3.8: Transition to Empire - States, Cities, and New Religions (600 to 321 BCE)
The sixth century begins a transitional period in India’s history marked by important developments. Some of these bring to fruition processes that gained momentum during the late Vedic Age. Out of the hazy formative stage of state development, sixteen powerful kingdoms and oligarchies emerged. By the end of this period, one will dominate. Accompanying their emergence, India entered a second stage of urbanization, as towns and cities become a prominent feature of northern India. Other developments were newer. The caste system took shape as an institution, giving Indian society one of its most distinctive traits. Lastly, new religious ideas were put forward that challenged the dominance of Brahmanism.
States and Cities
The kingdom of Magadha became the most powerful among the sixteen states that dominated this transitional period, but only over time. At the outset, it was just one of eleven located up and down the Ganges River (see Map \(\PageIndex{1}\)). The rest were established in the older northwest or central India. In general, larger kingdoms dominated the Ganges basin while smaller clan-based states thrived on the periphery. They all fought with each other over land and resources, making this a time of war and shifting alliances.
The victors were the states that could field the largest armies. To do so, rulers had to mobilize the resources of their realms. The Magadhan kings did this most effectively. Expansion began in 545 BCE under King Bimbisara. His kingdom was small, but its location to the south of the lower reaches of the Ganges River gave it access to fertile plains, iron ore, timber, and elephants. Governing from his inland fortress at Rajagriha , Bimbisara built an administration to extract these resources and used them to form a powerful military. After concluding marriage alliances with states to the north and west, he attacked and defeated the kingdom of Anga to the east. His son Ajatashatru, after killing his father, broke those alliances and waged war on the Kosala Kingdom and the Vrijji Confederacy. Succeeding kings of this and two more Magadhan dynasties continued to conquer neighboring states down to 321 BCE, thus forging an empire. But its reach was largely limited to the middle and lower reaches of the Ganges River.
To the northwest, external powers gained control. As we have seen, the mountain ranges defining that boundary contain passes permitting the movement of peoples. This made the northwest a crossroads, and, at times, the peoples crossing through were the armies of rulers who sought to control the riches of India. Outside powers located in Afghanistan, Iran, or beyond might extend political control into the subcontinent, making part of it a component in a larger empire.
One example is the Persian Empire (see Chapter 5). During the sixth century, two kings, Cyrus the Great and Darius I, made this empire the largest in its time. From their capitals on the Iranian Plateau, they extended control as far as the Indus River, incorporating parts of northwest India as provinces of the Persian Empire (see Map \(\PageIndex{2}\)). Another example is Alexander the Great (also see Chapter 5). Alexander was the king of Macedon, a Greek state. After compelling other Greeks to follow him, he attacked the Persian Empire, defeating it in 331 BCE. That campaign took his forces all the way to mountain ranges bordering India. Desiring to find the end of the known world and informed of the riches of India, Alexander took his army through the Khyber Pass and overran a number of small states and cities located in the Punjab. But to Alexander’s dismay, his soldiers refused to go any further, forcing him to turn back. They were exhausted from years of campaigning far from home and discouraged by news of powerful Indian states to the east. One of those was the kingdom of Magadha (see Map \(\PageIndex{3}\)).
Magadha’s first capital—Rajagriha—is one of many cities and towns with ruins dating back to this transitional period. Urban centers were sparse during the Vedic Age but now blossomed, much like they did during the mature phase of the Harappan Civilization. Similar processes were at work. As more forests were cleared and marshes drained, the agricultural economy of the Ganges basin produced ever more surplus food. Population grew, enabling more people to move into towns and engage in other occupations as craftsmen, artisans, and traders. Kings encouraged this economic growth as its revenue enriched their treasuries. Caravans of ox-drawn carts or boats laden with goods travelling from state to state could expect to encounter the king’s customs officials and pay tolls. So important were rivers to accessing these trade networks that the Magadhan kings moved their capital to Pataliputra , a port town located on the Ganges (see Map \(\PageIndex{1}\)). Thus, it developed as a hub of both political power and economic exchange. Most towns and cities began as one or the other, or as places of pilgrimage.
