23Nephrons . License : CC BY-SA: Attribution-ShareAlike
- renal. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/renal . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, The Kidneys and Osmoregulatory Organs. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44809/latest...e_41_03_01.jpg . License : CC BY: Attribution
- OpenStax College, The Kidneys and Osmoregulatory Organs. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44809/latest...e_41_03_02.png . License : CC BY: Attribution
- Loop of Henle. Provided by : Wikitionary. Located at : https://en.wiktionary.org/wiki/loop_of_Henle . License : CC BY-SA: Attribution-ShareAlike
- The Urinary System. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Human_...tem%23Nephrons . License : CC BY-SA: Attribution-ShareAlike
- The Kidneys. Provided by : OpebStax CNX. Located at<EMAIL_ADDRESS>. License : CC BY-SA: Attribution-ShareAlike
- Glomerulus. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Glomerulus . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, The Kidneys and Osmoregulatory Organs. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44809/latest...e_41_03_01.jpg . License : CC BY: Attribution
- OpenStax College, The Kidneys and Osmoregulatory Organs. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44809/latest...e_41_03_02.png . License : CC BY: Attribution
- The Kidneys and Osmoregulatory Organs Source: Boundless. u201cKidney Structure.u201d Boundless Biology. Boundless, 08 Jan. 2016. Retrieved 19 Feb. 2016 from www.boundless.com/biology/te...ure-860-12107/. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44809/latest...e_41_03_03.png . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Biology. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44809/latest...ol11448/latest . License : CC BY: Attribution
- countercurrent. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/countercurrent . License : CC BY-SA: Attribution-ShareAlike
- electrolyte. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/electrolyte . License : CC BY-SA: Attribution-ShareAlike
- arteriole. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/arteriole . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, The Kidneys and Osmoregulatory Organs. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44809/latest...e_41_03_01.jpg . License : CC BY: Attribution
- OpenStax College, The Kidneys and Osmoregulatory Organs. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44809/latest...e_41_03_02.png . License : CC BY: Attribution
- The Kidneys and Osmoregulatory Organs Source: Boundless. u201cKidney Structure.u201d Boundless Biology. Boundless, 08 Jan. 2016. Retrieved 19 Feb. 2016 from www.boundless.com/biology/te...ure-860-12107/. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44809/latest...e_41_03_03.png . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, The Kidneys and Osmoregulatory Organs. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44809/latest...e_41_03_04.jpg . License : CC BY: Attribution
- OpenStax College, The Kidneys and Osmoregulatory Organs. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44809/latest...e_41_03_05.png . License : CC BY: Attribution
Key Points
- Glomerular filtration, tubular reabsorption, and tubular secretion are the three primary steps in which kidneys filter blood and maintain proper electrolyte balance.
- Glomerular filtration removes solutes from the blood; it is the first step of urine formation.
- In tubular reabsoption, the second step of urine formation, almost all nutrients are reabsorbed in the renal tubule by active or passive transport.
- Tubular secretion is the last step of urine formation, where solutes and waste are secreted into the collecting ducts, ultimately flowing to the bladder in the form of urine.
Key Terms
- arteriole : one of the small branches of an artery, especially one that connects with capillaries
- countercurrent : a current that flows against the prevailing one
- electrolyte : any of the various ions (such as sodium or chloride) that regulate the electric charge on cells and the flow of water across their membranes


12.7: Publishing Protocol
To "publish" a book is a catch all phrase to represent the final step in releasing a newly constructed text for public consumption, typically from an author/remixer's sandbox. This entails several steps to complete properly.
-
Confirm that the author/remixer's campus hub exists (and doesn't exceed 5 if the campus is not a LibreNet member). If the hub does not exist, this requires creating one:
- Go to "Courses" section of the library
- Make a " New " page. Select Book/unit. Rename the new section to the campus name
- Find the campus Logo online that works. Scale/crop it to 150 points vertical (e.g., with Paint) and add it to the newly constructed (and empty) campus Hub's thumbnail in the Page Settings .
- Go to the unpublished book in the author/remixer sandbox and move it to the new hub (under Options)
- Once moved, then change the permission of the entire book by selecting "Restrict Access" under Options and then select Public AND apply changed to all subpages.
- To confirm the book is reasonably well put together, open up the Remixer in edit remix mode and run it (i.e., save to the server). This is a general clean up step.
- Under the top page of the book and only the top page select Set Cover Page/Add to Download Center under the Cover Page options of the Page settings
- The Cover specifications need to added to the top of the book (i.e., the designated Cover Page). This follows this format, which is quite specific
lulu@CoverTitle@CoverAuthor@CoverAffiliation@BinderTitle
- Optional : It is always recommended that the author add some text into the Cover Page's Summary (in PageSettings) since that goes on the back cover of the book
- Optional (for LibreNet Members): The default logo on the back of the book is the LibreTexts "The Future is Open" logo. LibreNet members have the rights to have their campus logo branded to the books from their campus. That requires uploading the Institutional Logo section ( https://chem.libretexts.org/Under_Construction/Development_Details/Project_Icons_and_Logos/Institution_Logos ) and then adding a new metatag to the Cover Page
luluCover@URLofLOGO
- Compile the book (you need batchaccess via membership in this group to access this button. if you do not have this membership, then the Compile button does not show up
- The above procedure will create Front and Back matter sections, if they do not already exist. If the two sections are at the end of the book (ei., the Front matter is not at the front), then you want to Unorder the top level of the book:
- If the Unorder scrambles the order of the chapters, then run the remixer again to renumber the pages accordingly. (This is rare, but happens from time to time). If you have to do this, then Compile the book again.
This workflow will handle most publishing efforts. The pages that are default in the Front Matter and Back Mater will be automatically constructed including the TitlePage, the InfoPage, the Table of Contents, the Index (and soon to be the Glossary). The TitlePage may require additional editing depending on author's desires.


13.9.1: Simplify Rational Expressions
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Summary
By the end of this section, you will be able to:
- Determine the values for which a rational expression is undefined
- Evaluate rational expressions
- Simplify rational expressions
- Simplify rational expressions with opposite factors
Are you Ready?
Before you get started, take this readiness quiz.
If you miss a problem, go back to the section listed and review the material.
In Chapter 1, we reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers, and the denominator is not zero.
In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call these rational expressions.
Definition: RATIONAL EXPRESSION
A rational expression is an expression of the form \(\frac{p(x)}{q(x)}\), where p and q are polynomials and \(q \ne 0\).
Here are some examples of rational expressions:
\[\begin{array}{cccc} {−\frac{13}{42}}&{\frac{7y}{8z}}&{\frac{5x+2}{x^2−7}}&{\frac{4x^2+3x−1}{2x−8}}\\ \nonumber \end{array}\]
Notice that the first rational expression listed above, \(−\frac{13}{42}\), is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero.
We will perform same operations with rational expressions that we do with fractions. We will simplify, add, subtract, multiply, divide, and use them in applications.
Determine the Values for Which a Rational Expression is Undefined
When we work with a numerical fraction, it is easy to avoid dividing by zero, because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.
If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator.
So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.
Definition: DETERMINE THE VALUES FOR WHICH A RATIONAL EXPRESSION IS UNDEFINED.
- Set the denominator equal to zero.
- Solve the equation in the set of reals, if possible.
Example \(\PageIndex{1}\)
Determine the values for which the rational expression is undefined:
- \(\frac{9y}{x}\)
- \(\frac{4b−3}{2b+5}\)
- \(\frac{x+4}{x^2+5x+6x}\)
Solution
The expression will be undefined when the denominator is zero.
| 1. | \(\frac{9y}{x}\) |
| Set the denominator equal to zero. Solve for the variable. | x=0 |
| \(\frac{9y}{x}\) is undefined for x=0. | |
| 2. |
\(\frac{4b−3}{2b+5}\) |
| Set the denominator equal to zero. Solve for the variable. | 2b+5=0 |
| 2b=−5 | |
| \(b=−\frac{5}{2}\) | |
| \(\frac{4b−3}{2b+5}\) is undefined for \(b=−\frac{5}{2}\). | |
| 3. | \(\frac{x+4}{x^2+5x+6x}\) |
| Set the denominator equal to zero. Solve for the variable. | \(x^2+5x+6x=0\) |
| \((x+2)(x+3)=0\) | |
| x+2=0 or x+3=0 | |
| x=−2 or x=−3 | |
| \(\frac{x+4}{x^2+5x+6x}\) is undefined for x=−2 or x=−3. |
Saying that the rational expression \(\frac{x+4}{x^2+5x+6x}\) is undefined for x=−2orx=−3 is similar to writing the phrase “void where prohibited” in contest rules.
Try It \(\PageIndex{2}\)
Determine the values for which the rational expression is undefined:
- \(\frac{3y}{x}\)
- \(\frac{8n−5}{3n+1}\)
- \(\frac{a+10}{a^2+4a+3a}\)
- Answer
-
- x=0
- \(n=−\frac{1}{3}\)
- a=−1, a=−3
Try It \(\PageIndex{3}\)
Determine the values for which the rational expression is undefined:
- \(\frac{4p}{5q}\)
- \(\frac{y−1}{3y+2}\)
- \(\frac{m−5}{m^2+m−6}\)
- Answer
-
- q=0
- \(y=−\frac{2}{3}\)
- m=2, m=−3
Evaluate Rational Expressions
To evaluate a rational expression, we substitute values of the variables into the expression and simplify, just as we have for many other expressions in this book.
Example \(\PageIndex{4}\)
Evaluate \(\frac{2x+3}{3x−5}\) for each value:
- x=0
- x=2
- x=−3
Solution
| 1. | |
| Simplify. | |
| 2. | |
| Simplify. | |
| 3. | |
| Simplify. | |
Try It \(\PageIndex{5}\)
Evaluate \(\frac{y+1}{2y−3}\) for each value:
- y=1
- y=−3
- y=0
- Answer
-
- −2
- \(\frac{2}{9}\)
- \(−\frac{1}{3}\)
Try It \(\PageIndex{6}\)
Evaluate \(\frac{5x−1}{2x+1}\) for each value:
- x=1
- x=−1
- x=0
- Answer
-
- \(\frac{4}{3}\)
- 6
- −1
Example \(\PageIndex{7}\)
Evaluate \(\frac{x^2+8x+7}{x^2−4}\) for each value:
- x=0
- x=2
- x=−1
Solution
| 1. | |
| Simplify. | |
| 2. | |
| Simplify. | |
| This rational expression is undefined for x = 2. | |
| 3. | |
| Simplify. | |
Try It \(\PageIndex{8}\)
Evaluate \(\frac{x^2+1}{x^2−3x+2}\) for each value:
- x=0
- x=−1
- x=3
- Answer
-
- \(\frac{1}{2}\)
- \(\frac{1}{3}\)
- 2
Try It \(\PageIndex{9}\)
Evaluate \(\frac{x^2+x−6}{x^2−9}\) for each value.
- x=0
- x=−2
- x=1
- Answer
-
- \(\frac{2}{3}\)
- \(\frac{4}{5}\)
- \(\frac{1}{2}\)
Remember that a fraction is simplified when it has no common factors, other than 1, in its numerator and denominator. When we evaluate a rational expression, we make sure to simplify the resulting fraction.
