3.1: Traction
Daniel M. Queiroz
Department of Agricultural Engineering
Universidade Federal de Vicosa
Viçosa, Minas Gerais, Brazil
John K. Schueller
Department of Mechanical and Aerospace Engineering
University of Florida
Gainesville, Florida, USA
| Key Terms |
| Mechanics of traction | Traction devices | Tractors |
| Engine power | Transport devices | Pulled implements |
| Tractive force |
Variables
Introduction
Tractors were created to reduce human and animal labor inputs and increase efficiency and productivity in crop production activities (Schueller, 2000). The main use of tractors is to pull implements such as tillage tools, planters, cultivators, and harvesters in the field and, to some extent, on the road (Renius, 2020). To pull implements efficiently, a tractor needs to generate traction between the tires and the soil surface. Traction is the way a vehicle uses force to move over a surface.
Quite early in tractor development, the direct transfer of power from tractors to implements was made possible by using power take-offs (PTOs) that transfer rotary power to implements and machines and by using hydraulic systems to lift and lower implements and to move parts of attached machines. Pulling implements is still the most common use of tractor power. The field capacity of agricultural machines, i.e., the field area that can be covered per unit time, has caused bigger implements to be developed and used. The increased sizes require greater traction from the pulling tractor. More efficient systems to create the tractive force are necessary to provide the large forces necessary to pull those implements.
The efficiency of how tractors convert the power generated by an engine to the power required to pull the implements depends on many variables associated with the tractor and the soil conditions. Traction is especially important in agriculture as field soils are not as firm as the roads used by cars and trucks. This chapter presents the basic principles of traction applied to agricultural machinery.
Outcomes
After reading this chapter, you should be able to:
- • Explain how tractors develop tractive force
- • Describe the effect of some important variables on the tractive force
- • Calculate how much power a tractor can develop when pulling an implement
- • Calculate the power requirements to match tractors to implements
Concepts
Traction and Transport Devices
According to the American Society of Agricultural and Biological Engineers (ASABE Standards, 2018), there are two types of surface contact devices associated with the motion of a vehicle: traction devices and transport devices. A traction device receives power from an engine and uses the reactions of forces from the supporting surface to propel the vehicle, while a transport device does not receive power, but is needed to support the vehicle on a surface while the vehicle is moving over that surface. Wheels, tires, and tracks can be traction devices if they are connected to an engine or other power source; if not connected, they are transport devices. The main components of an agricultural tractor are presented in Figure 3.1.1. In this example, the tractor is 2-wheel drive, so the large rear wheels, which receive power from the engine, are the traction devices, and the small front wheels are the transport devices. All wheels would be traction devices if the tractor were 4-wheel drive. The engine is connected to the traction device by the drive train, often consisting of a clutch, transmission, differential, axles, and other components. (The drive train is not discussed in this chapter.) The drawbar is an attachment point through which the tractor can apply pulling force to an implement.
Mechanics of Traction
The simplest way of analyzing the traction produced by a traction device, such as a wheel or track, is to consider friction forces that act at the contact between a traction device and the surface when the system is in equilibrium. For simplification the machine is assumed to be moving at a constant velocity on a non-variable surface (Figure 3.1.2). A traction device (hereafter simplified to the most common implementation as a “wheel”) has two main functions: to support the load acting on the wheel axle ( W ) and to produce a net tractive force ( H ). The force W is generally called the dynamic load acting on the wheel. The dynamic load depends on how the weight of the tractor at that point in time is distributed to each wheel. If the system is in equilibrium, the surface reacts to W by applying a vertical reaction force ( R ) to the wheel. In the contact between the surface and the wheel, a friction force ( F f ) is generated. To keep equilibrium in the horizontal direction, the magnitude of the net tractive force H is equal to the magnitude of the friction force F f . To produce a net tractive force H , the friction force needs to be overcome. This is done by applying a torque ( T ) to the wheel axle. This torque is proportional to the torque produced by the tractor engine according to the drive train, including the current transmission ratio.
When moving, the wheel (Figure 3.1.2) rotates with a constant angular velocity ( ω ), and this angular speed is proportional to the engine rotation speed, depending on the gearing ratio in the drive train. The wheel has an actual velocity v a , which is equal to the angular velocity multiplied by the wheel’s rolling radius reduced by the slip (as discussed below). In an equilibrium situation, ω and v a are constants. The power transferred to the wheel axle ( P w ) can be calculated as the product of the torque ( T ) and the angular velocity ( ω ), as shown in Equation 3.1.1. The tractive power developed by the wheel ( P t ) is the product of the net tractive force ( H ) and the actual velocity ( v a ), as shown in Equation 3.1.2. The tractive efficiency of the wheel ( T E ) can be calculated as the ratio between tractive power and the wheel axle power, as shown in Equation 3.1.3.
\[ P_{W}=T\omega \]
\[ P_{t}=H\nu_{a} \]
\[ T_{E}=\frac{P_{t}}{P_{W}} \]
where P w = power transferred to the wheel axle (W)
T = torque transferred to the wheel axle (N m)
ω = angular velocity of the wheel (rad s −1 )
P t = tractive power developed by the wheel (W)
H = net tractive force (N)
v a = actual velocity of the wheel (m s −1 )
T E = tractive efficiency of the wheel (dimensionless)
The friction force ( F f in Figure 3.1.2) is generated by the interaction between the wheel and the surface. The friction force can be calculated by multiplying the reaction force ( R ) by the equivalent friction coefficient ( μ ). Table 3.1.1 presents some typical values. Because R is equal to the dynamic load acting on the wheel axle ( W ) and the net tractive force is equal to the friction force, the tractive force can be calculated as the product of the equivalent coefficient of friction and the dynamic load, as:
| Surface type | Equivalent coefficient of friction ( μ )[a] |
|---|---|
|
Soft soil |
0.26–0.31 |
|
Medium soil |
0.40–0.46 |
|
Firm soil |
0.43–0.53 |
|
Concrete |
0.91–0.98 |
[a] These values were estimated based on data presented by Kolator and Bialobrzewski (2011).
\[ H= \mu W \]
where μ = coefficient of friction (dimensionless).
The theoretical velocity ( v t ) is determined by the wheel’s rotational velocity ( ω ) times the rolling radius ( r ) as shown in Equation 3.1.5, but the actual wheel velocity ( v a ) is less due to the relative motion at the interface between the wheel and the surface. This relative motion is the travel reduction ratio, commonly called slip, and is defined as the ratio of the loss of wheel velocity to the theoretical velocity, that is, the velocity that wheel would have if there was no loss. Equation 3.1.6 shows how the travel reduction ratio can be estimated:
\[ \nu_{t} = \omega r \]
\[ s= \frac{\nu_{t}-\nu_{a}}{\nu_{t}} \]
where v t = theoretical velocity of the wheel (m s −1 )
r = wheel rolling radius (m s −1 )
s = travel reduction ratio, or slip (dimensionless)
The travel reduction ratio is an important variable for wheel tractive force analysis. The travel reduction ratio of a wheel can vary from 0 to 1 depending on wheel and surface conditions. When the travel reduction ratio is equal to 0, there would be no relative motion between the periphery of the wheel and the surface. The wheel rotation causes a perfect translational motion relative to the surface. However, experience has shown that for a wheel to develop a tractive force, there must be relative motion (slip) between the wheel and the surface. Therefore, a wheel generating tractive force needs to have a travel reduction ratio greater than zero. When a wheel generates more tractive force, the travel reduction ratio increases, and the actual wheel velocity reduces. When the travel reduction ratio is equal to 1, the wheel does not move forward when it rotates. The models used to calculate the tractive force generally use the travel reduction ratio as one of the variables.
Another important concept when analyzing the traction process of a moving wheel is the motion resistance force ( F r ) (Figure 3.1.3). If a wheel is moving, the wheel and the surface deform. Energy is spent to produce this deformation. The resistance produced by the wheel and surface deformations must be overcome to allow the wheel to move. Considering the existence of the motion resistance force, in the contact between the wheel and the surface, it is necessary to generate a friction force greater than the motion resistance force at the wheel-surface contact to produce a tractive force. This friction force is now termed the gross tractive force (denoted by F ). Thus, the gross tractive force would be the net tractive force generated by the wheel if there was no motion resistance. Adding the concepts of motion resistance and gross tractive forces to Figure 3.1.2 results in Figure 3.1.3, which is an improved representation of forces acting on a wheel.
If the wheel represented in Figure 3.1.3 has no motion in the vertical direction (z axis), the wheel is in static equilibrium in this direction. In this condition, the summation of forces in the z (vertical) direction is zero. Therefore,
\[ \sum F_{z} = 0 \]
\[ R-W=0 \]
\[ R=W \]
where F z = any force applied to the wheel in z direction (N)
R = vertical reaction force of the wheel (N)
If the actual speed of the wheel represented in Figure 3.1.3 is constant, the horizontal forces are in static equilibrium in this direction and the sum of the horizontal forces is zero. Therefore,
\[ \sum F_{x}=0 \]
where F x = any force applied to the wheel in x direction (N)
F = gross tractive force (N)
F r = motion resistance force (N)
Based on Equation 3.1.12, the gross tractive force ( F ) must be the net tractive force ( H ) plus the motion resistance force ( F r ). If both sides of Equation 3.1.12 are divided by the dynamic load ( W ) acting on the wheel, resulting in Equation 3.1.13, three dimensionless numbers, i.e., μ n , μ g , and ρ , are created as shown in Equations 3.1.15, 3.1.16, and 3.1.17. The first one is the net traction ratio (μ n ), defined as the net tractive force divided by the dynamic load. The second one is the gross traction ratio ( μ g ), defined as the gross tractive force divided by the dynamic load. And the third one is the motion resistance ratio ( μ ), defined as the motion resistance force divided by the dynamic load.
\[ \frac{H}{W}=\frac{F}{W}-\frac{F_{r}}{W} \]
Equation 3.1.14 shows that μ n , μ g , and ρ are not independent. By using a technique called dimensional analysis, functions were developed to predict how μ g and ρ change as a function of the wheel variables and soil resistance. This analysis is presented by Goering et al. (2003) and is beyond the scope of this chapter. If μ g , ρ , and W are known, the tractive force generated by the wheel can be predicted using Equation 3.1.18:
\[ H= (\mu_{g}-\rho)W \]
The R , F , and F r forces (Figure 3.1.3) act on a point called the wheel resistance center. This point is not aligned with the direction of the dynamic load W but is a little bit ahead of it. This horizontal distance is called the horizontal offset ( e ). The static analysis of a towed wheel (Figure 3.1.4) shows that the wheel resistance center is not aligned with the direction of the dynamic wheel load. In a towed wheel, there is no torque applied to its axle. The soil reaction ( G ) at the resistance center is the resultant of the R and F r forces. The direction of the G force passes through the wheel center. To move the towed wheel at a constant actual velocity ( v a ), a net tractive force ( H ) equal to the motion resistance force ( F r ) needs to be applied to the wheel. For the wheel to keep an angular velocity constant, the sum of the momentums at the center of the wheels must equal zero. Goering et al. (2003) showed that the horizontal offset can be calculated with Equations 3.1.19-3.1.21.
\[ Re-F_{r}r = 0 \]
where e = horizontal offset (m).