The Caste System
As the population of northern India rose and the landscape was dotted with more villages, towns, and cities, society became more complex. The social life of a Brahmin priest who served the king differed from that of a blacksmith who belonged to a town guild, a rich businessman residing in style in a city, a wealthy property owner, or a poor agricultural laborer living in a village. Thus, the social identity of each member of society differed.
In ancient India, one measure of identity and the way people imagined their social life and how they fit together with others was the varna system of four social classes. Another was caste . Like the varnas , castes were hereditary social classifications; unlike them, they were far more distinct social groups. The four-fold varna system was more theoretical and important for establishing clearly who the powerful spiritual and political elites in society were: the Brahmins and Kshatriya. But others were more conscious of their caste. There were thousands of these, and each was defined by occupation, residence, marriage, customs, and language. In other words, because “I” was born into such-and-such a caste, my role in society is to perform this kind of work. “I” will be largely confined to interacting with and marrying members of this same group. Our caste members reside in this area, speak this language, hold these beliefs, and are governed by this assembly of elders. “I” will also be well aware of who belongs to other castes, and whether or not “I” am of a higher or lower status in relation to them, or more or less pure. On that basis, “I” may or may not be able, for instance, to dine with them. That is how caste defined an individual’s life.
The lowest castes were the untouchables. These were peoples who engaged in occupations considered highly impure, usually because they were associated with taking life; such occupations include corpse removers, cremators, and sweepers. So those who practiced such occupations were despised and pushed to the margins of society. Because members of higher castes believed touching or seeing them was polluting, untouchables were forced to live outside villages and towns, in separate settlements.
The Challenge to Brahmanism: Buddhism
During this time of transition, some individuals became dissatisfied with life and chose to leave the everyday world behind. Much like the sages of the Upanishads , these renunciants, as they were known, were people who chose to renounce social life and material things in order that they might gain deeper insight into the meaning of life. Some of them altogether rejected Brahmanism and established their own belief systems. The most renowned example is Siddhartha Gautama (c. 563 – 480 BCE), who is otherwise known as the Buddha .
Buddha means “Enlightened One” or “Awakened One,” implying that the Buddha was at one time spiritually asleep but at some point woke up and attained insight into the truth regarding the human condition. His life story is therefore very important to Buddhists, people who follow the teachings of the Buddha.
Siddhartha was born a prince, son to the chieftain of Shakya, a clan-based state located at the foothills of the Himalaya in northern India. His father wished for him to be a ruler like himself, but Siddhartha went in a different direction. At twenty-nine, after marrying and having a son, he left home.  Aging, sickness, and death posed the question of suffering for Siddhartha, leading him to pursue spiritual insight. For years, he sought instruction from other wanderers and experimented with their techniques for liberating the self from suffering through meditation and asceticism. But he failed to obtain the answers he sought.
Then, one day, while seated beneath a tree meditating for an extended period of time, a deepening calm descended upon Siddhartha, and he experienced nirvana. He also obtained insight into the reasons for human suffering and what was needful to end it. This insight was at the heart of his teachings for the remaining forty-five years of his life. During that time, he travelled around northern India teaching his dharma—his religious ideas and practices—and gained a following of students.