Example \(\PageIndex{10}\)
Evaluate \(\frac{a^2+2ab+b^2}{3ab}\) for each value.
- a=1, b=2
- a=−2, b=−1
- \(a=\frac{1}{3}\), b=0
Solution
| 1. | \(\frac{a^2+2ab+b^2}{3ab}\) when a=1, b=2 |
| Simplify. | |
| 2. | \(\frac{a^2+2ab+b^2}{3ab}\) when a=−2, b=−1 |
| Simplify. | |
| 3. | \(\frac{a^2+2ab+b^2}{3ab}\) when \(a=\frac{1}{3}\), b=0 |
| Simplify. | |
Try It \(\PageIndex{11}\)
Evaluate \(\frac{2a^{3}b}{a^2+2ab+b^2}\) for each value.
- a=−1, b=2
- a=0, b=−1
- a=1, \(b=\frac{1}{2}\)
- Answer
-
- −4
- 0
- \(\frac{4}{9}\)
Try It \(\PageIndex{12}\)
Evaluate \(\frac{a^2−b^2}{8ab^3}\) for each value:
- a=1, b=−1
- \(a=\frac{1}{2}\), b=−1
- a=−2, b=1
- Answer
-
- 0
- \(\frac{3}{16}\)
- \(\frac{3}{16}\)
Simplify Rational Expressions
Just like a fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator, a rational expression is simplified if it has no common factors, other than 1, in its numerator and denominator.
Definition: SIMPLIFIED RATIONAL EXPRESSION
A rational expression is considered simplified if there are no common factors in its numerator and denominator.
For example:
- \(\frac{2}{3}\) is simplified because there are no common factors of 2 and 3.
- \(\frac{2x}{3x}\) is not simplified because x is a common factor of 2 x and 3 x .
We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expression s.
Definition: EQUIVALENT FRACTIONS PROPERTY
If a , b , and c are numbers where \(b \ne 0\), \(c\ne 0\), then \(\frac{a}{b}=\frac{a·c}{b·c}\) and \(\frac{a·c}{b·c}=\frac{a}{b}\)
Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see \(b \ne 0\), \(c\ne 0\) clearly stated. Every time we write a rational expression, we should make a similar statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.
Let’s start by reviewing how we simplify numerical fractions.
Example \(\PageIndex{13}\)
Simplify: \(−\frac{36}{63}\).
Solution
| Rewrite the numerator and denominator showing the common factors. | |
| Simplify using the Equivalent Fractions Property. |
Notice that the fraction \(−\frac{4}{7}\) is simplified because there are no more common factors.
Try It \(\PageIndex{14}\)
Simplify: \(−\frac{45}{81}\).
- Answer
-
\(−\frac{5}{9}\)
Try It \(\PageIndex{15}\)
Simplify: \(−\frac{42}{54}\).
- Answer
-
\(−\frac{7}{9}\)
Throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, \(x \ne 0\) and \(y \ne 0\).
Example \(\PageIndex{16}\)
Simplify: \(\frac{3xy}{18x^{2}y^{2}}\).
Solution
| Rewrite the numerator and denominator showing the common factors. | |
| Simplify using the Equivalent Fractions Property. |
Did you notice that these are the same steps we took when we divided monomials in Polynomials ?
Try It \(\PageIndex{17}\)
Simplify: \(\frac{4x^{2}y}{12xy^2}\).
- Answer
-
\(\frac{x}{3y}\)
Try It \(PageIndex{18}\)
Simplify: \(\frac{16x^{2}y}{2xy^2}\).
- Answer
-
\(\frac{8x}{y}\)
Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.
Note that removing the x ’s from \(\frac{x+5}{x}\) would be like cancelling the 2’s in the fraction \(\frac{2+5}{2}\)!
How to Simplify Rational Binomials
Example \(\PageIndex{19}\)
Simplify: \(\frac{2x+8}{5x+20}\).
Solution
Try It \(\PageIndex{20}\)
Simplify: \(\frac{3x−6}{2x−4}\).
- Answer
-
\(\frac{3}{2}\)
Try It \(\PageIndex{21}\)
Simplify: \(\frac{7y+35}{5y+25}\).
- Answer
-
\(\frac{7}{5}\)
We now summarize the steps you should follow to simplify rational expressions.
Definition: SIMPLIFY A RATIONAL EXPRESSION.
- Factor the numerator and denominator completely.
- Simplify by dividing out common factors.
We’ll use the methods we covered in Factoring to factor the polynomials in the numerators and denominators in the following examples.
Example \(\PageIndex{22}\)
Simplify: \(\frac{x^2+5x+6}{x^2+8x+12}\).
Solution
| \(\frac{x^2+5x+6}{x^2+8x+12}\) | |
| Factor the numerator and denominator. | \(\frac{(x+2)(x+3)}{(x+2)(x+6)}\) |
| Remove the common factor x+2 from the numerator and the denominator. | \(\frac{x+3}{x+6}\) |
Can you tell which values of x must be excluded in this example?
Try It \(\PageIndex{23}\)
Simplify: \(\frac{x^2−x−2}{x^2−3x+2}\).
- Answer
-
\(\frac{x+1}{x−1}\)
Try It \(\PageIndex{24}\)
Simplify: \(\frac{x^2−3x−10}{x^2+x−2}\).
- Answer
-
\(\frac{x−5}{x−1}\)
Example \(\PageIndex{25}\)
Simplify: \(\frac{y^2+y−42}{y^2−36}\).
Solution
| \(\frac{y^2+y−42}{y^2−36}\). | |
| Factor the numerator and denominator. | \(\frac{(y+7)(y−6)}{(y+6)(y−6)}\) |
| Remove the common factor y−6 from the numerator and the denominator. | \(\frac{y+7}{y+6}\) |
Try It \(\PageIndex{26}\)
Simplify: \(\frac{x^2+x−6}{x^2−4}\).
- Answer
-
\(\frac{x+3}{x+2}\)
Try It \(\PageIndex{27}\)
Simplify: \(\frac{x^2+8x+7}{x^2−49}\).
- Answer
-
\(\frac{x+1}{x−7}\)
Example \(\PageIndex{28}\)
Simplify: \(\frac{p^3−2p^2+2p−4}{p^2−7p+10}\).
Solution
| \(\frac{p^3−2p^2+2p−4}{p^2−7p+10}\) | |
| Factor the numerator and denominator, using grouping to factor the numerator. | \(\frac{p^2(p−2)+2(p−2)}{(p−5)(p−2)}\) |
| \(\frac{(p^2+2)(p−2)}{(p−5)(p−2)}\) | |
| Remove the common factor p−2 from the numerator and the denominator. | \(\frac{p^2+2}{p−5}\) |
Try It \(\PageIndex{29}\)
Simplify: \(\frac{y^3−3y^2+y−3}{y^2−y−6}\).
- Answer
-
\(\frac{y^2+1}{y+2}\)
Try It \(\PageIndex{30}\)
Simplify: \(\frac{p^3−p^2+2p−2}{p^2+4p−5}\).
- Answer
-
\(\frac{p^2+2}{p+5}\)
Example \(\PageIndex{31}\)
Simplify: \(\frac{2n^2−14n}{4n^2−16n−48}\).
Solution
| \(\frac{2n^2−14n}{4n^2−16n−48}\) | |
| Factor the numerator and denominator, first factoring out the GCF. | \(\frac{2n(n−7)}{4(n^2−4n−12)}\) |
| \(\frac{2n(n−7)}{4(n−6)(n+2)}\) | |
| Remove the common factor, 2. | \(\frac{n(n−7)}{2(n−6)(n+2)}\) |
Try It \(\PageIndex{32}\)
Simplify: \(\frac{2n^2−10n}{4n^2−16n−20}\).
- Answer
-
\(\frac{n}{2(n+1)}\)
Try It \(\PageIndex{33}\)
Simplify: \(\frac{4x^2−16x}{8x^2−16x−64}\).
- Answer
-
\(\frac{x}{2(x+2)}\)
Example \(\PageIndex{34}\)
Simplify: \(\frac{3b^2−12b+12}{6b^2−24}\).
Solution
| \(\frac{3b^2−12b+12}{6b^2−24}\) | |
| Factor the numerator and denominator, first factoring out the GCF. | \(\frac{3(b^2−4b+4)}{6(b^2−4)}\) |
| \(\frac{3(b−2)(b−2)}{6(b−2)(b+2)}\) | |
| Remove the common factors of b−2 and 3. | \(\frac{3(b−2)}{2(b+2)}\) |
Try It \(\PageIndex{35}\)
Simplify: \(\frac{2x^2−12x+18}{3x^2−27}\).
- Answer
-
\(\frac{2(x−3)}{3(x+3)}\)
Try It \(PageIndex{36}\)
Simplify: \(\frac{5y^2−30y+25}{2y^2−50}\).
- Answer
-
\(\frac{5(x−1)}{2(x+5)}\)
Example \(\PageIndex{37}\)
Simplify: \(\frac{m^3+8}{m^2−4}\).
Solution
| \(\frac{m^3+8}{m^2−4}\) | |
| Factor the numerator and denominator, using the formulas for sum of cubes and difference of squares. | \(\frac{(m+2)(m^2−2m+4)}{(m+2)(m−2)}\) |
| Remove the common factors of m+2. | \(\frac{m^2−2m+4}{m−2}\) |
Try It \(\PageIndex{38}\)
Simplify: \(\frac{p^3−64}{p^2−16}\).
- Answer
-
\(\frac{p^2+4p+16}{p+4}\)
Try It \(\PageIndex{39}\)
Simplify: \(\frac{x^3+8}{x^2−4}\).
- Answer
-
\(\frac{x^2−2x+4}{x−2}\)
Simplify Rational Expressions with Opposite Factors
Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. Let’s start with a numerical fraction, say \(\frac{7}{−7}\).
We know this fraction simplifies to −1. We also recognize that the numerator and denominator are opposites.
In Foundations , we introduced opposite notation: the opposite of a is −a. We remember, too, that −a=−1·a
We simplify the fraction \(\frac{a}{−a}\)
\[\begin{array}{ll} {}&{\frac{a}{−a}}\\ {\text{We could rewrite this.}}&{\frac{1·a}{−1·a}}\\ {\text{Remove the common factors.}}&{\frac{1}{−1}}\\ {\text{Simplify.}}&{−1}\\ \nonumber \end{array}\]
So, in the same way, we can simplify the fraction \(\frac{x−3}{−(x−3)}\)
\[\begin{array}{ll} {}&{\frac{x−3}{−(x−3)}}\\ {\text{We could rewrite this.}}&{\frac{1·(x−3)}{−1·(x−3)}}\\ {\text{Remove the common factors.}}&{\frac{1}{−1}}\\ {\text{Simplify.}}&{−1}\\ \nonumber \end{array}\]
But the opposite of x−3 could be written differently:
\[\begin{array}{ll} {}&{−(x−3)}\\ {\text{Distribute.}}&{−x+3}\\ {\text{Rewrite.}}&{3−x}\\ \nonumber \end{array}\]
This means the fraction \(\frac{x−3}{3−x}\) simplifies to −1.