By using Equation 3.1.18, the tractive force can be predicted. The other important information in the wheel traction analysis is to predict how much torque needs to be transferred to the wheel axle to generate the tractive force ( H ). In Equation 3.1.5, the wheel radius is used to convert the rotational angular velocity to the theoretical wheel velocity. The wheel radius can also be used to calculate the torque necessary to produce the wheel tractive force. The torque ( T ) necessary to keep the angular velocity of the wheel constant and produce the net tractive force is the product of the gross tractive force and the torque radius of the wheel, as given by:
where r t = torque radius of the wheel (m).
The wheel radius defined in Equation 3.1.5 is different from the torque radius of the wheel defined in Equation 3.1.22 because of the interaction of the wheel and the surface, which varies on a soft soil surface. Generally, a rolling radius based on the distance from the center of the wheel axle to a hard surface is used. Therefore, Equation 3.1.23 can be used to estimate the torque acting at wheel axle:
\[ T=Fr \]
Engine Power Needed to Produce a Tractive Force
ASABE Standards (2015) presented a diagram (Figure 3.1.5) of the approximate typical power relationship for agricultural tractors. Tractors can be specified by their engine gross flywheel rated power ( P e ). One of the standards used to define the engine gross flywheel rated power is SAE J1995 (SAE, 1995). The rated power defined by this standard is the mechanical power produced by the engine without some of its accessories (such as the alternator, the radiator fan, and the water pump). Therefore, the engine gross flywheel rated power is greater than the net power produced by the engine. The approximate engine net flywheel power can be estimated by multiplying the gross flywheel power by 0.92. The power at the tractor PTO is about equal to the engine gross flywheel power multiplied by 0.83 or the engine net flywheel power multiplied by 0.90.
The power that the tractor can generate to pull implements, often termed drawbar power because many implements are attached to the tractor’s drawbar, depends on the tractor type, i.e., 2-wheel drive (2WD), mechanical front wheel drive (MFWD), 4-wheel drive (4WD), or tracked. The surface condition where the tractor is used has an even greater effect. Using these two pieces of information, coefficients that show estimates of the relationship between the drawbar power and the PTO power is given in Figure 3.1.5.
The drawbar power required to pull an implement is:
\[ P_{DB} = F_{i} \nu_{i} \]
where P DB = drawbar power (W)
F i = force required to pull an implement (N)
v i = implement velocity (m s −1 )
The force required to pull an implement depends on the implement. For example, the force required to pull a planter F p is the force required per row times the number of rows:
\[ F_{p} = f_{r}n_{r} \]
where f r = force required per row of planter (N row −1 )
n r = number of rows
Once the required drawbar power is determined, the values in Figure 3.1.5 can be used to calculate the estimated needed gross flywheel rated power of a tractor to pull the implement.
Applications
The concepts of traction and tractor power are necessary for properly matching the tractor to an implement. Agricultural operations cannot be performed if the tractor cannot develop enough power or traction to pull the implement. As implements have increased in size over the years, it is necessary that the tractors have enough power and enough traction for the tasks they have to perform. Choosing a tractor that is too large will negatively impact agricultural profitability because larger tractors cost more than smaller tractors. An oversize tractor may also increase fuel consumption and exhaust emissions. This is significant because even the most efficient tractors get less than 4 kWh of work per liter of diesel fuel.
Tractors range greatly in size (e.g., Figure 3.1.6). For example, one large contemporary manufacturer sells tractors from 17 to 477 kW. The weight of the tractor must be enough to generate sufficient traction force, as shown in Equation 3.1.18. However, besides the cost of adding weight, additional weight may increase soil compaction and depress crop yields. It is therefore necessary to understand these concepts to design tractors and implements. The capabilities of the tractor’s engine, power transmission elements, and wheels need to be appropriately scaled. There needs to be a trade-off between making them large and powerful with making them compact and inexpensive. The above analyses can be used to guide tractor choice and design.
The concepts are also applied to other types of agricultural machinery, such as self-propelled harvesters and sprayers. For these machines to be able to complete their tasks, they need to be able to move across agricultural soils. The same calculations can be used to determine if there is enough power and to design the various components on those machines. The wheels, axles, and power transmission components must be able to withstand the forces, torques, and power during the machines’ use.
Examples
Example \(\PageIndex{1}\)
Example 1: Tractive force
Problem:
Calculate the tractive force produced by a tractor wheel that works on a firm soil with a dynamic load of 5 kN. The wheel velocity is 2 m s −1 . If the tractive efficiency is 0.73, what is the power that needs to be transferred to the wheel axle?
Solution
Assume an equivalent coefficient of friction of 0.48, the mean value for firm soil presented in Table 3.1.1. Calculate the tractive force using Equation 3.1.4:
\( H= \mu W=0.48 \times 5 = 2.4 \text{ kN} \)
Now, calculate the tractive power for the tractor wheel using Equation 3.1.2:
\( P_{t} = H\nu_{a} = 2.4 \times 2 = 4.8 \text{ kW} \)
Calculate the power that needs to be transferred to the wheel axle for using Equation 3.1.3 with the given tractive efficiency of 0.73:
\( P_{W} = \frac{P_{t}}{T_{E}} = \frac{4.8}{0.73} = 6.58 \text{ kW} \)
This value of needed power can be used to design the various power transmission components. The power consumption can also be used to calculate the power demanded of the ultimate power source, probably an engine, to calculate fuel consumption and, thereby, costs of a particular field operation.
Example \(\PageIndex{2}\)
Example 2: Torque and travel reduction ratio, or slip
Problem:
A wheel on another tractor receives 40 kW from the tractor powertrain. The wheel rotates at 25 rpm, which is an angular velocity, ω, of 2.62 rad s −1 . (Note: 2π rad × 25 rpm/60 min s −1 = 2.62 rad s −1 .) If the rolling radius of the wheels is 0.81 m and the speed of the tractor is 2 m s −1 , calculate the torque acting on the wheel and the travel reduction ratio (commonly known as slip).
Solution
Calculate the torque acting on the wheel T for a power P w of 40 kW using Equation 3.1.1:
\( T=\frac{P_{W}}{\omega} = \frac{40}{2.62} = 15.28 \text{ Nm} \)
Calculate the power to be transferred to the wheel for producing 2.4 kN of tractive force at 2 m s −1 of wheel speed using Equation 3.1.3:
\( P_{W} = \frac{P_{t}}{T_{E}} = \frac{4.8}{0.73} =6.58 \text{ kW} \)
Calculate the theoretical velocity of the wheel v t for a rolling radius r of 0.81 m using Equation 3.1.5:
\( \nu_{t} = \omega r = 2.62 \times 0.81 = 2.12 \text{ m}s^{-1} \)
Since the actual velocity of the wheel is 2 m s −1 , which is less than the theoretical velocity of the wheel, calculate the travel reduction ratio s using Equation 3.1.6:
\( s= \frac{\nu_{t}-\nu_{a}}{\nu_{t}} = \frac{2.12-2.00}{2.12} = 0.057, \text{or} \ 5.7\% \)
In addition to providing guidance to the design of the agricultural machine and its power consumption, calculation of the slip is useful to determine how fast the operation will be performed. Excessive slip can also have adverse effects on the soil’s structure and inhibit plant growth.
Example \(\PageIndex{3}\)
Example 3: Tractive force and power
Problem:
Consider a wheel that works with a dynamic load of 10 kN, a motion resistance ratio of 0.08, and a gross traction ratio of 0.72. Find the tractive force that the wheel can develop. If this wheel rotates at 40 rpm and the rolling radius of the wheel is 0.71 m, how much power is necessary to move this wheel?
Solution
Calculate the gross tractive force developed by the wheel F using Equation 3.1.16:
\( F= \mu_{g} W = 0.72 \times 10 = 7.2 \text{ kN} \)
Calculate the motion resistance F r of this wheel using Equation 3.1.17:
\( F_{r} = \rho W = 0.08 \times 10 = 0.80 \text{ kN} \)
The tractive force H developed by the wheel, according to Equation 3.1.12, is the difference between the gross tractive force and the motion resistance:
\( H = F-F_{r} = 7.2 - 0.8 = 6.4 \text{ kN} \)
Calculate the torque necessary to move this wheel using Equation 3.1.23:
\( T = Fr = 7.2 \times 0.71 = 5.11 \text{ kN m} \)
Calculate the power P w necessary to turn the wheel using Equation 3.1.1:
\( P_{W} = T \omega = T \frac{2\pi N}{60} = 5.11 \times \frac{2 \times \pi \times 40}{60} = 21.4 \text{ kW} \)
Example \(\PageIndex{4}\)
Example 4: Engine gross flywheel power
Problem:
Calculate the necessary power of a MFWD tractor to pull a 30-row planter. According to ASABE Standards (2015), a force of 900 N per row is required to pull a drawn row crop planter if it is only performing the seeding operation. The speed of the tractor will be 8.1 km h −1 (2.25 m s −1 ). The soil is in the tilled condition. Consider that the tractor should have a power reserve of 20% to overcome unexpected overloads.
Solution
Calculate the drawbar force needed to pull the planter using Equation 3.1.25:
\( F_{p} = f_{r}n_{r} = 900 \times 30 = 27,000 \text{ N} \)
Calculate the drawbar power P DB needed to pull the planter using Equation 3.1.24:
\( P_{DB} = F_{p} \nu_{p} = 27,000 \times 2.25 = 60,750 \text{ W} \)
Therefore, the tractor needs to produce a drawbar power of 60.75 kW. From Figure 3.1.5, find that the coefficient that relates the drawbar power to the PTO power of the tractor for a MFWD tractor working on tilled soil condition is 0.72. Thus, the tractor PTO power P PTO should be:
\( P_{PTO} = \frac{P_{DB}}{0.72} = \frac{60.75}{0.72} = 84 \text{ kW} \)
Considering that the coefficient that relates the PTO power to the engine gross flywheel power is 0.83 (Figure 3.1.5), the engine gross flywheel power P e is:
\( P_{e} = \frac{P_{PTO}}{0.83} = \frac{84.375}{0.83} = 102 \text{ kW} \)
Considering a reserve of power of 20% to overcome unexpected overloads, the tractor selected should have an engine gross flywheel power at least 20% greater than that needed to pull the 30-row planter, or 1.2 × 102 kW = 122 kW.