The principal teaching of the Buddha, presented at his first sermon, is called the Four Noble Truths. The first is the noble truth of suffering. Based on his own experiences, the Buddha concluded that life is characterized by suffering not only in an obvious physical and mental sense, but also because everything that promises pleasure and happiness is ultimately unsatisfactory and impermanent. The second noble truth states that the origin of suffering is an unquenchable thirst. People are always thirsting for something more, making for a life of restlessness with no end in sight. The third noble truth is that there is a cure for this thirst and the suffering it brings: nirvana. Nirvana means “blowing out,” implying extinction of the thirst and the end of suffering. No longer striving to quell the restlessness with temporary enjoyments, people can awaken to “the city of nirvana, the place of highest happiness, peaceful, lovely, without suffering, without fear, without sickness, free from old age and death.” 1 The fourth noble truth is the Eight-Fold Path, a set of practices that leads the individual to this liberating knowledge. The Buddha taught that through a program of study of Buddhist teachings (right understanding, right attitude), moral conduct (right speech, right action, right livelihood), and meditation (right effort, right concentration, right mindfulness), anyone could become a Buddha. Everyone has the potential to awaken, but each must rely on his or her own determination.
After the Buddha died c. 480 BCE, his students established monastic communities known as the Buddhist sangha . Regardless of their varna or caste, both men and women could choose to leave home and enter a monastery as a monk or nun. They would shave their heads, wear ochre-colored robes, and vow to take refuge in the Buddha, dharma, and sangha . Doing so meant following the example of the Buddha and his teachings on morality and meditation, as well as living a simple life with like-minded others in pursuit of nirvana and an end to suffering.
Endnotes
1 Rupert Gethin, The Foundations of Buddhism (Oxford University Press, 1998), 79.


15.2: Viscous Damped Free Vibrations
Viscous damping is damping that is proportional to the velocity of the system. That is, the faster the mass is moving, the more damping force is resisting that motion. Fluids like air or water generate viscous drag forces.
We will only consider linear viscous dampers, that is where the damping force is linearly proportional to velocity. The equation for the force or moment produced by the damper, in either \(x\) or \(\theta\), is:
\[ \vec{F}_c = c \vec{\dot{x}}, \]
\[ \vec{M}_c = c \vec{\dot{\theta}}, \]
where \(c\) is the damping constant. This is a physical property of the damper based on the type of fluid, size of the piston, etc. Note that the units of \(c\) change depending on whether it is damping linear motion (N-s/m) or rotational motion (N-m s/rad).
When the system is at rest in the equilibrium position, the damper produced no force on the system (no velocity), while the spring can produce force on the system, such as in the hanging mass shown above. Recall that this is the equilibrium position, but the spring is NOT at its unstretched length, as the static mass produces an extention of the spring.
If we perturb the system (applying an initial displacement, an initial velocity, or both), the system will tend to move back to its equilibrium position. What that movement looks like will depend on the system parameters (\(m\), \(c\), and \(k\)).
To determine the equation of motion of the system, we draw a free body diagram of the system with perturbation and apply Newton's Second Law.
The process for finding the equation of motion of the system is again:
- Sketch the system with a small positive perturbation (\(x\) or \(\theta\)).
- Draw the free body diagram of the perturbed system. Ensure that the spring force and the damper force have directions opposing the perturbation.
- Find the one equation of motion for the system in the perturbed coordinate using Newton's Second Law. Keep the same positive direction for position, and assign positive acceleration in the same direction.
- Move all terms of the equation to one side, and check that all terms are positive. If all terms are not positive, there is an error in the direction of displacement, acceleration, and/or spring or damper force.
For the example system above, with mass \(m\), spring constant \(k\) and damping constant \(c\), we derive the following:
\[ \sum F_x = m a_x = m \ddot{x} \]
\[ -F_k - F_c = m \ddot{x} \]
\[ -kx - \dot{c(x)} = m \ddot{x} \]
\[ m \ddot{x} + \dot{c(x)} + kx = 0 \]
This gives us a differential equation that describes the motion of the system. We can rewrite it in normal form:
\[ m \ddot{x} + c \dot{x} + kx = 0 \]
\[ \Rightarrow \ddot{x} + \frac{c}{m} \dot{x} + \frac{k}{m} x = 0 \]
\[ \Rightarrow \ddot{x} + 2 \zeta \omega_n \dot{x} + \omega_n^2 x = 0 \]
As before, the term \(\omega_n\) is called the angular natural frequency of the system , and has units of rad/s.