In general, we could write the opposite of a−b as b−a. So the rational expression \(\frac{a−b}{b−a}\) simplifies to −1.
Definition: OPPOSITES IN A RATIONAL EXPRESSION
The opposite of a−b is b−a
\(\frac{a−b}{b−a}=−1\), \(a \ne b\)
An expression and its opposite divide to −1
We will use this property to simplify rational expressions that contain opposites in their numerators and denominators.
Example \(\PageIndex{40}\)
Simplify: \(\frac{x−8}{8−x}\).
Solution
| \(\frac{x−8}{8−x}\). | |
| Recognize that x−8 and 8−x are opposites | −1 |
Try It \(\PageIndex{41}\)
Simplify: \(\frac{y−2}{2−y}\).
- Answer
-
−1
Try It \(\PageIndex{42}\)
Simplify: \(\frac{n−9}{9−n}\).
- Answer
-
−1
Remember, the first step in simplifying a rational expression is to factor the numerator and denominator completely.
Example \(\PageIndex{43}\)
Simplify: \(\frac{14−2x}{x^2−49}\).
Solution
| Factor the numerator and denominator. | |
| Recognize 7−x and x−7 are opposites. | |
| Simplify. |
Try It \(\PageIndex{44}\)
Simplify: \(\frac{10−2y}{y^2−25}\).
- Answer
-
\(−\frac{2}{y+5}\)
Try It \(\PageIndex{45}\)
Simplify: \(\frac{3y−27}{81−y^2}\).
- Answer
-
\(−\frac{3}{9+y}\)
Example \(\PageIndex{46}\)
Simplify: \(\frac{x^2−4x−32}{64−x^2}\).
Solution
| Factor the numerator and denominator. | |
| Recognize the factors that are opposites. | |
| Simplify. |
Try It \(\PageIndex{47}\)
Simplify: \(\frac{x^2−4x−5}{25−x^2}\).
- Answer
-
\(−\frac{x+1}{x+5}\)
Try It \(\PageIndex{48}\)
Simplify: \(\frac{x^2+x−2}{1−x^2}\).
- Answer
-
\(−\frac{x+2}{x+1}\)
Key Concepts
-
Determine the Values for Which a Rational Expression is Undefined
- Set the denominator equal to zero.
- Solve the equation, if possible.
-
Simplified Rational Expression
- A rational expression is considered simplified if there are no common factors in its numerator and denominator.
-
Simplify a Rational Expression
- Factor the numerator and denominator completely.
- Simplify by dividing out common factors.
-
Opposites in a Rational Expression
-
The opposite of a−b is b−a
\(\frac{a−b}{b−a}=−1\) \(a \ne b\), \(b \ne 0\), \(a \ne b\)
-
The opposite of a−b is b−a
Practice Makes Perfect
In the following exercises, determine the values for which the rational expression is undefined.
Example \(\PageIndex{49}\)
- \(\frac{2x}{z}\)
- \(\frac{4p−1}{6p−5}\)
- \(\frac{n−3}{n^2+2n−8}\)
- Answer
-
- z=0
- \(p=\frac{5}{6}\)
- n=−4, n=2
Example \(\PageIndex{50}\)
- \(\frac{10m}{11n}\)
- \(\frac{6y+13}{4y−9}\)
- \(\frac{b−8}{b^2−36}\)
Example \(\PageIndex{51}\)
- \(\frac{4x^{2}y}{3y}\)
- \(\frac{3x−2}{2x+1}\)
- \(\frac{u−1}{u^2−3u−28}\)
- Answer
-
- y=0
- \(x=−\frac{1}{2}\)
- u=−4, u=7
Example \(\PageIndex{52}\)
- \(\frac{5pq^{2}}{9q}\)
- \(\frac{7a−4}{3a+5}\)
- \(\frac{1}{x^2−4}\)
In the following exercises, evaluate the rational expression for the given values.
Example \(\PageIndex{53}\)
\(\frac{2x}{x−1}\)
- x=0
- x=2
- x=−1
- Answer
-
- 0
- 4
- 1
Example \(\PageIndex{54}\)
\(\frac{4y−1}{5y−3}\)
- y=0
- y=2
- y=−1
Example \(\PageIndex{55}\)
\(\frac{2p+3}{p^2+1}\)
- p=0
- p=1
- p=−2
- Answer
-
- 3
- \(\frac{5}{2}\)
- \(−\frac{1}{5}\)
Example \(\PageIndex{56}\)
\(\frac{x+3}{2−3x}\)
- x=0
- x=1
- x=−2
Example \(\PageIndex{57}\)
\(\frac{y^2+5y+6}{y^2−1}\)
- y=0
- y=2
- y=−2
- Answer
-
- −6
- \(\frac{20}{3}\)
- 0
Example \(\PageIndex{58}\)
\(\frac{z^2+3z−10}{z^2−1}\)
- z=0
- z=2
- z=−2
Example \(\PageIndex{59}\)
\(\frac{a^2−4}{a^2+5a+4}\)
- a=0
- a=1
- a=−2
- Answer
-
- −1
- \(−\frac{3}{10}\)
- 0
Example \(\PageIndex{60}\)
\(\frac{b^2+2}{b^2−3b−4}\)
- b=0
- b=2
- b=−2
Example \(\PageIndex{61}\)
\(\frac{x^2+3xy+2y^2}{2x^{3}y}\)
- x=1, y=−1
- x=2, y=1
- x=−1, y=−2
- Answer
-
- 0
- \(\frac{3}{4}\)
- \(\frac{15}{4}\)
Example \(\PageIndex{62}\)
\(\frac{c^2+cd−2d^2}{cd^{3}}\)
- c=2, d=−1
- c=1, d=−1
- c=−1, d=2
Example \(\PageIndex{63}\)
\(\frac{m^2−4n^2}{5mn^3}\)
- m=2, n=1
- m=−1, n=−1
- m=3, n=2
- Answer
-
- 0
- \(−\frac{3}{5}\)
- \(−\frac{7}{20}\)
Example \(\PageIndex{64}\)
\(\frac{2s^{2}t}{s^2−9t^2}\)
- s=4, t=1
- s=−1, t=−1
- s=0, t=2
In the following exercises, simplify.
Example \(\PageIndex{65}\)
\(−\frac{4}{52}\)
- Answer
-
\(−\frac{1}{13}\)
Example \(\PageIndex{66}\)
\(−\frac{44}{55}\)
Example \(\PageIndex{67}\)
\(\frac{56}{63}\)
- Answer
-
\(\frac{8}{9}\)
Example \(\PageIndex{68}\)
\(\frac{65}{104}\)
Example \(\PageIndex{69}\)
\(\frac{6ab^{2}}{12a^{2}b}\)
- Answer
-
\(\frac{b}{2ab}\)
Example \(\PageIndex{70}\)
\(\frac{15xy^{3}}{x^{3}y^{3}}\)
Example \(\PageIndex{71}\)
\(\frac{8m^{3}n}{12mn^2}\)
- Answer
-
\(\frac{2m^2}{3n}\)
Example \(\PageIndex{72}\)
\(\frac{36v^{3}w^2}{27vw^3}\)
Example \(\PageIndex{73}\)
\(\frac{3a+6}{4a+8}\)
- Answer
-
\(\frac{3}{4}\)
Example \(\PageIndex{74}\)
\(\frac{5b+5}{6b+6}\)
Example \(\PageIndex{75}\)
\(\frac{3c−9}{5c−15}\)
- Answer
-
\(\frac{3}{5}\)
Example \(\PageIndex{76}\)
\(\frac{4d+8}{9d+18}\)
Example \(\PageIndex{77}\)
\(\frac{7m+63}{5m+45}\)
- Answer
-
\(\frac{7}{5}\)
Example \(\PageIndex{78}\)
\(\frac{8n−96}{3n−36}\)
Exercise \(\PageIndex{79}\)
\(\frac{12p−240}{5p−100}\)
- Answer
-
\(\frac{12}{5}\)
Example \(\PageIndex{80}\)
\(\frac{6q+210}{5q+175}\)
Example \(\PageIndex{81}\)
\(\frac{a^2−a−12}{a^2−8a+16}\)
- Answer
-
\(\frac{a+3}{a−4}\)
Example \(\PageIndex{82}\)
\(\frac{x^2+4x−5}{x^2−2x+1}\)
Example \(\PageIndex{83}\)
\(\frac{y^2+3y−4}{y^2−6y+5}\)
- Answer
-
\(\frac{y+4}{y−5}\)
Example \(\PageIndex{84}\)
\(\frac{v^2+8v+15}{v^2−v−12}\)
Example \(\PageIndex{85}\)
\(\frac{x^2−25}{x^2+2x−15}\)
- Answer
-
\(\frac{x−5}{x−3}\)
Example \(\PageIndex{86}\)
\(\frac{a^2−4}{a^2+6a−16}\)
Example \(\PageIndex{87}\)
\(\frac{y^2−2y−3}{y^2−9}\)
- Answer
-
\(\frac{y+1}{y+3}\)
Example \(\PageIndex{88}\)
\(\frac{b^2+9b+18}{b^2−36}\)
Example \(\PageIndex{89}\)
\(\frac{y^3+y^2+y+1}{y^2+2y+1}\)
- Answer
-
\(\frac{y^2+1}{y+1}\)
Example \(\PageIndex{90}\)
\(\frac{p^3+3p^2+4p+12}{p^2+p−6}\)
Example \(\PageIndex{91}\)
\(\frac{x^3−2x^2−25x+50}{x^2−25}\)
- Answer
-
x−2
Example \(\PageIndex{92}\)
\(\frac{q^3+3q^2−4q−12}{q^2−4}\)
Example \(\PageIndex{93}\)
\(\frac{3a^2+15a}{6a^2+6a−36}\)
- Answer
-
\(\frac{a(a+5)}{2(a+3)(a−2)}\)
Example \(\PageIndex{94}\)
\(\frac{8b^2−32b}{2b^2−6b−80}\)
Example \(\PageIndex{95}\)
\(\frac{−5c^2−10c}{−10c^2+30c+100}\)
- Answer
-
\(\frac{c}{2(c−5)}\)
Example \(\PageIndex{96}\)
\(\frac{4d^2−24d}{2d^2−4d−48}\)
Example \(\PageIndex{97}\)
\(\frac{3m^2+30m+75}{4m^2−100}\)
- Answer
-
\(\frac{3(m+5)}{4(m−5)}\)
Example \(\PageIndex{98}\)
\(\frac{5n^2+30n+45}{2n^2−18}\)
Example \(\PageIndex{99}\)
\(\frac{5r^2+30r−35}{r^2−49}\)
- Answer
-
\(\frac{5(r−1)}{r+7}\)
Example \(\PageIndex{100}\)
\(\frac{3s^2+30s+24}{3s^2−48}\)
Example \(\PageIndex{101}\)
\(\frac{t^3−27}{t^2−9}\)
- Answer
-
\(\frac{t^2+3t+9}{t+3}\)
Example \(\PageIndex{102}\)
\(\frac{v^3−1}{v^2−1}\)
Example \(\PageIndex{103}\)
\(\frac{w^3+216}{w^2−36}\)
- Answer
-
\(\frac{w^2−6w+36}{w−6}\)
Example \(\PageIndex{104}\)
\(\frac{v^3+125}{v^2−25}\)
Simplify Rational Expressions with Opposite Factors
In the following exercises, simplify each rational expression.