These calculations will help the farm manager select the proper tractor for the operation.
Image Credits
Figure 1. Queiroz, D. (CC By 4.0). (2020). Schematic view of a two-wheel drive agricultural tractor.
Figure 2. Queiroz, D. (CC By 4.0). (2020). Simplified diagram of the variables related to a wheel developing a net tractive force.
Figure 3. Queiroz, D. (CC By 4.0). (2020). Diagram of the variables related to a wheel developing a net tractive force ( H ) including the gross tractive force ( F ) and the motion resistance force ( F r ).
Figure 4. Queiroz, D. (CC By 4.0). (2020). Diagram of forces acting on a towed wheel.
Figure 5. ASABE Standard ASAE D497.7 (CC By 4.0). (2020). Diagram of the approximate power relationships in agricultural tractors (types are defined in the main text) and soil conditions.
Figure 6. Schueller, J. (CC By 4.0). (2020). Typical contemporary (a) small and (b) large tractors.
References
ASABE Standards. (2018). ANSI/ASAE S296.5 DEC2003 (R2018): General terminology for traction of agricultural traction and transport devices and vehicles. St. Joseph, MI: ASABE.
ASABE Standards. (2015). ASAE D497.7 MAR2011 (R2015): Agricultural machinery management data. St. Joseph, MI: ASABE.
Goering, C. E., Stone, M. L., Smith, D. W., & Turnquist, P. K. (2003). Traction and transport devices. In Off-road vehicle engineering principles (pp. 351-382). St. Joseph, MI: ASAE.
Kolator, B., & Białobrzewski, I. (2011). A simulation model of 2WD tractor performance. Comput. Electron. Agric. 76 (2): 231-239.
Renius, K. T. (2020). Fundamentals of tractor design. Cham, Switzerland: Springer Nature.
SAE. (1995). SAE J1995_199506: Engine power test code—Spark ignition and compression ignition—Gross power rating. Troy, MI: SAE.
Schueller, J. K. (2000). In the service of abundance: Agricultural mechanization provided the nourishment for the 20th century’s extraordinary growth. Mech. Eng. 122 (8):58-65.


11.3E: Exercises
Practice Makes Perfect
In the following exercises, graph each equation by using properties.
- \(y=-x^{2}+4 x-3\)
- \(y=-x^{2}+8 x-15\)
- \(y=6 x^{2}+2 x-1\)
- \(y=8 x^{2}-10 x+3\)
- Answer
-
1.
3.
In the following exercises,
- Write the equation in standard form and
- Use properties of the standard form to graph the equation.
- \(y=-x^{2}+2 x-4\)
- \(y=2 x^{2}+4 x+6\)
- \(y=-2 x^{2}-4 x-5\)
- \(y=3 x^{2}-12 x+7\)
- Answer
-
1.
- \(y=-(x-1)^{2}-3\)
3.
- \(y=-2(x+1)^{2}-3\)
In the following exercises, graph each equation by using properties.
- \(x=-2 y^{2}\)
- \(x=3 y^{2}\)
- \(x=4 y^{2}\)
- \(x=-4 y^{2}\)
- \(x=-y^{2}-2 y+3\)
- \(x=-y^{2}-4 y+5\)
- \(x=y^{2}+6 y+8\)
- \(x=y^{2}-4 y-12\)
- \(x=(y-2)^{2}+3\)
- \(x=(y-1)^{2}+4\)
- \(x=-(y-1)^{2}+2\)
- \(x=-(y-4)^{2}+3\)
- \(x=(y+2)^{2}+1\)
- \(x=(y+1)^{2}+2\)
- \(x=-(y+3)^{2}+2\)
- \(x=-(y+4)^{2}+3\)
- \(x=-3(y-2)^{2}+3\)
- \(x=-2(y-1)^{2}+2\)
- \(x=4(y+1)^{2}-4\)
- \(x=2(y+4)^{2}-2\)
- Answer
-
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
In the following exercises,
- Write the equation in standard form and
- Use properties of the standard form to graph the equation.
- \(x=y^{2}+4 y-5\)
- \(x=y^{2}+2 y-3\)
- \(x=-2 y^{2}-12 y-16\)
- \(x=-3 y^{2}-6 y-5\)
- Answer
-
1.
- \(x=(y+2)^{2}-9\)
3.
- \(x=-2(y+3)^{2}+2\)
In the following exercises, match each graph to one of the following equations:
- \(x^{2}+y^{2}=64\)
- \(x^{2}+y^{2}=49\)
- \((x+5)^{2}+(y+2)^{2}=4\)
- \((x-2)^{2}+(y-3)^{2}=9\)
- \(y=-x^{2}+8 x-15\)
- \(y=6 x^{2}+2 x-1\)
1.
2.
3.
4.
5.
6.
- Answer
-
1. a
3. b
5. d
Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Use the lower left side of the bridge as the origin \((0, 0)\).
1.
2.
3.
4.
- Answer
-
1. \(y=-\frac{1}{15}(x-15)^{2}+15\)
3. \(y=-\frac{1}{10}(x-30)^{2}+90\)
- In your own words, define a parabola.
- Is the parabola \(y=x^{2}\) a function? Is the parabola \(x=y^{2}\) a function? Explain why or why not.
- Write the equation of a parabola that opens up or down in standard form and the equation of a parabola that opens left or right in standard form. Provide a sketch of the parabola for each one, label the vertex and axis of symmetry.
- Explain in your own words, how you can tell from its equation whether a parabola opens up, down, left or right.
- Answer
-
1. Answers may vary
3. Answers may vary
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. After reviewing this checklist, what will you do to become confident for all objectives?


3.2: Health, Wellness, Sickness, and Disease
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- Elizabeth B. Pearce & Amy Huskey
- Linn-Benton Community College via OpenOregon
“It is health that is real wealth and not pieces of gold and silver.” Mahatma Gandhi
Health is the “state of complete physical, mental and social well-being and not merely the absence of disease or infirmity,” as defined by the World Health Organization in its Constitution. [1] The organization goes on to state that families and communities are able to thrive when individuals are able to maintain health.
While the status of health is very real, it is also important to examine the aspects of health and illness that are socially constructed. Society shares an assumption of reality that creates a definition of both health and illness. Illness has a biological component, yet it also embodies an independent element that is experienced by the person and observed by those outside of the illness. Societies construct the idea of “health” differently from place to place, and over time. For example, many societies consider health and health care to be a human right that all human beings are entitled to, but this is not universally true. What is your viewpoint?
The social construction of the idea of family, as discussed in the first chapter of this text, plays a special role in the health of families in the United States. In all other industrialized countries, health care is considered a human right, and all individuals access comparable care regardless of family status. But in the U.S. health care insurance is accessed through a variety of systems, all of which have age, income, employment status, and family relationships as criteria. How do definitions of “family” impact the access of any individual to health care?
Society informs the definitions for when an illness can be considered a disability, eligibility for insurance and medical coverage, what illnesses are
perceived as legitimate, when the reality of an illness is questioned, and what illnesses are stigmatized. [2] These social constructs can, in themselves, contribute to differentiation in individual health as well as stereotypes, prejudice, and discrimination. Sociologist Erving Goffman said, “Stigma is a process by which the reaction of others spoils normal identity.” A disease or illness that is “stigmatized” is one in which there is some societal disapproval or questioning of the integrity of people who have the disease, which can also include medical professionals and the person with the disease stigmatizing themselves.
A YouTube element has been excluded from this version of the text. You can view it online here: https://openoregon.pressbooks.pub/families/?p=408
The Khan Academy Medicine channel describes the interrelationship among stigma, self, and society in the seven minute video above.
There are many socially stigmatized diseases which can include mental disorders, AIDS, sexually transmitted diseases, substance abuse disorders, skin conditions, and diseases that are less understood by the medical community such as auto-immune disorders and chronic fatigue syndrome. Individuals may be less likely to seek treatment when experiencing symptoms of a stigmatized condition. In the United States, where health insurance creates another layer of bureaucracy, less understood illnesses and treatments are less likely to be covered.Even if health care is accessed, there may be fewer social supports available, and greater chances of isolation and discrimination in the health care setting and beyond. Management of a disability because of the deterioration of health due to a stigmatized illness can prove to be a tremendous hardship. [3] It is important to consider how marginalized groups experience health inequities, including those experienced due to the stigma and prejudices linked with the societal perception of specific illnesses.
In this chapter we will examine the overall health of families in the United States, with attention to comparisons amongst families’ experiences with health, illness, injury and diseases as well as comparisons with other countries. We will describe the daily life factors and resources that affect health, including the effects of stigma, prejudice and discrimination. We will look closely at how families access health care services, including the health care insurance system. We will discuss the question of health care as a human right.
In Focus: Lung Cancer, Stigma, and Intersectionality
In a 2014 report, Addressing the Stigma of Lung Cancer, the American Lung Association (ALA) described how stigma can contribute to patients avoiding or delaying care when symptoms are experienced and how they may resist disclosing health information to providers. Patients could also experience fewer social supports, feel more distress related to the disease, experience additional conflicts in relationships, and receive lower quality of health care. The report then explained how stigma could be related to lung cancer being hard to detect early, and then being seen as a death sentence with little hope for survival. In addition to this, being labelled a smokers disease could contribute to society’s feeling that people with lung cancer are personally responsible for their disease, placing perceived shame and blame on those with the illness.
In the survey conducted by the ALA, public knowledge regarding the prevalence and mortality of the disease was lacking, as well as knowledge about the funding levels for research. Responders also did not have as much knowledge of factors other than smoking that play a part in the development of lung cancer, like genetics, occupational or environmental exposures. Lack of knowledge could play an important role in perpetuating some of the negative associations and assumptions related to lung cancer.