Example \(\PageIndex{105}\)
\(\frac{a−5}{5−a}\)
- Answer
-
−1
Example \(\PageIndex{106}\)
\(\frac{b−12}{12−b}\)
Example \(\PageIndex{107}\)
\(\frac{11−c}{c−11}\)
- Answer
-
−1
Example \(\PageIndex{108}\)
\(\frac{5−d}{d−5}\)
Example \(\PageIndex{109}\)
\(\frac{12−2x}{x^2−36}\)
- Answer
-
\(−\frac{2}{x+6}\)
Example \(\PageIndex{110}\)
\(\frac{20−5y}{y^2−16}\)
Example \(\PageIndex{111}\)
\(\frac{4v−32}{64−v^2}\)
- Answer
-
\(−\frac{4}{8+v}\)
Example \(\PageIndex{112}\)
\(\frac{7w−21}{9−w^2}\)
Example \(\PageIndex{113}\)
\(\frac{y^2−11y+24}{9−y^2}\)
- Answer
-
\(−\frac{y−8}{3+y}\)
Example \(\PageIndex{114}\)
\(\frac{z^2−9z+20}{16−z^2}\)
Example \(\PageIndex{115}\)
\(\frac{a^2−5a−36}{81−a^2}\)
- Answer
-
\(−\frac{a+4}{9+a}\)
Example \(\PageIndex{116}\)
\(\frac{b^2+b−42}{36−b^2}\)
Everyday Math
Example \(\PageIndex{117}\)
Tax Rates For the tax year 2015, the amount of tax owed by a single person earning between $37,450 and $90,750, can be found by evaluating the formula 0.25x−4206.25, where x is income. The average tax rate for this income can be found by evaluating the formula \(\frac{0.25x−4206.25}{x}\). What would be the average tax rate for a single person earning $50,000?
- Answer
-
16.5%
Example \(\PageIndex{118}\)
Work The length of time it takes for two people for perform the same task if they work together can be found by evaluating the formula \(\frac{xy}{x+y}\). If Tom can paint the den in x=45 minutes and his brother Bobby can paint it in y=60 minutes, how many minutes will it take them if they work together?
Writing Exercises
Example \(\PageIndex{119}\)
Explain how you find the values of x for which the rational expression \(\frac{x^2−x−20}{x^2−4}\) is undefined.
Example \(\PageIndex{120}\)
Explain all the steps you take to simplify the rational expression \(\frac{p^2+4p−21}{9−p^2}\).
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!
…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.
Glossary
- rational expression
- A rational expression is an expression of the form \(\frac{p}{q}\), where p and q are polynomials and \(q \ne 0\).


11.1: Quiz 0-Check Yourself
1. OSHA's mission is to:
a. Protect the safety and health of the general public.
b. Protect the safety and health of America's workers,
c. Ensure that all workers receive the federal minimum wage,
d. Ensure that all workers receive adequate workers' compensation payments.
2. The creation of OSHA provided this important right to workers:
a. The right to equal employment opportunities,
b. The right to privacy,
c. The right to pension benefits.
d. The right to a safe and healthful workplace.
3. A Safety Data Sheet (SDS) gives information about:
a. Hazardous chemicals,
b. Injuries in the workplace,
c. Medical examinations,
d. Machinery maintenance.
4. Among the rights related to OSHA recordkeeping, workers have the right to review:
a. All first aid treatment forms.
b. All Workers Compensation forms.
c. The OSHA 300 Log and the OSHA 300A Summary.
d. Medical and exposure records for all workers.
5. During an OSHA inspection:
a. Worker representatives are not permitted to accompany the inspector,
b. You may not describe safety and health concerns you have to the inspector,
c. Employers determine which workers are interviewed.
d. You have the right to talk to the inspector privately.
6. The right of workers to seek safety and health on the job without fear of punishment is spelled out in:
a. State laws,
b. The OSHA standards.
c. Section ll(c) of the OSH Act.
d. The General Duty Clause.
7. One of the main responsibilities employers have under OSHA is to:
a. Provide training required by OSHA standards.
b. Reduce air pollution in the environment.
c. Conduct energy audits.
d. Notify OSHA of any workplace injury or illness.
8. OSHA requires that employers pay for most required personal protective equipment (PPE), including:
a. Uniforms.
b. Logging boots.
c. Hard hats.
d. Weather-related gear.
9. The OSHA standards for Construction and General Industry are also known as:
a. Part 1926 and Part 1910.
b. Part 1915 and Part 1917.
c. Codes 501 through 1000.
d. Construction and GI Registers.
10. What type of OSHA inspection is conducted when immediate death or serious harm is likely?
a. Complaint.
b. Programmed.
c. Referral.
d. Imminent danger.
11. When the employer receives an OSHA citation, it must be:
a. Copied and mailed to each worker.
b. Posted for 3 days or until the violation is fixed.
c. Contested and filed with the courts.
d. Signed and returned to OSHA.
12. If you feel that an OSHA inspection is needed to get hazards corrected at your workplace, which is your best option?
a. File a complaint online.
b. Submit a written, signed complaint with specific hazard information.
c. Request a Health Hazard Evaluation from NIOSH.
d. Submit an unsigned complaint form to OSHA.


18.9: Personality Theory in Real Life
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Media Violence and Its Effects on Children
The second Bobo doll study conducted by Bandura and his colleagues presented videotapes of the models behaving aggressively toward the Bobo doll (Bandura, Ross, & Ross, 1963a). The observation in this study that children did, indeed, imitate the aggressive behavior of models seen on film, but not in person, has become one the most influential articles in history, both in terms of stimulating further research on the modeling of aggression and in terms of practical applications (e.g., ratings on television programs, video games, and CDs with controversial lyrics). In their introduction to an issue of Psychological Science in the Public Interest ( PSPI ) on media violence and its effects on children, Ceci and Bjork (2003) describe how the authors of this particular PSPI issue were a collection of experts brought together to prepare a portion of the U.S. Surgeon General’s report on youth violence. However, political concerns led the Surgeon General’s office to radically alter the report. The authors protested, and attempted to satisfy the concerns of the politicians. Ultimately, however, this portion of the report was dropped from the final version. So, the editors of PSPI chose to publish this valuable psychological information, which clearly causes political concerns for some, so that psychologists, psychiatrists, law enforcement officials, teachers and school administrators, government policy makers, parents, indeed anyone who has a vested interest in youth violence, might have the most current information available on the effects of media violence on America’s youth. I would also like to acknowledge an important point made by the authors themselves. This issue of PSPI was a collaborative project by eight experts on the effects of media violence. The authors decided that the fairest way to list their names was alphabetical order. Thus, as I refer to the article as Anderson, et al. (2003), please note that the “et al.” includes widely respected experts, chosen by the National Institute of Mental Health for this project, each of whom contributed significantly to this article.
It would be difficult to do justice to this report by summarizing it here, since the report, which is a review itself, is nearly thirty pages long and cites 245 research studies. Clearly, the attention drawn by this question, the role of media violence in affecting violent behavior among children, provides a testament to the significance of this research. Anderson, et al. (2003) provide an overview of the empirical research, offer theoretical explanations, they address moderating effects, media use and content, and they examine research on interventions.
Empirical research on media violence and aggression has covered of wide variety of media types, including dramatic television and movies, television news violence, music videos and their lyrics, video games, and the internet. The review of this extensive body of research has shown a statistically significant association between exposure to media violence and aggression and violence among youth. Exposure to media violence, and the findings are consistent across the various types of media, increases the likelihood that children have aggressive thoughts and, indeed, engage in aggressive behavior. Most importantly, longitudinal studies have consistently shown that exposure to media violence in childhood, even beginning in late adolescence, are predictors of increased aggression and violent behavior in adulthood. Although the effects are at best small to medium, the authors note that they are as high as other problems that are considered significant public health issues, such as cigarette smoking and exposure to asbestos.
Several theories have been put forth to explain the manner in which exposure to media violence increases aggression in children, not the least of which is Bandura’s theory of observational learning. Observational learning appears to be so important to the humans, and other primate species, that we have developed a specific neurological system to learn from watching others: the mirror-neuron system (see Rizzolatti & Craighero, 2004). Additional theories suggest that exposure to media violence prepares an individual to engage in aggressive behavior. This can occur either because the media violence primes the individual by activating aggressive thoughts and scripts or by generally arousing the individual, and that arousal then carries over into heightened aggressive behavior in situations that provoke aggressive actions. Finally, there is the widely used, and often misunderstood, term “ desensitization .” Anderson et al. (2003) prefer the term emotional desensitization to refer to a reduction in distress-related reactions to observing or thinking about violence. Although emotional desensitization may result in an individual being more willing to engage in aggressive or violent behavior, there is no evidence that it stimulates aggression. Thus, although emotional desensitization may be related to the disinhibition discussed in the chapter (it may even be the same thing), its role remains unclear.
Not all children are affected by media violence in the same way, and not all examples of media violence are equally effective in enhancing aggression. Thus, there must be moderating factors that come into play in the relationship between exposure to media violence and subsequent aggressive behavior. For example, the effects of media violence decline as children grow older, media violence is less likely to affect children who are less aggressive to begin with, and, although there is not a direct effect of intelligence, children of lower intelligence are more likely to watch television and to be at risk for other factors enhancing aggressiveness. The way in which the violence is presented in the media is also important, as is any portrayed justification for the violence as well as its consequences. However, all of these relationships are complex, and have not been studied in great detail. Thus, there is much more work to be done. Finally, one of the most important moderating factors is parental control. When parents discuss the content of media violence with their children, when they comment regularly on the reality of violence, children are much less likely to demonstrate aggressive tendencies.
This entire discussion would be moot, if not for the prevalence of media in our society and the content of that media. Data reported by Anderson, et al. (2003) from three national surveys provide an amazing view of media availability in America. Virtually all families with children have a television set, most have at least one VCR or DVD player, and about three quarters of those families subscribe to cable or satellite television. Approximately 70 percent of families with children have a video game system and a computer, and most American children have a television in their room (including 30 percent of children age 0 to 3 years old!). Watching television is the third most common activity for children, after sleeping and going to school. A significant number of children watch more than 40 hours of television a week, and children ages 0 to 6 years spend more time engaged in media entertainment than engaged in reading, being read to, and playing outside combined! And what’s included in that media content? Plenty of violence: 61 percent of programs contain violence, and only 4 percent of programs contain an anti-violence theme:
…put in another way, 96% of all violent television programs use aggression as a narrative, cinematic device for simply entertaining the audience…Moreover, most aggression on television is glamorized and trivialized…and nearly 75% of all violent scenes feature no immediate punishment or condemnation for violence. (pg. 101; Anderson, et al., 2003)
Similar findings have been reported for violence in video games, both in America and in Japan. Parents seldom recognize the popularity of those violent games, and only about one third of parents even knew the name of their child’s favorite video game (Anderson, et al., 2003).
Despite the wealth of information on the relationship between exposure to media violence and aggressive behavior, and a number of potential approaches for intervention, there has been little empirical research on potential interventions. In general, there are three approaches to reducing the influence of violent media: changing attitudes to counter violent messages, encouraging parental monitoring and guidance, and providing education on the content and effects of various forms of media. But once again, none of these potential interventions has been studied in detail. It must also be remembered that this is only one of many factors that contribute to aggressive and violent behavior amongst youth, though it may be the least expensive to address, since it involves little more than making choices about the media children are exposed to:
…However, the troubling truth is that violent media are entering the home and inviting active participation of even very young children - often with little parental supervision…Although additional research to address unresolved questions is needed, it is clear that media violence is a causal risk factor that should be addressed in thoughtful ways…parents can reduce and shape their children’s consumption of violent media…from a public-health perspective, today’s consumption patterns are far from optimal. And for many youths, they are clearly harmful. (pgs. 105-106; Anderson, et al., 2003)
Research on this important issue certainly did not end with the publication of the PSPI issue in 2003. In a more recent study, Carnagey and Anderson (2005) had college students play one of three versions of a race-car video game: one version rewarded all violence, one punished all violence, and one was non-violent. Overall, rewarding violent actions in the game increased hostile emotion, aggressive thinking, and aggressive behavior. In contrast, punishing violence still increased hostile emotion, but did not increase aggressive thoughts or behaviors. The authors suggest that an important mechanism through which violent media increases aggressive behavior is to first increase aggressive thoughts and states of mind. Another interesting result in this study was that the non-violent game did not result in as much violence as the game in which violence was rewarded, suggesting that the violent behavior in the game was not simply the result of competitiveness by the players. Of course, not everyone responds aggressively when aggression is modeled. It appears that individuals who score high on the trait of agreeableness (one of the “Big Five” traits, see Chapter 13) are able to short-circuit the effects of aggression-related cues and curb their aggression (Meier, Robinson, & Wilkowski, 2006). Meier, et al. go on to suggest that teaching other people to associate aggression-related cues with prosocial behavior might become a valuable intervention in continued efforts to reduce aggression and violence in our society. First, however, it might prove important to reduce the number of weapons present in American society (or at least our relationship with them). Klinesmith, Kasser, and McAndrew (2006) recently demonstrated that when male college students were exposed to a gun there was an increase in both their testosterone levels and their aggressive behavior. Media violence is clearly not the only aggression-related cue present for most people. However, as noted above, it may well be one area in which a distinct reduction in such cues can easily be accomplished without substantial financial costs (but certainly with substantial benefits) to society.


11.2: Interactions of Skeletal Muscles, Their Fascicle Arrangement, and Their Lever Systems
By the end of this section, you will be able to:
- Compare and contrast agonist and antagonist muscles
- Describe how fascicles are arranged within a skeletal muscle
- Explain the major events of a skeletal muscle contraction within a muscle in generating force
To move the skeleton, the tension created by the contraction of the fibers in most skeletal muscles is transferred to the tendons. The tendons are strong bands of dense, regular connective tissue that connect muscles to bones. The bone connection is why this muscle tissue is called skeletal muscle.
Interactions of Skeletal Muscles in the Body
To pull on a bone, that is, to change the angle at its synovial joint, which essentially moves the skeleton, a skeletal muscle must also be attached to a fixed part of the skeleton. The moveable end of the muscle that attaches to the bone being pulled is called the muscle’s insertion , and the end of the muscle attached to a fixed (stabilized) bone is called the origin . During forearm flexion —bending the elbow—the brachioradialis assists the brachialis.
Although a number of muscles may be involved in an action, the principal muscle involved is called the prime mover , or agonist . To lift a cup, a muscle called the biceps brachii is actually the prime mover; however, because it can be assisted by the brachialis, the brachialis is called a synergist in this action (Figure \(\PageIndex{1}\) ). A synergist can also be a fixator that stabilizes the bone that is the attachment for the prime mover’s origin.
A muscle with the opposite action of the prime mover is called an antagonist . Antagonists play two important roles in muscle function: (1) they maintain body or limb position, such as holding the arm out or standing erect; and (2) they control rapid movement, as in shadow boxing without landing a punch or the ability to check the motion of a limb.
For example, to extend the knee, a group of four muscles called the quadriceps femoris in the anterior compartment of the thigh are activated (and would be called the agonists of knee extension). However, to flex the knee joint, an opposite or antagonistic set of muscles called the hamstrings is activated.
As you can see, these terms would also be reversed for the opposing action. If you consider the first action as the knee bending, the hamstrings would be called the agonists and the quadriceps femoris would then be called the antagonists. See Table \(\PageIndex{1}\) for a list of some agonists and antagonists.
| Agonist | Antagonist | Movement |
|---|---|---|
| Biceps brachii: in the anterior compartment of the arm | Triceps brachii: in the posterior compartment of the arm | The biceps brachii flexes the forearm, whereas the triceps brachii extends it. |
| Hamstrings: group of three muscles in the posterior compartment of the thigh | Quadriceps femoris: group of four muscles in the anterior compartment of the thigh | The hamstrings flex the leg, whereas the quadriceps femoris extend it. |
| Flexor digitorum superficialis and flexor digitorum profundus: in the anterior compartment of the forearm | Extensor digitorum: in the posterior compartment of the forearm | The flexor digitorum superficialis and flexor digitorum profundus flex the fingers and the hand at the wrist, whereas the extensor digitorum extends the fingers and the hand at the wrist. |
There are also skeletal muscles that do not pull against the skeleton for movements. For example, there are the muscles that produce facial expressions. The insertions and origins of facial muscles are in the skin, so that certain individual muscles contract to form a smile or frown, form sounds or words, and raise the eyebrows. There also are skeletal muscles in the tongue, and the external urinary and anal sphincters that allow for voluntary regulation of urination and defecation, respectively. In addition, the diaphragm contracts and relaxes to change the volume of the pleural cavities but it does not move the skeleton to do this.
Everyday Connection
Exercise and Stretching
When exercising, it is important to first warm up the muscles. Stretching pulls on the muscle fibers and it also results in an increased blood flow to the muscles being worked. Without a proper warm-up, it is possible that you may either damage some of the muscle fibers or pull a tendon. A pulled tendon, regardless of location, results in pain, swelling, and diminished function; if it is moderate to severe, the injury could immobilize you for an extended period.
Recall the discussion about muscles crossing joints to create movement. Most of the joints you use during exercise are synovial joints, which have synovial fluid in the joint space between two bones. Exercise and stretching may also have a beneficial effect on synovial joints. Synovial fluid is a thin, but viscous film with the consistency of egg whites. When you first get up and start moving, your joints feel stiff for a number of reasons. After proper stretching and warm-up, the synovial fluid may become less viscous, allowing for better joint function.
Patterns of Fascicle Organization
Skeletal muscle is enclosed in connective tissue scaffolding at three levels. Each muscle fiber (cell) is covered by endomysium and the entire muscle is covered by epimysium. When a group of muscle fibers is “bundled” as a unit within the whole muscle by an additional covering of a connective tissue called perimysium, that bundled group of muscle fibers is called a fascicle . Fascicle arrangement by perimysia is correlated to the force generated by a muscle; it also affects the range of motion of the muscle. Based on the patterns of fascicle arrangement, skeletal muscles can be classified in several ways. What follows are the most common fascicle arrangements.
Parallel muscles have fascicles that are arranged in the same direction as the long axis of the muscle (Figure \(\PageIndex{2}\)). The majority of skeletal muscles in the body have this type of organization. Some parallel muscles are flat sheets that expand at the ends to make broad attachments. Other parallel muscles are rotund with tendons at one or both ends. Muscles that seem to be plump have a large mass of tissue located in the middle of the muscle, between the insertion and the origin, which is known as the central body. A more common name for this muscle is belly . When a muscle contracts, the contractile fibers shorten it to an even larger bulge. For example, extend and then flex your biceps brachii muscle; the large, middle section is the belly (Figure \(\PageIndex{3}\)). When a parallel muscle has a central, large belly that is spindle-shaped, meaning it tapers as it extends to its origin and insertion, it sometimes is called fusiform .
Circular muscles are also called sphincters (see Figure \(\PageIndex{2}\)). When they relax, the sphincters’ concentrically arranged bundles of muscle fibers increase the size of the opening, and when they contract, the size of the opening shrinks to the point of closure. The orbicularis oris muscle is a circular muscle that goes around the mouth. When it contracts, the oral opening becomes smaller, as when puckering the lips for whistling. Another example is the orbicularis oculi, one of which surrounds each eye. Consider, for example, the names of the two orbicularis muscles (orbicularis oris and oribicularis oculi), where part of the first name of both muscles is the same. The first part of orbicularis, orb (orb = “circular”), is a reference to a round or circular structure; it may also make one think of orbit, such as the moon’s path around the earth. The word oris (oris = “oral”) refers to the oral cavity, or the mouth. The word oculi (ocular = “eye”) refers to the eye.
There are other muscles throughout the body named by their shape or location. The deltoid is a large, triangular-shaped muscle that covers the shoulder. It is so-named because the Greek letter delta looks like a triangle. The rectus abdominis (rector = “straight”) is the straight muscle in the anterior wall of the abdomen, while the rectus femoris is the straight muscle in the anterior compartment of the thigh.
When a muscle has a widespread expansion over a sizable area, but then the fascicles come to a single, common attachment point, the muscle is called convergent . The attachment point for a convergent muscle could be a tendon, an aponeurosis (a flat, broad tendon), or a raphe (a very slender tendon). The large muscle on the chest, the pectoralis major, is an example of a convergent muscle because it converges on the greater tubercle of the humerus via a tendon. The temporalis muscle of the cranium is another.
Pennate muscles (penna = “feathers”) blend into a tendon that runs through the central region of the muscle for its whole length, somewhat like the quill of a feather with the muscle arranged similar to the feathers. Due to this design, the muscle fibers in a pennate muscle can only pull at an angle, and as a result, contracting pennate muscles do not move their tendons very far. However, because a pennate muscle generally can hold more muscle fibers within it, it can produce relatively more tension for its size. There are three subtypes of pennate muscles.
In a unipennate muscle, the fascicles are located on one side of the tendon. The extensor digitorum of the forearm is an example of a unipennate muscle. A bipennate muscle has fascicles on both sides of the tendon. In some pennate muscles, the muscle fibers wrap around the tendon, sometimes forming individual fascicles in the process. This arrangement is referred to as multipennate . A common example is the deltoid muscle of the shoulder, which covers the shoulder but has a single tendon that inserts on the deltoid tuberosity of the humerus.
Because of fascicles, a portion of a multipennate muscle like the deltoid can be stimulated by the nervous system to change the direction of the pull. For example, when the deltoid muscle contracts, the arm abducts (moves away from midline in the sagittal plane), but when only the anterior fascicle is stimulated, the arm will abduct and flex (move anteriorly at the shoulder joint).
The Lever System of Muscle and Bone Interactions
Skeletal muscles do not work by themselves. Muscles are arranged in pairs based on their functions. For muscles attached to the bones of the skeleton, the connection determines the force, speed, and range of movement. These characteristics depend on each other and can explain the general organization of the muscular and skeletal systems.