Lung cancer stigma has real consequences. People with lung cancer may hide their condition, and experience anxiety and depression, while also holding a perception that healthcare providers are judgmental and biased towards them and would treat them differently as a result. Research, advocacy, and public policy are also dimensions in which stigma can have negative consequences for advancing progress toward the treatment of this disease. For instance, lung cancer kills far more people than any other cancer type and contributes to over 25% of all cancer deaths per year. [4] Yet, a study of nonprofit research funding by cancer type determined that lung cancer ranked fourth in its level of funding and is underfunded in relation to its burden on society. [5] Not only are funding levels influenced by stigma, but celebrities, community leaders, and volunteers may be less likely to lend their name or volunteer their time due to negative associations with this disease. [6]
This example provides a way of analyzing how intersectionality, when multiple social identities can overlap to create unique forms of discrimination, can be illustrated through the lens of the social construction of illness. The LGBTQ+ community, for instance, already face challenges in finding health care providers who are culturally competent and have adequate knowledge in LGBTQ+ health. [7] If you couple this with the stigma related to a lung cancer diagnosis, then you are compounding the obstacles that stand in the way of receiving quality health care. One could also examine how socioeconomic status may further complicate this scenario by realizing that people living in poverty are more likely to have no health insurance, therefore restricting access even more. [8] Even geographical location, like living in a rural area, or in the Southern portion of the United States, will reduce accessibility to healthcare. [9] Every time an individual is grouped into a marginalized group, including illness status, it intensifies the effects of discrimination and resulting health disparities.
As you finish reading this section, notice if you blame or hold people more responsible knowing that they have lung cancer, than if they have another cancer such as breast or prostate cancer. What effect does the stigma itself have on people who are sick, and the healthcare that they receive?
- World Health Organization. (2005, September 15). Constitution. Retrieved December 30, 2019, from https://www.who.int/about/who-we-are/constitution ↵
- OpenStax. (2015, April 24). Introduction to Sociology 2e. Retrieved March 8, 2020, from https://openstax.org/books/introduct...tion-of-health ↵
- Sartorius, N. (2007). Stigmatized illnesses and health care. Croatian Medical Journal, 48(3), 396–397. ↵
- American Cancer Society. (2020, January 8). Lung cancer statistics: How common is lung cancer. www.cancer.org/cancer/lung-c...statistics.htm ↵
- Kamath, S. D., Kircher, S. M., & Benson, A. B. (2019). Comparison of cancer burden and nonprofit organization funding reveals disparities in funding across cancer types. Journal of the National Comprehensive Cancer Network, 17(7), 849–854. https://doi.org/10.6004/jnccn.2018.7280 ↵
- American Lung Association. (2014, April). Addressing the stigma of lung cancer. Retrieved March 8, 2020, from www.lung.org/our-initiatives...ng-cancer.html ↵
- U.S. Dept. of Health and Human Services. (n.d.) Lesbian, gay, bisexual, and transgender health. Healthy People 2020. Retrieved March 8, 2020, from https://www.healthypeople.gov/2020/t...nder-health#31 ↵
- U.S. Census Bureau. (2019, May 23). Who are the uninsured? Retrieved March 8, 2020, from https://www.census.gov/library/stori...uninsured.html ↵
- Warshaw, R. (2017, October 31). Health Disparities Affect Millions in Rural U.S. Communities. Retrieved March 8, 2020, from https://www.aamc.org/news-insights/h...us-communities ↵


3.3.2: The Number and Counting System of the Inca Civilization
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Background
There is generally a lack of books and research material concerning the historical foundations of the Americas. Most of the “important” information available concentrates on the eastern hemisphere, with Europe as the central focus. The reasons for this may be twofold: first, it is thought that there was a lack of specialized mathematics in the American regions; second, many of the secrets of ancient mathematics in the Americas have been closely guarded.[i] The Peruvian system does not seem to be an exception here. Two researchers, Leland Locke and Erland Nordenskiold, have carried out research that has attempted to discover what mathematical knowledge was known by the Incas and how they used the Peruvian quipu, a counting system using cords and knots, in their mathematics. These researchers have come to certain beliefs about the quipu that we will summarize here.
Counting Boards
It should be noted that the Incas did not have a complicated system of computation. Where other peoples in the regions, such as the Mayans, were doing computations related to their rituals and calendars, the Incas seem to have been more concerned with the simpler task of record-keeping. To do this, they used what are called the “quipu” to record quantities of items. (We will describe them in more detail in a moment.) However, they first often needed to do computations whose results would be recorded on quipu. To do these computations, they would sometimes use a counting board constructed with a slab of stone. In the slab were cut rectangular and square compartments so that an octagonal (eight-sided) region was left in the middle. Two opposite corner rectangles were raised. Another two sections were mounted on the original surface of the slab so that there were actually three levels available. In the figure shown, the darkest shaded corner regions represent the highest, third level. The lighter shaded regions surrounding the corners are the second highest levels, while the clear white rectangles are the compartments cut into the stone slab.
Pebbles were used to keep accounts and their positions within the various levels and compartments gave totals. For example, a pebble in a smaller (white) compartment represented one unit. Note that there are 12 such squares around the outer edge of the figure. If a pebble was put into one of the two (white) larger, rectangular compartments, its value was doubled. When a pebble was put in the octagonal region in the middle of the slab, its value was tripled. If a pebble was placed on the second (shaded) level, its value was multiplied by six. And finally, if a pebble was found on one of the two highest corner levels, its value was multiplied by twelve. Different objects could be counted at the same time by representing different objects by different colored pebbles.
Example 1
Suppose you have the following counting board with two different kind of pebbles places as illustrated.  How many dogs are being represented?
Solution
- There are two black pebbles in the outer square regions…these represent 2 dogs.
- There are three black pebbles in the larger (white) rectangular compartments. These represent 6 dogs.
- There is one black pebble in the middle region…this represents 3 dogs.
- There are three black pebbles on the second level…these represent 18 dogs.
- Finally, there is one black pebble on the highest corner level…this represents 12 dogs.
We then have a total of \(2+6+3+18+12 = 41\) dogs.
Try it Now 1
How many cats are represented on this board?
- Answer
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\(1+6 \times 3+3 \times 6+2 \times 12=61\) cats.
The Quipu
This kind of board was good for doing quick computations, but it did not provide a good way to keep a permanent recording of quantities or computations. For this purpose, they used the quipu. The quipu is a collection of cords with knots in them. These cords and knots are carefully arranged so that the position and type of cord or knot gives specific information on how to decipher the cord.
A quipu is made up of a main cord which has other cords (branches) tied to it. See pictures to the right.[ii]
Locke called the branches H cords. They are attached to the main cord. B cords, in turn, were attached to the H cords. Most of these cords would have knots on them. Rarely are knots found on the main cord, however, and tend to be mainly on the H and B cords. A quipu might also have a “totalizer” cord that summarizes all of the information on the cord group in one place.
Locke points out that there are three types of knots, each representing a different value, depending on the kind of knot used and its position on the cord. The Incas, like us, had a decimal (base-ten) system, so each kind of knot had a specific decimal value. The Single knot, pictured in the middle of the diagram[iii] was used to denote tens, hundreds, thousands, and ten thousands. They would be on the upper levels of the H cords. The figure-eight knot on the end was used to denote the integer “one.” Every other integer from 2 to 9 was represented with a long knot, shown on the left of the figure. (Sometimes long knots were used to represents tens and hundreds.) Note that the long knot has several turns in it…the number of turns indicates which integer is being represented. The units (ones) were placed closest to the bottom of the cord, then tens right above them, then the hundreds, and so on.
In order to make reading these pictures easier, we will adopt a convention that is consistent. For the long knot with turns in it (representing the numbers 2 through 9), we will use the following notation:
The four horizontal bars represent four turns and the curved arc on the right links the four turns together. This would represent the number 4.
We will represent the single knot with a large dot ( · ) and we will represent the figure eight knot with a sideways eight ( \( \infty \)).
Example 2
What number is represented on the cord shown?
Solution
On the cord, we see a long knot with four turns in it…this represents four in the ones place. Then 5 single knots appear in the tens position immediately above that, which represents 5 tens, or 50. Finally, 4 single knots are tied in the hundreds, representing four 4 hundreds, or 400. Thus, the total shown on this cord is 454.
Try it Now 2
What numbers are represented on each of the four cords hanging from the main cord?
- Answer
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From left to right:
\(\begin{array}{l}
\text { Cord } 1=2,162 \\
\text { Cord } 2=301 \\
\text { Cord } 3=0 \\
\text { Cord } 4=2,070
\end{array}\)
The colors of the cords had meaning and could distinguish one object from another. One color could represent llamas, while a different color might represent sheep, for example. When all the colors available were exhausted, they would have to be re-used. Because of this, the ability to read the quipu became a complicated task and specially trained individuals did this job. They were called Quipucamayoc, which means keeper of the quipus. They would build, guard, and decipher quipus.
As you can see from this photograph of an actual quipu, they could get quite complex.
There were various purposes for the quipu. Some believe that they were used to keep an account of their traditions and history, using knots to record history rather than some other formal system of writing. One writer has even suggested that the quipu replaced writing as it formed a role in the Incan postal system.[iv] Another proposed use of the quipu is as a translation tool. After the conquest of the Incas by the Spaniards and subsequent “conversion” to Catholicism, an Inca supposedly could use the quipu to confess their sins to a priest. Yet another proposed use of the quipu was to record numbers related to magic and astronomy, although this is not a widely accepted interpretation.
The mysteries of the quipu have not been fully explored yet. Recently, Ascher and Ascher have published a book, The Code of the Quipu: A Study in Media, Mathematics, and Culture , which is “ an extensive elaboration of the logical-numerical system of the quipu.”[v] For more information on the quipu, you may want to check out the following Internet link:
www.anthropology.wisc.edu/salomon/Chaysimire/khipus.php
We are so used to seeing the symbols 1, 2, 3, 4, etc. that it may be somewhat surprising to see such a creative and innovative way to compute and record numbers. Unfortunately, as we proceed through our mathematical education in grade and high school, we receive very little information about the wide range of number systems that have existed and which still exist all over the world. That’s not to say our own system is not important or efficient. The fact that it has survived for hundreds of years and shows no sign of going away any time soon suggests that we may have finally found a system that works well and may not need further improvement, but only time will tell that whether or not that conjecture is valid or not. We now turn to a brief historical look at how our current system developed over history.
[i] Diana, Lind Mae; The Peruvian Quipu in Mathematics Teacher, Issue 60 (Oct., 1967), p. 623-28.
[ii] Diana, Lind Mae; The Peruvian Quipu in Mathematics Teacher, Issue 60 (Oct., 1967), p. 623-28.
[iii] wiscinfo.doit.wisc.edu/chaysimire/titulo2/khipus/what.htm
[iv] Diana, Lind Mae; The Peruvian Quipu in Mathematics Teacher, Issue 60 (Oct., 1967), p. 623-28.


2.8: Conclusion and References
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- Gina Peterson and Emily Elam
- College of the Canyons
Research suggests that an enriched learning environment, along with meaningful interactions and experiences can significantly enhance a child’s overall development (Center for American Progress, 2017). Not only do children deserve high-quality learning experiences, parents need the assurance of knowing that their children are in safe, healthy and nurturing environments. Unfortunately, providing high-quality early childhood education services can be particularly challenging especially for the estimated four million children who are living in poverty and considered at-risk. Thus, the achievement gap continues to be a concern and the quest for quality is ongoing. As early childhood educators, we must strive to regularly observe, collect data and assess our programs, and reflect on our teaching practices to be more proactive in our quest for quality. While a score is important to look at and improve on, it is not the end all, be all, it merely serves to guide our practices as it relates to a whole picture. Ongoing observation and documentation help to keep us informed of how to best serve the children and families in our care.
References
- Crosswalk 15 minute service suites classroom. (n.d.). U.S. Department of Health and Human Services, Administration for Children and Families. Retrieved from https://eclkc.ohs.acf.hhs.gov/professional-development/article/crosswalk-15-minute-service-suites-classr
- Early Childhood Mentored Field Observations by Susan Eliason is licensed under CC BY-NC-SA 4.0
- Hamre, B. (n.d.). Using Classroom Observation to Gauge Teacher Effectiveness: Classroom Assessment Scoring System (CLASS). Retrieved from https://cepr.harvard.edu/files/cepr/files/ncte-conference-class-hamre.pdf
- Indiana Institute on Disability and Community. (n.d.). Retrieved from https://iidcweb.indiana.edu/styles/iidc/defiles/ECC/CLASS%20Overview%20for%20EEMG%20participants.pdf
- Mashburn, A. (n.d.). Class: Optimizing Reliability and Validity. Retrieved from https://curry.virginia.edu/faculty-research/centers-labs-projects/castl/class-optimizing-reliability-and-validity
- MiraCosta College. (n.d.). Retrieved from https://www.miracosta.edu/instruction/childdevelopmentcenter/downloads/5.4CLASSSummaryPreschool.pdf
- Slot, P. (2018). Structural characteristics and process quality in early childhood education and care: A literature review, OECD Education Working Paper No. 176. Retrieved from http://www.oecd.org/officialdocuments/publicdisplaydocumentpdf/?cote=EDU/WKP(2018)12&docLanguage=En
- Slot, P., Leseman, P., Verhagen, J., Mulder, H. (2015). Associations between structural quality aspects and process quality in Dutch early childhood education and care settings. Early Childhood Research Quarterly, Volume 33, 4th Quarter 2015, 64-76. Retrieved from https://www.sciencedirect.com/science/article/pii/S0885200615000599
- The Environment as the Third Teacher. (n.d.). Retrieved from https://files.eric.ed.gov/fulltext/ED493517.pdf
- University of Virginia. (n.d.). Retrieved from https://curry.virginia.edu/uploads/resourceLibrary/CASTL_practioner_Part1_single.pdf
- University of Virginia. (n.d.). Retrieved from https://curry.virginia.edu/uploads/resourceLibrary/CASTL_practioner_Part4_single.pdf
- University of Virginia. (n.d.). Retrieved from https://curry.virginia.edu/uploads/resourceLibrary/CASTL_practioner_Part5_single.pdf
- The National Center on Early Childhood Development, Teaching and Learning. (n.d.). Understanding and Using CLASS for Program Improvement. Retrieved from https://eclkc.ohs.acf.hhs.gov/sites/default/files/pdf/class-brief-understanding-using-class-program-improvement.pdf
- The National Center on Quality Teaching and Learning. (2013). Improving Teacher–Child Interactions: Using the CLASS™ in Head Start Preschool Programs. Retrieved from https://eclkc.ohs.acf.hhs.gov/sites/default/files/pdf/using-the-class-in-hs-preschool-programs.pdf
- Washington Office of Superintendent of Public Instruction. (n.d.). Assessment. Retrieved from
- ( http://www.k12.wa.us/EarlyLearning/pubdocs/assessment_print.pdf
- Wechsler, M., Melnick, H., Maier, A., Bishop, J. (2016). The Building Blocks of High-Quality Early Childhood Education Programs. Learning Policy Institute. Retrieved from https://learningpolicyinstitute.org/product/building-blocks-high-quality-early-childhood-education-programs


6.1: Complex Numbers
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Outcomes
- Understand the geometric significance of a complex number as a point in the plane.
- Prove algebraic properties of addition and multiplication of complex numbers, and apply these properties. Understand the action of taking the conjugate of a complex number.
- Understand the absolute value of a complex number and how to find it as well as its geometric significance.
Although very powerful, the real numbers are inadequate to solve equations such as \(x^2+1=0\), and this is where complex numbers come in. We define the number \(i\) as the imaginary number such that \(i^2 = -1\), and define complex numbers as those of the form \(z = a + bi\) where \(a\) and \(b\) are real numbers. We call this the standard form, or Cartesian form, of the complex number \(z\). Then, we refer to \(a\) as the real part of \(z\), and \(b\) as the imaginary part of \(z\). It turns out that such numbers not only solve the above equation, but in fact also solve any polynomial of degree at least 1 with complex coefficients. This property, called the Fundamental Theorem of Algebra, is sometimes referred to by saying \(\mathbb{C}\) is algebraically closed. Gauss is usually credited with giving a proof of this theorem in 1797 but many others worked on it and the first completely correct proof was due to Argand in 1806.
Just as a real number can be considered as a point on the line, a complex number \(z = a + bi\) can be considered as a point \(\left( a,b\right)\) in the plane whose \(x\) coordinate is \(a\) and whose \(y\) coordinate is \(b.\) For example, in the following picture, the point \(z = 3+2i\) can be represented as the point in the plane with coordinates \(\left( 3,2\right) .\)
Addition of complex numbers is defined as follows. \[\left( a+bi\right) +\left( c+di\right) =\left( a+c\right) +\left( b+d\right)i\nonumber \]
This addition obeys all the usual properties as the following theorem indicates.
Theorem \(\PageIndex{1}\): Properties of Addition of Complex Numbers
Let \(z,w,\) and \(v\) be complex numbers. Then the following properties hold.
- Commutative Law for Addition \[z+w=w+z\nonumber\]
- Additive Identity \[z+0=z\nonumber \]
- Existence of Additive Inverse \[\begin{array}{l} \mbox{For each} \; z\in \mathbb{C}, \mbox{there exists}\; -z\in \mathbb{C} \mbox{ such that}\; z+\left( -z\right) =0 \\ \mbox{In fact if } z=a+bi, \mbox{ then } -z=-a-bi. \end{array}\nonumber\]
- Associative Law for Addition \[\left( z+w\right) +v= z +\left( w+v\right)\nonumber \]
- Proof
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The proof of this theorem is left as an exercise for the reader.
Now, multiplication of complex numbers is defined the way you would expect, recalling that \(i^{2} = -1\). \[\begin{aligned} \left( a+bi\right) \left( c+di\right) &=ac+adi+bci+i^{2}bd \\ &=\left( ac-bd\right) +\left( ad + bc \right)i \end{aligned}\]
Consider the following examples.
The following are important properties of multiplication of complex numbers.
Theorem \(\PageIndex{2}\): Properties of Multiplication of Complex Numbers
Let \(z,w\) and \(v\) be complex numbers. Then, the following properties of multiplication hold.
- Commutative Law for Multiplication \[zw=wz\nonumber\]
- Associative Law for Multiplication \[\left( zw\right) v=z\left( wv\right)\nonumber\]
- Multiplicative Identity \[1z=z\nonumber\]
- Existence of Multiplicative Inverse \[\mbox{For each}\; z\neq 0, \mbox{there exists}\; z^{-1} \mbox{ such that}\; zz^{-1}=1\nonumber\]
- Distributive Law \[z\left( w+v\right) =zw+zv\nonumber\]
You may wish to verify some of these statements. The real numbers also satisfy the above axioms, and in general any mathematical structure which satisfies these axioms is called a field. There are many other fields, in particular even finite ones particularly useful for cryptography, and the reason for specifying these axioms is that linear algebra is all about fields and we can do just about anything in this subject using any field. Although here, the fields of most interest will be the familiar field of real numbers, denoted as \(\mathbb{R}\), and the field of complex numbers, denoted as \(\mathbb{C}\).
An important construction regarding complex numbers is the complex conjugate denoted by a horizontal line above the number, \(\overline{z}\). It is defined as follows.
Geometrically, the action of the conjugate is to reflect a given complex number across the \(x\) axis. Algebraically, it changes the sign on the imaginary part of the complex number. Therefore, for a real number \(a\), \(\overline{a} = a\).
Consider the following computation.
\[\begin{aligned} \left( \overline{a+bi}\right) \left( a+bi\right) &= \left( a-bi\right) \left( a+bi\right) \\[4pt] &= a^{2}+b^{2}-\left( ab-ab\right)i =a^{2}+b^{2}\end{aligned}\]
Notice that there is no imaginary part in the product, thus multiplying a complex number by its conjugate results in a real number.
Theorem \(\PageIndex{3}\): Properties of the Conjugate
Let \(z\) and \(w\) be complex numbers. Then, the following properties of the conjugate hold.
- \(\overline{z\pm w} = \overline{z} \pm \overline{w}\).
- \(\overline{(zw)} = \overline{z}~ \overline{w}\).
- \(\overline{(\overline{z})}=z\).
- \(\overline{\left(\frac{z}{w}\right)} = \frac{\overline{z}}{\overline{w}}\).
- \(z\) is real if and only if \(\overline{z}=z\).
Division of complex numbers is defined as follows. Let \(z=a+bi\) and \(w=c+di\) be complex numbers such that \(c,d\) are not both zero. Then the quotient \(z\) divided by \(w\) is
\[\begin{aligned} \frac{z}{w} &= \frac{a+bi}{c+di} \\[4pt] &= \frac{a+bi}{c+di}\times \frac{c-di}{c-di} \\[4pt] &= \frac{(ac+bd)+(bc-ad)i}{c^2+d^2} \\[4pt] & = \frac{ac+bd}{c^2+d^2} +\frac{bc-ad}{c^2+d^2}i.\end{aligned}\]
In other words, the quotient \(\frac{z}{w}\) is obtained by multiplying both top and bottom of \(\frac{z}{w}\) by \(\overline{w}\) and then simplifying the expression.