The skeleton and muscles act together to move the body. Have you ever used the back of a hammer to remove a nail from wood? The handle acts as a lever and the head of the hammer acts as a fulcrum, the fixed point that the force is applied to when you pull back or push down on the handle. The effort applied to this system is the pulling or pushing on the handle to remove the nail, which is the load, or “resistance” to the movement of the handle in the system. Our musculoskeletal system works in a similar manner, with bones being stiff levers and the articular endings of the bones—encased in synovial joints—acting as fulcrums. The load would be an object being lifted or any resistance to a movement (your head is a load when you are lifting it), and the effort, or applied force, comes from contracting skeletal muscle.


1.1: Overview
Cosmologists speak with a high degree of confidence about conditions that existed billions of years ago when the universe was quite different from how we find it today: \(10^9\) times hotter than today, over \(10^{30}\) times denser, and much much smoother, with variations in density from one place to another only as large as one part in 100,000. We claim to know the composition of the universe at this early time, dominated almost entirely by thermal distributions of photons and subatomic particles called neutrinos. We know in detail many aspects of the evolutionary process that connects this early universe to the current one. Our models of this evolution have been highly predictive and enormously successful.
In this chapter we provide an overview of our subject, broken into two parts. The first is focused on the discovery of the expansion of the universe in 1929, and the theoretical context for this discovery, which is given by Einstein's general theory of relativity (GR). The second is on the implications of this expansion for the early history of the universe, and relics from that period observable today: the cosmic microwave background and the lightest chemical elements. The consistency of such observations with theoretical predictions is why we speak confidently about such early times, approximately 14 billion years in our past.
Overview Part I: The Expansion of Space and the Contents of the Universe
Space is not what you think it is. It can curve and it can expand over time. We can observe the consequences of this expansion over time, and from these observations infer some knowledge of the Universe's contents.
Newton-Maxwell Incompatibility and Einstein's Theory
It would be difficult to overstate the impact that Einstein’s 1915 General Theory of Relativity has had on the field of cosmology. So we begin our review with some discussion of the General Theory, and its origin in conflicts between Newtonian physics and Maxwell’s theory of electric and magnetic fields. We end our discussion of Einstein’s theory with its prediction that the universe must be either expanding or contracting.
Newton’s laws of motion and gravitation have amazing explanatory powers. Relatively simple laws describe, almost perfectly, the motions of the planets and the moon, as well as the motions of bodies here on Earth -- at least at speeds much lower than the speed of light. The discovery of the planet Neptune provides us with an example of their predictive power. The discovery began with calculations by Urbain Le Verrier. Using Newton’s theory, he was able to explain the observed motion of Uranus only if he posited the existence of a planet with a particular orbit, an extra planet beyond those known at the time. Without the gravitational pull of this not-yet-seen planet, Newton's theory could not account for the motion of Uranus. Following up on his prediction, made public in 1846, Johannes Gottfried Galle looked for a new planet where Le Verrier said it was to be found and, indeed, there it was, what we now call Neptune. The time from prediction to confirmation was less than a month.
Less than 20 years after the discovery of Neptune, a triumph of the Newtonian theory, came a great inductive synthesis: Maxwell’s theory of electric and magnetic fields. The experiments that led to this synthesis, and the synthesis itself, have enabled the development of a great range of technologies we now take for granted such as electric motors, radio, television, cellular communication networks and microwave ovens. More important for our subject, they also led to radical changes to our conception of space and time.
These radical changes arose from conflicts between the synthesis of Maxwell with that of Newton. For example, in the Newtonian theory velocities add: if A sees B move at speed v to the west, and B sees C moving at speed v to the west relative to B, then A sees C moving at speed v+v = 2v to the west. But one solution to the Maxwell equations is a propagating disturbance in the electromagnetic fields that travels with a fixed speed of about 300,000 km/sec. Without modification, the Maxwell equations predict that both A and B would see electromagnetic wave C moving away from them at a speed of 300,000 km/sec, violating the velocity addition rule that one can derive from Newtonian concepts of space and time.
Einstein’s solution to these inconsistencies includes an abandonment of Newtonian concepts of space and time. This abandonment, and discovery of the replacement principles consistent with the Maxwell theory, happened over a considerable amount of time. A solution valid in the absence of gravitation came out first in 1905, with Einstein’s paper titled (in translation from German) “On the Electrodynamics of Moving Bodies.” His effort to reconcile gravitational theory with Maxwell’s theory did not fully come together until November of 1915, with a series of lectures in Berlin where he presented his General Theory of Relativity.
One indicator that Einstein was on the right track was his realization, in September 1915, that his theory provided an explanation for a longstanding problem in solar system dynamics known as the anomalous perihelion precession of Mercury. Given Newtonian theory, and an absence of other planets, Mercury would orbit the Sun in an ellipse shape. However, the influence of the other planets is to make Mercury follow almost an ellipse, in a pattern that is well approximated as that of a slowly rotating ellipse. One way of expressing this rotation is to say how rapidly the location of closest approach, called perihelion, is rotating around the Sun. Mercury’s perihelion precession is quite slow. In fact, it’s less than one degree per century. More precisely, it’s 575 seconds of arc per century, where a second of arc is a degree divided by 3,600 (just as a second is an hour divided by 3,600).
Urbain Le Verrier, following his success with Neptune, took up the question of this motion of Mercury: could the perihelion precession be understood as resulting from the pulls on Mercury from the other planets. He found that he could ascribe about 532 seconds of arc to the other planets, but not the entire 575. There is an additional, unexplained (“anomalous”) precession of 43 seconds of arc per century. Le Verrier, of course, knew how to handle situations like this. He proposed that this motion is caused by a not-yet-discovered planet. This planet was proposed to have an orbit closer to the Sun than Mercury’s and eventually had the name Vulcan.
But “Vulcan, Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto” is probably not the list of planets you learned in elementary school! Unlike the success with Neptune, Newtonian gravity was not going to be vindicated by discovery of another predicted planet. Rather than an unaccounted-for planet, the anomalous precession of Mercury, we now know, as Einstein figured out in September of 1915, is due to a failure of Newton’s theory of gravity. At slow speeds and for weak gravitational fields the Newtonian theory is an excellent approximation to Einstein’s theory -- so the largest errors in the Newtonian theory show up for the fastest-moving planet orbiting closest to the Sun.
Amazingly, thinking about a theory that explains experiments with electricity and magnets, and trying to reconcile it with Newton’s laws of gravitation and motion, had led to a solution to this decades-old problem in solar system dynamics. The theory has gone on from success to success since that time, most recently with the detection of gravitational waves first reported in 2016. It is of practical importance in the daily lives of many of us: GPS software written based on Newtonian theory rather than Einstein’s theory would be completely useless.
The Expansion of Space
More important to our subject, Einstein’s theory allowed for better-informed speculation about the history of the universe as a whole. In the years following Einstein’s November 1915 series of lectures, a number of theoreticians calculated solutions of the Einstein equations for highly-idealized models of the universe. The Einstein field equations are extremely difficult to solve in generality. The first attempts at solving these equations for the universe as a whole thus involved extreme idealization. They used what you might call “the most spherical cow approximation of all time.” They approximated the whole universe as completely homogeneous; i.e., absolutely the same everywhere.
We now know that on very large scales, this is a good approximation to our actual universe. To illustrate what we mean by homogeneity being a good approximation on very large scales, we have the figure below which shows a slice from a large-volume simulation of the large-scale structure of our universe. In these images, brighter regions are denser regions. The image has two sets of sub-boxes: large ones and small ones. We can see that the universe appears different in the small boxes. Box 4 is under dense, Box 5 is over dense, and Box 6 is about average. If we look at the larger boxes, the universe appears more homogeneous. Each box looks about the same. This is the sense by which we mean that on large scales the universe is highly homogeneous.
On large scales the Universe is highly homogeneous. The large boxes (boxes 1, 2, and 3) are about 200 Mpc across (that's about 600 million light years). No matter where you put down such a large box, the contents look similar. For the smaller boxes this is not the case. Images are from the Millenium Simulation .
Gif of an expanding grid (below) Image by Alex Eisner
D vs. z: Low z
In 1929, Edwin Hubble made an important observation by measuring distances to various galaxies and by measuring their "redshifts." Hubble inferred distances to galaxies by using standard candles, which are objects with predictable luminosities. Since farther away objects appear dimmer, one can predict distances by comparing the object’s expected luminosity with how bright it appears. The redshift of a galaxy, which cosmologists label as “\( z \)”, tells us how the wavelength of light has shifted during its propagation. Mathematically, \(1+z\) is equal to the ratio of the wavelength of observed light to the wavelength of emitted light: \( (1+z = \lambda_{\rm observed}\)/\(\lambda_{\rm emitted}) \). At least for \(z << 1\), one can think of this as telling us how fast the galaxy is moving away from us according to the Doppler effect: a higher redshift indicates a larger velocity with relationship \(v=cz\) with \(c\) the speed of light. If a galaxy were instead moving towards us, its light would appear blueshifted. Hubble found that not only were nearly all the galaxies redshifted, but there was a linear relationship between the galaxies’ distances and redshifts. This is represented by Hubble’s law, v=\(H_0\)d. Hubble's observations, which were evidence for this simple law, had profound consequences for our understanding of the cosmos: they indicated that the universe was expanding.
To understand this, take a look at the image of an expanding grid. Each of the two red points is “stationary”; i.e., they each have a specific defined location on the grid and do not move from that location. However, as the grid itself expands, the distance between the two points grows, and they appear to move away from each other. If you lived anywhere on this expanding grid, you would see all other points moving away from you. It's easiest to see this is true for the central location on the grid. Try placing your mouse over different points on the grid and following them as they expand. You will notice that points farther from the center will move away faster than points close to the center. This is how Hubble’s law and the expansion of space works. Instead of viewing redshifts as being caused by a Doppler effect from galaxies moving through space, we will come to understand the redshift as the result of the ongoing creation of new space.
Hubble’s constant, \(H_0\), tells us how fast the universe is expanding in the current epoch. It was first estimated to be about 500 km/s/Mpc, but after eliminating some gross systematic errors and obtaining more accurate measurements, we know this initial estimate was off, and not by any small amount. It's wrong by about a factor of 7! The exact value of the Hubble constant is actually somewhat controversial today, but everyone agrees it's somewhere between 66 and 75 km/s/Mpc. This means that on large length scales, where the approximation of a homogeneous universe becomes more accurate, about 70 kilometers of new space are created each second in every Megaparsec.
D vs. z: High z
As we will see, from Einstein’s theory of space and time we can expect the rate of expansion of space to change over time. The history of these changes to the expansion rate leave their impact on the distance-redshift relation if we trace it out to sufficiently large distances and redshifts. As we measure out to larger distances, the relationship is no longer governed by cz = \(H_0\) d. Instead, we will show that the redshift tells us how much the universe has expanded since light left the object we are observing.
\[1+z =\dfrac{ \lambda_{observed}}{\lambda_{emitted}} = \dfrac{a_0}{a_e}\]
where a is the “scale factor” that parameterizes the expansion of space, \(a_0\) is the scale factor today and \(a_e\) is the scale factor when the light was emitted. We can observe quasars so far away that the universe has expanded by a factor of 7 since the light left them that we are receiving now. For such a quasar we have \(a_0\)/\(a_e\) = 7 so the wavelength of light has been stretched by a factor of 7, and by definition of redshift z we have z = 6.