Example \(\PageIndex{3}\): Division of Complex Numbers
\[\frac{1}{i} = \frac{1}{i}\times \frac{-i}{-i} =\frac{-i}{-i^2}=-i\nonumber\]
\[\frac{2-i}{3+4i} = \frac{2-i}{3+4i}\times \frac{3-4i}{3-4i} =\frac{(6-4)+(-3-8)i}{3^2+4^2} =\frac{2-11i}{25} =\frac{2}{25} - \frac{11}{25}i\nonumber\]
\[\frac{1-2i}{-2+5i} = \frac{1-2i}{-2+5i}\times \frac{-2-5i}{-2-5i} =\frac{(-2-10) + (4-5)i}{2^2+5^2} =-\frac{12}{29}-\frac{1}{29}i\nonumber\]
Interestingly every nonzero complex number \(a+bi\) has a unique multiplicative inverse. In other words, for a nonzero complex number \(z\), there exists a number \(z^{-1}\) (or \(\frac{1}{z}\)) so that \(zz^{-1} = 1\). Note that \(z=a+bi\) is nonzero exactly when \(a^{2}+b^{2}\neq 0\), and its inverse can be written in standard form as defined now.
Definition \(\PageIndex{2}\): Inverse of a Complex Number
Let \(z = a+bi\) be a complex number. Then the multiplicative inverse of \(z\), written \(z^{-1}\) exists if and only if \(a^{2}+b^{2}\neq 0\) and is given by
\[z^{-1} = \frac{1}{a+bi} = \frac{1}{a+bi}\times \frac{a-bi}{a-bi}=\frac{a-bi}{a^{2}+b^{2}}=\frac{a}{a^{2}+b^{2}}-i\frac{b}{ a^{2}+b^{2}}\nonumber\]
Note that we may write \(z^{-1}\) as \(\frac{1}{z}\). Both notations represent the multiplicative inverse of the complex number \(z\). Consider now an example.
Example \(\PageIndex{4}\): Inverse of a Complex Number
Consider the complex number \(z = 2 + 6i\). Then \(z^{-1}\) is defined, and
\[\begin{aligned} \frac{1}{z} &= \frac{1}{2+6i} \\[4pt] &= \frac{1}{2+6i}\times \frac{2-6i}{2-6i} \\[4pt] &= \frac{2-6i}{2^2+6^2} \\[4pt] &= \frac{2-6i}{40} \\[4pt] &= \frac{1}{20} - \frac{3}{20}i \end{aligned}\]
You can always check your answer by computing \(zz^{-1}\).
Another important construction of complex numbers is that of the absolute value, also called the modulus. Consider the following definition.
Thus, if \(z\) is the complex number \(z=a+bi\), it follows that \[\left| z\right| =\left( z\overline{z}\right) ^{1/2}\nonumber\]
Also from the definition, if \(z=a+bi\) and \(w=c+di\) are two complex numbers, then \(\left\vert zw\right\vert =\left\vert z\right\vert \left\vert w\right\vert .\) Take a moment to verify this.
The triangle inequality is an important property of the absolute value of complex numbers. There are two useful versions which we present here, although the first one is officially called the triangle inequality.
Proposition \(\PageIndex{1}\): Triangle Inequality
Let \(z,w\) be complex numbers.
The following two inequalities hold for any complex numbers \(z,w\): \[\begin{array}{l} \left| z+w\right| \leq \left| z\right| +\left| w\right| \\ \left| \left| z\right| -\left| w\right| \right| \leq \left| z-w\right| \end{array}\nonumber\] The first one is called the Triangle Inequality .
- Proof
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Let \(z=a+bi\) and \(w=c+di\). First note that \[z \overline{w}=\left( a+bi\right) \left( c-di\right) =ac+bd+\left( bc-ad\right)i\nonumber\] and so \(\left\vert ac+bd\right\vert \leq \left\vert z\overline{w}\right\vert =\left\vert z\right\vert \left\vert w\right\vert .\)
Then, \[\left\vert z+w\right\vert ^{2}=\left( a+c+i\left( b+d\right) \right) \left( a+c-i\left( b+d\right) \right)\nonumber\] \[=\left( a+c\right) ^{2}+\left( b+d\right) ^{2}=a^{2}+c^{2}+2ac+2bd+b^{2}+d^{2}\nonumber\] \[\leq \left\vert z\right\vert ^{2}+\left\vert w\right\vert ^{2}+2\left\vert z\right\vert \left\vert w\right\vert =\left( \left\vert z\right\vert +\left\vert w\right\vert \right) ^{2}\nonumber\]
Taking the square root, we have that \[\left\vert z+w\right\vert \leq \left\vert z\right\vert +\left\vert w\right\vert\nonumber\] so this verifies the triangle inequality.
To get the second inequality, write \[z=z-w+w,\;w=w-z+z\nonumber\] and so by the first form of the inequality we get both: \[\left\vert z\right\vert \leq \left\vert z-w\right\vert +\left\vert w\right\vert ,\;\left\vert w\right\vert \leq \left\vert z-w\right\vert +\left\vert z\right\vert\nonumber\]
Hence, both \(\left\vert z\right\vert -\left\vert w\right\vert\) and \(\left\vert w\right\vert -\left\vert z\right\vert\) are no larger than \(\left\vert z-w\right\vert\). This proves the second version because \(\left\vert \left\vert z\right\vert -\left\vert w\right\vert \right\vert\) is one of \(\left\vert z\right\vert -\left\vert w\right\vert\) or \(\left\vert w\right\vert -\left\vert z\right\vert\).
With this definition, it is important to note the following. You may wish to take the time to verify this remark.
Let \(z=a+bi\) and \(w=c+di.\) Then
\[\left| z-w\right| =\sqrt{\left( a-c\right) ^{2}+\left( b-d\right) ^{2}}. \nonumber\]
Thus the distance between the point in the plane determined by the ordered pair \(\left( a,b\right)\) and the ordered pair \(\left( c,d\right)\) equals \(\left| z-w\right|\) where \(z\) and \(w\) are as just described.
For example, consider the distance between \(\left( 2,5\right)\) and \(\left( 1,8\right) .\) Letting \(z=2+5i\) and \(w=1+8i,\) \(z-w=1-3i\), \(\left( z-w\right) \left( \overline{z-w}\right) =\left( 1-3i\right) \left( 1+3i\right) = 10\) so \(\left\vert z-w\right\vert =\sqrt{10}\).
Recall that we refer to \(z=a+bi\) as the standard form of the complex number. In the next section, we examine another form in which we can express the complex number.


2.20: Economics
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- Susan Rahman, Prateek Sunder, and Dahmitra Jackson
- CC ECHO
Economics is a social science concerned with the production, distribution, and consumption of goods and services. It studies how individuals, businesses, governments, and nations make choices about how to allocate resources. Allocation of resources for national prosperity and growth sets a foundation for the nation's success (Hayes, 2020). Ethos like rugged individualism and ironically, pulling yourself up by your bootstraps, which is of course impossible, serve to enforce a myth of meritocracy that allows for judegment to be cast for those who do not fit the societal perception of “success,” and make it harder to identify the ways in which this myth is harmful to individuals and the larger society. When only some of a nation’s population are economically secure, the entirety of the nation and its future is at risk.
The United States has a particular problem with income inequality and the ways in which economists address these matters. In most fields of study, disaggregation, the breaking down of differing variables of data, is needed in order to see patterns often masked by aggregate data. Traditionally, the focus of economic measures has been on aggregate success/failure of the economy, but when you disaggregate you get to see a clearer picture of who is economically successful and comfortable and who is not. Disaggregation shows that economic prosperity and hardship falls down racial lines. Using this data to create economic policy is where the most economic impact can happen if programs are then targeted based on the data. Typically, neoliberal approaches seem to still permeate the field of economics. The notion that“a rising tide lifts all boats,” has not proven to be the case. Inequities of employment, income, and wealth persist for Black Americans no matter their educational attainment. In, How Economic Assumptions Uphold Racist Systems, Gamble (2020) writes, “Even if economic growth lifts all baseline living standards, persistent inequity, even at a higher standard, is still unjust”(Gamble, 2020). According to Gamble, there is a need to interrogate assumptions made about economic research and pedagogy in order to address racial wealth disparities, specifically:
●The assumption that value equals price. Taking the example of essential workers under Covid-19 times. While classified as such, they are often underpaid by comparison and predominantly made up of people of color. Market power of employers, outdated labor standards. and lack of union organizing all contribute to this. As a result, “the wealth gap between Black and white families is as wide now as it was in 1968. And extreme wealth, which is overwhelmingly white,has increasingly more control over political agendas as inequality increases.”
●The assumption that behavior represents the independent and rational preferences of individuals. This assumption does not sufficiently consider history, power and institutions in the shaping of behavior (Gamble, 2020).
These lived understandings of economics and the ways in which inequality plays out are missing in the neoliberal approach. Economists Anna Gifty Opoku-Agyeman and Fanta Traore wanted to know the percentage of Black women in the field. What they found is that it has been declining since 1995 and there are very few, 4% of all Economics BAs and 0.5% of all PhDs are earned by Black women. There is only 1 Black woman economist at the federal reserve out of a total of 409 economists, which raises questions regarding economic and fiscal policy making(Weingarten, 2020). This lack of representation prompted Opoku-Agyeman and Traore to form The Sadie Collective , an organization which works to get more Black women into economics related fields by illuminating the many roadblocks they face and attempting to remove those barriers. They cite structural racism in the field as the reason the discipline is so white. A future with more economists of color will have major implications on the lives of Black and Brown people. These future economists will be part of the financial decision making and have a voice and role regarding future fiscal policies. In the past, this has not been the case, and recessions have hurt marginalized communities at a significantly greater rate. By placing Black women in positions of prevalence and authority, our society begins to allow for representation of voices that are typically left out of economic policy discussions (Weingarten, 2020).
A White Paper was created by The Sadie Collective to address the need to diversify the field. This white paper addresses how to use financial commitments from high profile corporations and the newfound awareness of the racial inequity to upend systemic racism and improve access for underrepresented groups. Education statistics show that Black women in particular are underrepresented in the fields of math, economics, accounting and finance.There must be a commitment to cultivate a pipeline for Black women that begins earlier than their college careers to expose them to opportunities in these fields and support them along the way (The Sadie Collective, 2020). As there are few Black women majoring in these fields, in turn, there are few of them working in the corresponding professions. The global pandemic disproportionately hurt Black women who now sit at over 12% unemployment rate, much higher than the national average(Weingarten, 2020). Add on structural racism being evidenced lately by police killings of Black women further disadvantages them. The corporate world is at acritical juncture as it seeks to find ways to bring Black women into the field.The paper cites specific barriers to Black women’s entry into the field:
● Access to Information
● Discrimination in the Field
● Lack of role models
The Sadie Collection is a valuable resource for changing this. They offer a network of over 1000 Black women and allies in the field, offer mentorship, host conferences and create and partner on other vital outreach measures.
They offer two key ways to address the pipeline problem:
● Adopting the Black Women Best Framework
● Allocating Annual Funding for Black Women Led Initiatives That Support The Pipeline
The corporate world is at a turning point along with much of the nation and The Sadie Collective seeks to center Black women as a part of the change (The Sadie Collective, 2020).


8.3: Preparing Alkyl Halides from Alkanes - Radical Halogenation
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Objectives
After completing this section, you should be able to
- explain why the radical halogenation of alkanes is not usually a particularly good method of preparing pure samples of alkyl halides.
- use \(\ce{\sf{C–H}}\) bond energies to account for the fact that in radical chlorinations, the reactivity of hydrogen atoms decreases in the order \[\text{tertiary} > \text{secondary} > \text{primary}. \nonumber \]
- predict the approximate ratio of the expected products from the monochlorination of a given alkane.
Study Notes
The following terms are synonymous:
- methyl hydrogens, primary hydrogens, and 1° hydrogens.
- methylene hydrogens, secondary hydrogens, and 2° hydrogens.
- methine hydrogens, tertiary hydrogens, and 3° hydrogens.
Note that in radical chlorination reactions, the reactivity of methine, methylene and methyl hydrogens decreases in the ratio of approximately 5 : 3.5 : 1. This will aid in the prediction of expected products from the monochlorination of a given alkane.
Radical Halogenation
Alkanes (the simplest of all organic compounds) undergo very few reactions. One of these reactions is halogenation, or the substitution of a single hydrogen on the alkane for a single halogen (Cl 2 or Br 2 ) to form a haloalkane . This reaction is very important in organic chemistry because it functionalizes alkanes which opens a gateway to further chemical reactions.
General Reaction
\[CH_4 + Cl_2 + energy → CH_3Cl + HCl \nonumber \]
Radical Chain Mechanism
The reaction proceeds through the radical chain mechanism which is characterized by three steps: initiation , propagation, and termination . Initiation requires an input of energy but after that the reaction is self-sustaining.
Step 1: Initiation
During the initiation step free radicals are created when ultraviolet light or heat causes the X-X halogen bond to undergo homolytic to create two halogen free radicals. It is important to note that this step is not energetically favorable and cannot occur without some external energy input. After this step, the reaction can occur continuously (as long as reactants provide) without input of more energy.
Step 2: Propagation
The next two steps in the mechanism are called propagation steps. In the first propagation step, a chlorine radical abstracts hydrogen atom from methane. This gives hydrochloric acid (HCl, the inorganic product of this reaction) and the methyl radical. In the second propagation step, the methyl radical reacts with more of the chlorine starting material (Cl 2 ). One of the chlorine atoms becomes a radical and the other combines with the methyl radical to form the alkyl halide product.
Step 3: Termination
In the three termination steps of this mechanism, radicals produced in the mechanism an undergo radical coupling to form a sigma bond. These are called termination steps because a free radical is not produced as a product, which prevents the reaction from continuing. Combining the two types of radicals produced can be combined to from three possible products. Two chlorine radicals and couple to form more halogen reactant (Cl 2 ). A chlorine radical and a methyl radical can couple to form more product (CH 3 Cl). An finally, two methyl radicals can couple to form a side product of ethane (CH 3 CH 3 ).
This reaction is a poor synthetic method due to the formation of polyhalogenated side products. The desired product occurs when one of the hydrogen atoms in the methane has been replaced by a chlorine atom. However, the reaction doesn't stop there, and all the hydrogens in the methane can in turn be replaced by chlorine atoms to produce a mixture of chloromethane, dichloromethane, trichloromethane and tetrachloromethane.
Energetics
Why do these reactions occur? Is the reaction favorable? A way to answer these questions is to look at the change in enthalpy ΔH that occurs when the reaction takes place.
ΔH = (Energy put into reaction) – (Energy given off from reaction)
If more energy is put into a reaction than is given off, the ΔH is positive, the reaction is endothermic and not energetically favorable. If more energy is given off in the reaction than was put in, the ΔH is negative, the reaction is said to be exothermic and is considered favorable. The figure below illustrates the difference between endothermic and exothermic reactions.
ΔH can also be calculated using bond dissociation energies (ΔH°):
\[\Delta{H} = \sum \Delta{H^°} \text{ of bonds broken} - \sum \Delta{H^°} \text{ of bonds formed} \nonumber \]
Let’s look at our specific example of the chlorination of methane to determine if it is endothermic or exothermic:
Since, the ΔH for the chlorination of methane is negative, the reaction is exothermic. Energetically this reaction is favorable. In order to better understand this reaction we need to look at the mechanism ( a detailed step by step look at the reaction showing how it occurs) by which the reaction occurs.
Chlorination of Other Alkanes
When alkanes larger than ethane are halogenated, isomeric products are formed. Thus chlorination of propane gives both 1-chloropropane and 2-chloropropane as mono-chlorinated products. The halogenation of propane discloses an interesting feature of these reactions. All the hydrogens in a complex alkane do not exhibit equal reactivity . For example, propane has eight hydrogens, six of them being structurally equivalent primary , and the other two being secondary . If all these hydrogen atoms were equally reactive, halogenation should give a 3:1 ratio of 1-halopropane to 2-halopropane mono-halogenated products, reflecting the primary/secondary numbers. This is not what we observe. Light-induced gas phase chlorination at 25 ºC gives 45% 1-chloropropane and 55% 2-chloropropane.
CH 3 -CH 2 -CH 3 + Cl 2 → 45% CH 3 -CH 2 -CH 2 Cl + 55% CH 3 -CH Cl -CH 3
These results suggest strongly that 2º-hydrogens are inherently more reactive than 1º-hydrogens, by a factor of about 3.5:1. Further experiments showed that 3º-hydrogens are about 5 times more toward halogen atoms 1º-hydrogens. Thus, light-induced chlorination of 2-methylpropane gave predominantly (65%) 2-chloro-2-methylpropane, the substitution product of the sole 3º-hydrogen, despite the presence of nine 1º-hydrogens in the molecule.
(CH 3 ) 3 CH + Cl 2 → 65% (CH 3 ) 3 C Cl + 35% (CH 3 ) 2 CHCH 2 Cl
The Relative Reactivity of Hydrogens to
Radical Chlorination
This difference in reactivity can only be attributed to differences in C-H bond dissociation energies. In our previous discussion of bond energy we assumed average values for all bonds of a given kind, but now we see that this is not strictly true. In the case of carbon-hydrogen bonds, there are significant differences, and the specific dissociation energies (energy required to break a bond homolytically) for various kinds of C-H bonds have been measured. These values are given in the following table.
| R (in R–H) | methyl | ethyl | i-propyl | t-butyl |
|---|---|---|---|---|
|
Bond Dissociation Energy
(kcal/mole) |
103 | 98 | 95 | 93 |
This data shows that a tertiary C-H bond (93 kcal/mole) is easier to break than a secondary (95 kcal/mole) and primary (98 kcal/mole) C-H bond. These bond dissociation energies can be used to estimate the relative stability of the radicals formed after homolytic cleavage. Because a tertiary C-H bond requires less energy to undergo homolytic cleavage than a secondary or primary C-H bond, it can be inferred that a tertiary radical is more stable than secondary or primary.
Relative Stability of Free Radicals
Exercise \(\PageIndex{1}\)
Write out the complete mechanism for the chlorination of methane.
- Answer
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The answer to this problem is actually above in the initiation, propagation and termination descriptions.
Exercise \(\PageIndex{2}\)
Explain, in your own words, how the first propagation step can occur without input of energy if it is energetically unfavorable.
- Answer
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Since the second step in propagation is energetically favorable and fast, it drives the equilibrium toward products, even though the first step is not favorable.
Exercise \(\PageIndex{3}\)
Which step of the radical chain mechanism requires outside energy? What can be used as this energy?
- Answer
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Initiation step requires energy which can be in the form of light or het.
Exercise \(\PageIndex{4}\)
Having learned how to calculate the change in enthalpy for the chlorination of methane apply your knowledge and using the table provided below calculate the change in enthalpy for the bromination of ethane.
| Compound | Bond Dissociation Energy (kcal/mol) |
| CH 3 CH 2 -H | 101 |
| CH 3 CH 2 -Br | 70 |
| H-Br | 87 |
| Br 2 | 46 |
- Answer
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To calculate the enthalpy of reaction, you subtract the BDE of the bonds formed from the BDE of the bonds broken.
Bonds broken are C-H and Br-Br.
Bonds formed are H-Br adn C-Br.
Bonds broken - bonds formed = change in enthalpy
(101 kcal/mol + 46 kcal/mol) - (87 kcal/mol + 70 kcal/mol) = change in enthalpy
-10 kcal/mol = change in enthalpy for bromination of ethane.
Exercise \(\PageIndex{5}\)
1) Predict the mono-substituted halogenated product(s) of chlorine gas reacting with 2-methylbutane.
2) Predict the relative amount of each mono-brominated product when 3-methylpentane is reacted with Br 2 . Consider 1°, 2°, 3° hydrogen.
3) For the following compounds, give all possible monochlorinated derivatives.
- Answer
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1)
2)
.
3)


LEARNING OBJECTIVES
By the end of this section, you will be able to:
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Know the definition of international business.
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Comprehend how strategic management is related to international business.
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Understand how entrepreneurship is related to international business.
The Definition of International Business
International business relates to any situation where the production or distribution of goods or services crosses country borders. Globalization—the shift toward a more interdependent and integrated global economy—creates greater opportunities for international business. Such globalization can take place in terms of markets, where trade barriers are falling and buyer preferences are changing. It can also be seen in terms of production, where a company can source goods and services easily from other countries. Some managers consider the definition of international business to relate purely to “business,” as suggested in the Google case. However, a broader definition of international business may serve you better both personally and professionally in a world that has moved beyond simple industrial production. International business encompasses a full range of cross-border exchanges of goods, services, or resources between two or more nations. These exchanges can go beyond the exchange of money for physical goods to include international transfers of other resources, such as people, intellectual property (e.g., patents, copyrights, brand trademarks, and data), and contractual assets or liabilities (e.g., the right to use some foreign asset, provide some future service to foreign customers, or execute a complex financial instrument). The entities involved in international business range from large multinational firms with thousands of employees doing business in many countries around the world to a small one-person company acting as an importer or exporter. This broader definition of international business also encompasses for-profit border-crossing transactions as well as transactions motivated by nonfinancial gains (e.g., triple bottom line, corporate social responsibility, and political favor) that affect a business’s future.