The distance from us to such a quasar depends on how long it took for the universe to expand by a factor of 7. If the expansion rate were slower over this time, then it would have taken longer, so the quasar must be further away. Measurements of distance vs. redshift are thus sensitive to the history of the expansion rate.
As we will see, how the expansion rate changes over time depends on what the universe is made out of. Therefore, studying D vs. z out to high distances and redshifts can help us determine the composition of the universe. We can see such measurements in Fig. B, together with some model curves. The models all have the same expansion rate today, \(H_0\), but differ in the mix of different kinds of matter/energy in the universe. A model that’s purely non-relativistic matter, the green dashed line "CDM Model", does a very poor job of fitting the data. The data seem to require a contribution to the energy density that we call “vacuum energy” or “the cosmological constant.” The “Lambda CDM” or "LCDM" model has a mixture of this cosmological constant and non-relativistic matter. The cosmological constant causes the expansion rate to accelerate. Thus, compared to the model without a cosmological constant (the "CDM Model"), the expansion rate in the past was slower; it thus took a longer time for the universe to expand by a factor of 1+z, and thus objects with a given z are at further distances. This acceleration of the expansion rate, discovered via D vs. z measurements at the end of the 20th century, was a great surprise to most cosmologists. Why there is a cosmological constant, or whether there is something else causing the acceleration, is one of the great mysteries of modern cosmology.
Overview Part II: The Hot Big Bang and its Relics
The expansion of the universe implies that it must have been much smaller, and much denser, in the past. If this is true, we should be able to see some consequences from the very high density period of the early universe. One such consequence is the relative abundances of light elements we see today. Most of the Helium, the second-most-common element in the Universe, was created when the expansion was less than a few minutes old. Trace amounts of other light elements were also created in this early period in a process we call Big Bang Nucleosynthesis (BBN). Observation of the abundances of these light elements can tell us about conditions at such early times. To achieve consistency between predictions and observations requires that the big bang was very hot, and thereby led to the prediction of what we call the cosmic microwave background. The discovery of this background in 1964 led to the establishment of the hot big bang model as the standard cosmological paradigm.
Our First Relic: Light Elements
In 1948, Gamow explored the very early universe as a source of elements heavier than hydrogen. He extrapolated Einstein’s theory of an expanding universe with certain assumptions about the composition of the universe and concluded that it was infinitely dense at a finite time in the past. He theorized that this early universe could be a prodigious source of heavier elements. If the universe began at infinitely high density and temperatures and underwent rapid expansion and cooling, atomic nuclei would form not too early, when high-energy radiation did not permit any nuclei to survive, and not too late, when temperatures were too low for the nuclear collisions to overcome the Coulomb repulsion, but at a just-right intermediate stage.
As we will see, almost all of the hydrogen and helium in the universe originated in the big bang.  In our chapters on big bang nucleosnythesis we will explore the theory of the production of helium, and compare the predicted amounts of helium and trace amounts of deuterium and lithium with attempts to infer those abundances from observations.
Our Next Relic: The Cosmic Microwave Background
An important aspect of Gamow's work was that in order to avoid overproduction of helium and other heavy elements, the ratio of nucleons to photons at this just-right epoch had to be very small. Since the number of photons in black body radiation is proportional to temperature cubed, this means that the “Big Bang” had to be very, very hot. Pushing this chain of logic forward in a follow-up paper, Alpher and Herman theorized that we should see a background of heat and light from this period of high temperature and photon density. This background was discovered later in 1964 and is known as the Cosmic Microwave Background (CMB).
The CMB is an enormous gift from nature to cosmologists. By measuring both the spectrum of this light, and mapping its variation in intensity and polarization across the sky, we have a highly sensitive probe of conditions in the universe from times of thousands of years to hundreds of thousands of years. What's more, the conditions in the universe at this time were such that one can calculate, with great accuracy, the outcomes for the CMB spectrum, intensity, and polarization measurements, to be expected for some particular model of the cosmos. Physical systems with such extreme calculability and utility are rare. The CMB is to cosmology what the solar system was for science and Newtonian gravitational theory three hundred plus years ago.
The following image is a projection of intensity variations across the whole (curved) sky on to a flat map. To better understand how this relates to what we observe from Earth, explore the virtual CMB planetarium below it by clicking and dragging. This shows how the CMB would look over a tree-lined horizon, if human eyes were extremely sensitive to light at millimeter wavelengths.
CMB Planetarium
As we will see, the consistency between the statistical properties of CMB anisotropy and polarization maps, and the predictions of the standard cosmological model, LCDM, are quite extraordinary. They give us some confidence that we know what we are talking about when we describe conditions in the universe when it is just a few hundred thousand years old and even younger.
Other Relics of the Big Bang: the Cosmic Neutrino Background and Dark Matter
The hot and dense conditions of the big bang not only led to thermal production of a background of photons (the CMB), but also a background of neutrinos, and possibly dark matter as well. According to our standard cosmological model, most of the mass/energy density of the universe was at one time (just prior to the epoch of Big Bang Nucleosythesis) in the Cosmic Neutrino Background (CNB). This background has yet to be directly detected, but has been detected via its gravitational influence on the photons.
To explain a long list of different observed phenomena, our standard cosmological model also includes a significant amount of non-relativistic matter that does not interact with atoms, nuclei, and light, that we call "dark matter." The amount of dark matter in a large representative sample of the universe appears to be about six times the density of matter made of out of atomic nuclei and electrons. To date we only know of this dark matter via its gravitational influence; it has not yet been directly detected. Dark matter might also be a thermal relic of the Big Bang.
Conclusion
The agreement between cosmological predictions and observations give us a high degree of confidence that our models are capturing some important and true things about the nature of the cosmos. However, we still don’t know what most of the universe is made of, or how it came into existence. There is work left to do!
Our measurements are constantly improving, and one never knows when a combination of precise measurements and predictions will reveal something new and interesting about the universe, or when a theoretical insight will resolve some long-standing puzzles and shine a light for us to follow towards a more perfect understanding.
The adventure continues. Maybe you will join us in our quest!


19.2: Boolean Algebras
Let us investigate the example of the power set, \({\mathcal P}(X)\text{,}\) of a set \(X\) more closely. The power set is a lattice that is ordered by inclusion. By the definition of the power set, the largest element in \({\mathcal P}(X)\) is \(X\) itself and the smallest element is \(\emptyset\text{,}\) the empty set. For any set \(A\) in \({\mathcal P}(X)\text{,}\) we know that \(A \cap X = A\) and \(A \cup \emptyset = A\text{.}\) This suggests the following definition for lattices. An element \(I\) in a poset \(X\) is a largest element if \(a \preceq I\) for all \(a \in X\text{.}\) An element \(O\) is a smallest element of \(X\) if \(O \preceq a\) for all \(a \in X\text{.}\)
Let \(A\) be in \({\mathcal P}(X)\text{.}\) Recall that the complement of \(A\) is
We know that \(A \cup A' = X\) and \(A \cap A' = \emptyset\text{.}\) We can generalize this example for lattices. A lattice \(L\) with a largest element \(I\) and a smallest element \(O\) is complemented if for each \(a \in L\text{,}\) there exists an \(a'\) such that \(a \vee a' = I\) and \(a \wedge a' = O\text{.}\)
In a lattice \(L\text{,}\) the binary operations \(\vee\) and \(\wedge\) satisfy commutative and associative laws; however, they need not satisfy the distributive law
however, in \({\mathcal P}(X)\) the distributive law is satisfied since
for \(A, B, C \in {\mathcal P}(X)\text{.}\) We will say that a lattice \(L\) is distributive if the following distributive law holds:
for all \(a, b, c \in L\text{.}\)
Theorem \(19.15\)
A lattice \(L\) is distributive if and only if
for all \(a, b, c \in L\text{.}\)
- Proof
-
Let us assume that \(L\) is a distributive lattice.
\begin{align*} a \vee ( b \wedge c ) & = [a \vee (a \wedge c) ] \vee ( b \wedge c )\\ & = a \vee [(a \wedge c) \vee ( b \wedge c )]\\ & = a \vee [(c \wedge a) \vee ( c \wedge b )]\\ & = a \vee [c \wedge ( a \vee b )]\\ & = a \vee [( a \vee b ) \wedge c ]\\ & = [( a \vee b ) \wedge a ] \vee [(a \vee b) \wedge c ]\\ & = ( a \vee b ) \wedge ( a \vee c )\text{.} \end{align*}The converse follows directly from the Duality Principle.
A Boolean algebra is a lattice \(B\) with a greatest element \(I\) and a smallest element \(O\) such that \(B\) is both distributive and complemented. The power set of \(X\text{,}\) \({\mathcal P}(X)\text{,}\) is our prototype for a Boolean algebra. As it turns out, it is also one of the most important Boolean algebras. The following theorem allows us to characterize Boolean algebras in terms of the binary relations \(\vee\) and \(\wedge\) without mention of the fact that a Boolean algebra is a poset.
Theorem \(19.16\)
A set \(B\) is a Boolean algebra if and only if there exist binary operations \(\vee\) and \(\wedge\) on \(B\) satisfying the following axioms.
- \(a \vee b = b \vee a\) and \(a \wedge b = b \wedge a\) for \(a, b \in B\text{.}\)
-
\(a \vee ( b \vee c) = (a \vee b) \vee c\) and \(a \wedge ( b \wedge c) = (a \wedge b) \wedge c\) for \(a, b, c \in B\text{.}\)
-
\(a \wedge ( b \vee c ) = (a \wedge b ) \vee ( a \wedge c )\) and \(a \vee ( b \wedge c ) = (a \vee b ) \wedge ( a \vee c )\) for \(a, b, c \in B\text{.}\)
- There exist elements \(I\) and \(O\) such that \(a \vee O = a\) and \(a \wedge I = a\) for all \(a \in B\text{.}\)
- For every \(a \in B\) there exists an \(a' \in B\) such that \(a \vee a' = I\) and \(a \wedge a' = O\text{.}\)
- Proof
-
Let \(B\) be a set satisfying (1)–(5) in the theorem. One of the idempotent laws is satisfied since
\begin{align*} a & = a \vee O\\ & = a \vee (a \wedge a')\\ & = (a \vee a) \wedge (a \vee a')\\ & = (a \vee a ) \wedge I\\ & = a \vee a\text{.} \end{align*}Observe that
\[ I \vee b = (b \vee b' ) \vee b = (b' \vee b ) \vee b = b' \vee (b \vee b) = b' \vee b = I\text{.} \nonumber \]Consequently, the first of the two absorption laws holds, since
\begin{align*} a \vee (a \wedge b) & = (a \wedge I) \vee (a \wedge b)\\ & = a \wedge (I \vee b)\\ & = a \wedge I\\ & = a\text{.} \end{align*}The other idempotent and absorption laws are proven similarly. Since \(B\) also satisfies (1)–(3), the conditions of Theorem \(19.14\) are met; therefore, \(B\) must be a lattice. Condition (4) tells us that \(B\) is a distributive lattice.