Strategic Management and Entrepreneurship
A knowledge of both strategic management and entrepreneurship will enhance your understanding of international business. Strategic management is the body of knowledge that answers questions about the development and implementation of good strategies and is mainly concerned with the determinants of firm performance. A strategy, in turn, is the central, integrated, and externally oriented concept of how an organization will achieve its performance objectives. One of the basic tools of strategy is a SWOT (strengths, weaknesses, opportunities, threats) assessment. The SWOT tool helps you take stock of an organization’s internal characteristics—its strengths and weaknesses—to formulate an action plan that builds on what it does well while overcoming or working around weaknesses. Similarly, the external part of SWOT—the opportunities and threats—helps you assess those environmental conditions that favor or threaten the organization’s strategy. Because strategic management is concerned with organizational performance—be that social, environmental, or economic—your understanding of a company’s SWOT will help you better assess how international business factors should be accounted for in the firm’s strategy.
Entrepreneurship, in contrast, is defined as the recognition of opportunities (i.e., needs, wants, problems, and challenges) and the use or creation of resources to implement innovative ideas for new, thoughtfully planned ventures. An entrepreneur is a person who engages in entrepreneurship. Entrepreneurship, like strategic management, will help you to think about the opportunities available when you connect new ideas with new markets. For instance, given Google’s current global presence, it’s difficult to imagine that the company started out slightly more than a decade ago as the entrepreneurial venture of two college students. Google was founded by Larry Page and Sergey Brin, students at Stanford University. It was first incorporated as a privately held company on September 4, 1998. Increasingly, as the Google case study demonstrates, international businesses have an opportunity to create positive social, environmental, and economic values across borders. An entrepreneurial perspective will serve you well in this regard.
Spotlight on International Strategy and Entrepreneurship
Hemali Thakkar and three of her fellow classmates at Harvard found a way to mesh the power of play with electrical power. The foursome invented “a soccer ball with the ability to generate electricity,” Thakkar said. Every kick of the ball creates a current that’s captured for future use. Fifteen minutes of play lights a lamp for three hours.
Called the sOccket, the soccer ball can bring off-grid electricity to developing countries. Even better, the soccer ball can replace kerosene lamps. Burning kerosene is not only bad for the environment because of carbon dioxide emissions but it’s also a health hazard: according to the World Bank, breathing kerosene fumes indoors has the same effects as smoking two packs of cigarettes per day.
How did the idea of sOccket emerge? All four students (Jessica Lin, Jessica Matthews, Julia Silverman, and Hemali Thakkar) had experience with developing countries, so they knew that kids love playing soccer (it’s the world’s most popular sport). They also knew that most of these kids lived in homes that had no reliable energy.
As of November 2010, the sOccket prototype cost $70 to manufacture, but the team hopes to bring the cost down to $10 when production is scaled up. One ingenious way to bring costs down is to set up facilities where developing-world entrepreneurs assemble and sell the balls themselves.
At this point it’s also important to introduce you to the concepts of intrapreneurship and the intrapreneur. Intrapreneurship is a form of entrepreneurship that takes place inside a business that is already in existence. An intrapreneur, in turn, is a person within the established business who takes direct responsibility for turning an idea into a profitable finished product through assertive risk taking and innovation. An entrepreneur is starting a business, while an intrapreneur is developing a new product or service in an already existing business. Thus, the ideas of entrepreneurship can be applied not only in new ventures but also in the context of existing organizations—even government.
KEY TAKEAWAYS
International business encompasses a full range of cross-border exchanges of goods, services, or resources between two or more nations. These exchanges can go beyond the exchange of money for physical goods to include international transfers of other resources, such as people, intellectual property (e.g., patents, copyrights, brand trademarks, and data), and contractual assets or liabilities (e.g., the right to use some foreign asset, provide some future service to foreign customers, or execute a complex financial instrument).
Strategic management is the body of knowledge that answers questions about the development and implementation of good strategies and is mainly concerned with the determinants of firm performance. Because strategic management is concerned with organizational performance, your understanding of a company’s SWOT (strengths, weaknesses, opportunities, threats) helps you better assess how international business factors should be accounted for in the firm’s strategy.
Entrepreneurship is the recognition of opportunities (i.e., needs, wants, problems, and challenges) and the use or creation of resources to implement innovative ideas. Entrepreneurship helps you think about the opportunities available when you connect new ideas with new markets.
EXERCISES
(AACSB: Reflective Thinking, Analytical Skills)
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What is international business?
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Why is an understanding of strategy management important in the context of international business?
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Why is an understanding of entrepreneurship important in the context of international business?
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1.7.2: Reading and Review Questions
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How does Smith’s account of his experiences in the New World compare with the descriptions and assumptions of Columbus, Cabeza de Vaca, or Harriot?
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What grounds Smith’s assessment of events that he recounts, events like the men’s starving at the settlement? Does he ground his assessment in cause and effect? In Providence? Why, either way, do you think?
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Compare the way the Native Americans treat Smith after capturing him with the way his own men treat him before he leaves to follow the Council’s bidding. What, if anything, causes the differences, do you think?
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How does Smith use scientific knowledge against the Native Americans or for his own defense? Why?
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What introduction, if any, does Smith give to Pocahontas? Why? What purpose lies behind Smith’s probably inventing his rescue by Pocahontas? What role, if any, does it play in legitimizing a cultural as well as military conquest?


1.1: What is Abstract Algebra?
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. This course is an introduction to abstract algebra. We will spend most of our time studying groups. Group theory is the study of symmetry, and is one of the most beautiful areas in all of mathematics. It arises in puzzles, visual arts, music, nature, the physical and life sciences, computer science, cryptography, and of course, throughout mathematics. This book covers the basic concepts of group theory, and a special effort will be made to emphasize the intuition behind the concepts and motivate the subject matter. In the final chapter, we will also introduce rings and fields.
Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.
- Bertrand Russell, philosopher & mathematician


3.4: At Home
Day: 0
Time: 16h00
Place: At home
“Trixie stop barking,” Erin calls. She gets up from the couch slowly. “I can’t believe how tired I am.”
Taking a few steps towards the back door to let Trixie out, Erin stops at the corner of the kitchen island and puts a hand out to steady herself on the counter.
“Oh my. Can’t catch. My breath. Trixie. Stop barking.”
Remembering it was her late husband who took care of the dog, her eyes tear up slightly.
“I miss him so,” she thinks.
Moving toward the back door, Erin reaches down and lifts Trixie up onto the washing machine to place the leash on her.
“You stink, Trixie. Your bath will have to wait till I feel better. Not sure what is happening.”
Trixie, finally leashed, is lifted down and out they go through the back door into the cold winter air.
Erin gets down the steps and leans against the house to catch her breath. Meanwhile, Trixie relieves herself against a flower pot.
After about a minute, Erin begins to walk very slowly, with Trixie pulling on the leash. After about five minutes walking, Erin slows to a stop.
Looking back, Erin thinks to herself, “I have only walked about 50 meters. I am not sure I can even walk back to the house.”
Erin takes out her cell phone and calls her son at work.
“Thomas, I don’t feel well. You need to come home.”
“Mom, I’m at work. What’s up?” Thomas asks.
“I can’t. Catch. My. Breath. I think. I need. To go. To the. Hospital.”
“I will be there in ten minutes Mom.”


2.3.6: Confidentiality
Confidentiality means keeping what is supposed to be private, private. It means not disclosing (telling) other people who do not have a right to know information about the patient. Protecting a patient’s confidentiality means not sharing information with others (outside of the health care team) who the patient does not want to know the information. Home Health Aides/Personal Care Aides will come to know intimate details about their patient’s body and lives. They must take every step that they can to protect this privileged information. They have an ethical and a legal responsibility for protecting the privacy of their patient. A good rule of thumb to use before sharing patient information with members outside of the healthcare team is to ask :
- Does this person have a right to know this information?
- Did the patient give me permission to share information with this person?
- Would sharing this information be harmful to the patient?
- Would I want someone to know this information if it were me?”
Health Insurance Portability and Accountability Act (HIPAA)
In 1996, the Health Insurance Portability and Accountability Act (HIPAA) was passed by Congress. This law is designed to protect people’s privacy with regard to their health and health care. This law gives people rights over the medical information, including the right to receive a copy of their medical records, the right to correct incorrect information in their medical record, and the right to control who has access to their records. You can read more about privacy rights of HIPAA at the U.S. Department of Health and Human Services website: http://www.hhs.gov/ocr/privacy/hipaa/understanding/
Legal Ramifications for HIPAA Violations
If health care workers violate a patient’s privacy, they and/or their agency or the organization for which they work can be fined or imprisoned. The American Recovery and Reinvestment Act of 2009 established a penalty structure for HIPAA violations. Civil and criminal penalties can result if an individual health care worker and/or a health care agency does not comply with HIPAA.
According to the American Medical Association (n.d), fines can range from $100 per incident in a case where the worker was not aware of making a violation to $50,000 for violations which are willful (on purpose) and are not corrected within a specified time frame. Imprisonment sentences can range from one year to up to ten years. You can read more about specific fines for violations of HIPAA at: http://www.ama –assn.org/ama/pub/physician–resources/solutions–managing–your–practice/coding–billing–insurance/hipaahealth–insurance–portability–accountability–act/hipaa–violations–enforcement.page
Identify which of these scenarios is a HIPAA violation by indicating yes or no.
Telling your best friend to meet you at your patient’s house to go to lunch.  ______
Calling the nurse while you are at your patient’s house to let her know the patient has run out of medications. ______
Telling your fellow co-worker details about your patient’s health care while you are at the movies. _______
Telling your patient’s son details about your patient’s health care after your patient told you not to. ______
- Answer
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1. Yes
2. No
3. No
4. Yes
5. Yes
Feedback :
1. It is a HIPAA violation to tell your family and friends about a patient. Having them meet you at a patient’s house violates the patient’s privacy.
2. It is not a HIPAA violation to inform your supervisor about your patient’s condition. Any person who is part of the patient’s treatment team is allowed to have information about the patient in order to provide the best care.
3. It is not a HIPAA violation to call the nurse from your patient’s home as the nurse is part of the treatment team and is allowed access to patient information.
4. It is a HIPAA violation to talk about a patient with your co-workers in a public place. Other people who are not involved in the patient’s care can easily overhear your conversation. This would be a violation of the patient’s privacy.
5. It is a HIPAA violation to inform anyone about the details of your patient or their care, unless the patient has given you consent to do so. This includes family members. If a patient does not want a family member to know the details of their care, that is their choice and you must uphold their confidence and privacy.
Self-Check Activity M2-17
Identify which of these scenarios is a HIPAA violation by indicating yes or no.
Telling your best friend to meet you at your patient’s house to go to lunch.  ______
Calling the nurse while you are at your patient’s house to let her know the patient has run out of medications. ______
Telling your fellow co-worker details about your patient’s health care while you are at the movies. _______
Telling your patient’s son details about your patient’s health care after your patient told you not to. ______
Check your answers!