For \(a \in B\text{,}\) \(O \vee a = a\text{;}\) hence, \(O \preceq a\) and \(O\) is the smallest element in \(B\text{.}\) To show that \(I\) is the largest element in \(B\text{,}\) we will first show that \(a \vee b = b\) is equivalent to \(a \wedge b = a\text{.}\) Since \(a \vee I = a\) for all \(a \in B\text{,}\) using the absorption laws we can determine that
\[ a \vee I =(a \wedge I) \vee I = I \vee ( I \wedge a) = I \nonumber \]or \(a \preceq I\) for all \(a\) in \(B\text{.}\) Finally, since we know that \(B\) is complemented by (5), \(B\) must be a Boolean algebra.
Conversely, suppose that \(B\) is a Boolean algebra. Let \(I\) and \(O\) be the greatest and least elements in \(B\text{,}\) respectively. If we define \(a \vee b\) and \(a \wedge b\) as least upper and greatest lower bounds of \(\{ a, b\}\text{,}\) then \(B\) is a Boolean algebra by Theorem \(19.14\), Theorem \(19.15\), and our hypothesis.
Many other identities hold in Boolean algebras. Some of these identities are listed in the following theorem.
Theorem \(19.17\)
Let \(B\) be a Boolean algebra. Then
- \(a \vee I = I\) and \(a \wedge O = O\) for all \(a \in B\text{.}\)
-
If \(a \vee b = a \vee c\) and \(a \wedge b = a \wedge c\) for \(a, b, c \in B\text{,}\) then \(b = c\text{.}\)
- If \(a \vee b = I\) and \(a \wedge b = O\text{,}\) then \(b = a'\text{.}\)
- \((a')'=a\) for all \(a \in B\text{.}\)
- \(I' = O\) and \(O' = I\text{.}\)
- \((a \vee b)' = a' \wedge b'\) and \((a \wedge b)' = a' \vee b'\) (De Morgan's Laws).
- Proof
-
We will prove only (2). The rest of the identities are left as exercises. For \(a \vee b = a \vee c\) and \(a \wedge b = a \wedge c\text{,}\) we have
\begin{align*} b & = b \vee (b \wedge a)\\ & = b \vee (a \wedge b)\\ & = b \vee (a \wedge c)\\ & = ( b \vee a) \wedge ( b \vee c)\\ & = ( a \vee b) \wedge ( b \vee c)\\ & = ( a \vee c) \wedge ( b \vee c)\\ & = ( c \vee a ) \wedge ( c\vee b )\\ & = c \vee (a \wedge b)\\ & = c \vee ( a \wedge c )\\ & = c \vee ( c \wedge a )\\ & = c\text{.} \end{align*}
Finite Boolean Algebras
A Boolean algebra is a finite Boolean algebra if it contains a finite number of elements as a set. Finite Boolean algebras are particularly nice since we can classify them up to isomorphism.
Let \(B\) and \(C\) be Boolean algebras. A bijective map \(\phi : B \rightarrow C\) is an isomorphism of Boolean algebras if
for all \(a\) and \(b\) in \(B\text{.}\)
We will show that any finite Boolean algebra is isomorphic to the Boolean algebra obtained by taking the power set of some finite set \(X\text{.}\) We will need a few lemmas and definitions before we prove this result. Let \(B\) be a finite Boolean algebra. An element \(a \in B\) is an atom of \(B\) if \(a \neq O\) and \(a \wedge b = a\) for all \(b \in B\) with \(b \neq O\text{.}\) Equivalently, \(a\) is an atom of \(B\) if there is no \(b \in B\) with \(b \neq O\) distinct from \(a\) such that \(O \preceq b \preceq a\text{.}\)
Lemma \(19.18\)
Let \(B\) be a finite Boolean algebra. If \(b\) is a element of \(B\) with \(b \neq O\text{,}\) then there is an atom \(a\) in \(B\) such that \(a \preceq b\text{.}\)
- Proof
-
If \(b\) is an atom, let \(a =b\text{.}\) Otherwise, choose an element \(b_1\text{,}\) not equal to \(O\) or \(b\text{,}\) such that \(b_1 \preceq b\text{.}\) We are guaranteed that this is possible since \(b\) is not an atom. If \(b_1\) is an atom, then we are done. If not, choose \(b_2\text{,}\) not equal to \(O\) or \(b_1\text{,}\) such that \(b_2 \preceq b_1\text{.}\) Again, if \(b_2\) is an atom, let \(a = b_2\text{.}\) Continuing this process, we can obtain a chain
\[ O \preceq \cdots \preceq b_3 \preceq b_2 \preceq b_1 \preceq b\text{.} \nonumber \]Since \(B\) is a finite Boolean algebra, this chain must be finite. That is, for some \(k\text{,}\) \(b_k\) is an atom. Let \(a = b_k\text{.}\)
Lemma \(19.19\)
Let \(a\) and \(b\) be atoms in a finite Boolean algebra \(B\) such that \(a \neq b\text{.}\) Then \(a \wedge b = O\text{.}\)
- Proof
-
Since \(a \wedge b\) is the greatest lower bound of \(a\) and \(b\text{,}\) we know that \(a \wedge b \preceq a\text{.}\) Hence, either \(a \wedge b = a\) or \(a \wedge b = O\text{.}\) However, if \(a \wedge b = a\text{,}\) then either \(a \preceq b\) or \(a = O\text{.}\) In either case we have a contradiction because \(a\) and \(b\) are both atoms; therefore, \(a \wedge b = O\text{.}\)
Lemma \(19.20\)
Let \(B\) be a Boolean algebra and \(a, b \in B\text{.}\) The following statements are equivalent.
- \(a \preceq b\text{.}\)
- \(a \wedge b' = O\text{.}\)
- \(a' \vee b = I\text{.}\)
- Proof
-
(1) \(\Rightarrow\) (2). If \(a \preceq b\text{,}\) then \(a \vee b = b\text{.}\) Therefore,
\begin{align*} a \wedge b' & = a \wedge (a \vee b)'\\ & = a \wedge ( a' \wedge b')\\ & = ( a \wedge a') \wedge b'\\ & = O \wedge b'\\ & = O\text{.} \end{align*}(2) \(\Rightarrow\) (3). If \(a \wedge b' = O\text{,}\) then \(a' \vee b = (a \wedge b')' = O' = I\text{.}\)
(3) \(\Rightarrow\) (1). If \(a' \vee b = I\text{,}\) then
\begin{align*} a & = a \wedge (a' \vee b)\\ & = (a \wedge a') \vee (a \wedge b)\\ & = O \vee (a \wedge b)\\ & = a \wedge b\text{.} \end{align*}Thus, \(a \preceq b\text{.}\)
Lemma \(19.21\)
Let \(B\) be a Boolean algebra and \(b\) and \(c\) be elements in \(B\) such that \(b \not\preceq c\text{.}\) Then there exists an atom \(a \in B\) such that \(a \preceq b\) and \(a \not\preceq c\text{.}\)
- Proof
-
By Lemma \(19.20\), \(b \wedge c' \neq O\text{.}\) Hence, there exists an atom \(a\) such that \(a \preceq b \wedge c'\text{.}\) Consequently, \(a \preceq b\) and \(a \not\preceq c\text{.}\)
Lemma \(19.22\)
Let \(b \in B\) and \(a_1, \ldots, a_n\) be the atoms of \(B\) such that \(a_i \preceq b\text{.}\) Then \(b = a_1 \vee \cdots \vee a_n\text{.}\) Furthermore, if \(a, a_1, \ldots, a_n\) are atoms of \(B\) such that \(a \preceq b\text{,}\) \(a_i \preceq b\text{,}\) and \(b = a \vee a_1 \vee \cdots \vee a_n\text{,}\) then \(a = a_i\) for some \(i = 1, \ldots, n\text{.}\)
- Proof
-
Let \(b_1 = a_1 \vee \cdots \vee a_n\text{.}\) Since \(a_i \preceq b\) for each \(i\text{,}\) we know that \(b_1 \preceq b\text{.}\) If we can show that \(b \preceq b_1\text{,}\) then the lemma is true by antisymmetry. Assume \(b \not\preceq b_1\text{.}\) Then there exists an atom \(a\) such that \(a \preceq b\) and \(a \not\preceq b_1\text{.}\) Since \(a\) is an atom and \(a \preceq b\text{,}\) we can deduce that \(a = a_i\) for some \(a_i\text{.}\) However, this is impossible since \(a \preceq b_1\text{.}\) Therefore, \(b \preceq b_1\text{.}\)
Now suppose that \(b = a_1 \vee \cdots \vee a_n\text{.}\) If \(a\) is an atom less than \(b\text{,}\)
\[ a = a \wedge b = a \wedge( a_1 \vee \cdots \vee a_n ) = (a \wedge a_1) \vee \cdots \vee ( a \wedge a_n )\text{.} \nonumber \]But each term is \(O\) or \(a\) with \(a \wedge a_i\) occurring for only one \(a_i\text{.}\) Hence, by Lemma \(19.19\), \(a = a_i\) for some \(i\text{.}\)
Theorem \(19.23\)
Let \(B\) be a finite Boolean algebra. Then there exists a set \(X\) such that \(B\) is isomorphic to \({\mathcal P}(X)\text{.}\)
- Proof
-
We will show that \(B\) is isomorphic to \({\mathcal P}(X)\text{,}\) where \(X\) is the set of atoms of \(B\text{.}\) Let \(a \in B\text{.}\) By Lemma \(19.22\), we can write \(a\) uniquely as \(a = a_1 \vee \cdots \vee a_n\) for \(a_1, \ldots, a_n \in X\text{.}\) Consequently, we can define a map \(\phi : B \rightarrow {\mathcal P}(X)\) by
\[ \phi(a) = \phi( a_1 \vee \cdots \vee a_n ) = \{a_1, \ldots, a_n \}\text{.} \nonumber \]Clearly, \(\phi\) is onto.
Now let \(a = a_1 \vee \cdots \vee a_n\) and \(b = b_1 \vee \cdots \vee b_m\) be elements in \(B\text{,}\) where each \(a_i\) and each \(b_i\) is an atom. If \(\phi(a) = \phi(b)\text{,}\) then \(\{a_1, \ldots, a_n \} = \{b_1, \ldots, b_m \}\) and \(a = b\text{.}\) Consequently, \(\phi\) is injective.
The join of \(a\) and \(b\) is preserved by \(\phi\) since
\begin{align*} \phi(a \vee b) & = \phi( a_1 \vee \cdots \vee a_n \vee b_1 \vee \cdots \vee b_m )\\ & = \{ a_1, \ldots, a_n, b_1, \ldots, b_m \}\\ & = \{ a_1, \ldots, a_n \} \cup \{ b_1, \ldots, b_m \}\\ & = \phi( a_1 \vee \cdots \vee a_n ) \cup \phi( b_1 \wedge \cdots \vee b_m )\\ & = \phi(a) \cup \phi(b)\text{.} \end{align*}Similarly, \(\phi( a \wedge b ) = \phi(a) \cap \phi(b)\text{.}\)
We leave the proof of the following corollary as an exercise.
Corollary \(19.24\)
The order of any finite Boolean algebra must be \(2^n\) for some positive integer \(n\text{.}\)