\[ \omega_n ^2 = \frac{k}{m}\, ; \quad \omega_n = \sqrt{\frac{k}{m}} \]
\(\zeta\) (zeta) is called the damping ratio. It is a dimensionless term that indicates the level of damping, and therefore the type of motion of the damped system.
\[ \zeta = \frac{c}{c_c} = \frac{\text{actual damping}}{\text{critical damping}} \]
The expression for critical damping comes from the solution of the differential equation. The solution to the system differential equation is of the form
\[ x(t) = a e^{rt}, \]
where \(a\) is constant and the value(s) of \(r\) can be can be obtained by differentiating this general form of the solution and substituting into the equation of motion.
\[ m r^2 e^{rt} + c r e^{rt} + k e^{rt} = 0 \]
\[ \Rightarrow (mr^2 + cr + k) e^{rt} = 0 \]
Because the exponential term is never zero, we can divide both sides by that term and get:
\[ m r^2 + cr + k = 0. \]
Using the quadratic formula, we can find the roots of the equation:
\[ r_{1,2} = \frac{-c \pm \sqrt{c^2 - 4mk}}{2m} \]
Critical damping occurs when the term under the square root sign equals zero:
\[ c_c ^2 = 4 mk \]
\[ c_c = 2 \sqrt{mk} = 2 m \omega_n \]
Four Viscous Damping Cases:
There are four basic cases for the damping ratio. For the solutions that follow in each case, we will assume that the initial perturbation displacement of the system is \(x_0\) and the initial perturbation velocity of the system is \(v_0\).
1. \(\zeta = 0\): Undamped
\[ c = 0 \]
This is the case covered in the previous section. Undamped systems oscillate about the equilibrium position continuously, unless some other force is applied.
2. \(\zeta > 1\): Overdamped
\[ c^2 > 4 mk \]
Roots are both real and negative, but not equal to each other. Overdamped systems move slowly toward equilibrium without oscillating.
The response for an overdamped system is:
\[ x(t) = a_1 e^{ \left( \frac{-c + \sqrt{c^2 - 4mk}}{2m} \right) t} + a_2 e^{ \left( \frac{-c - \sqrt{c^2 - 4mk}}{2m} \right) t}, \]
\[ \text{where} \,\, a_1 = \frac{-v_0 + r_2 x_0}{r_2 - r_1} \,\, \text{and} \,\, a_2 = \frac{v_0 + r_1 x_0}{r_2 - r_1}. \]
3. \(\zeta = 1\): Critically damped
\[ c^2 = 4 mk ( = c_c^2) \]
Roots are real and both equal to \(- \omega_n\). Critically-damped systems will allow the fastest return to equilibrium without oscillation.
The solution for a critically-damped system is:
\[ x(t) = (A + Bt) e^{-\omega_n t} , \]
\[ \text{where} \,\, A = x_0 \,\, \text{and} \,\, B = v_0 + x_0 \omega_n. \]
4. \(\zeta < 1\): Underdamped
\[ c^2 < 4 mk \]
The roots are complex numbers. Underdamped systems do oscillate around the equilibrium point; unlike undamped systems, the amplitude of the oscillations diminishes until the system eventually stops moving at the equilibrium position.
The solution for an underdamped system is:
\[ x(t) = [ C_1 \sin (\omega_d t) + C_2 \cos (\omega_d t) ] e^{- \omega_n \zeta t} , \]
\[ \text{where} \,\, C_1 = \frac{v_0 + \omega_n \zeta x_0}{\omega_d}, \,\, C_2 = x_0, \,\, \text{and} \,\, \zeta = \frac{c}{2 m \omega_n}. \]
\(\omega_d\) is called the damped natural frequency of the system. It is always less than \(\omega_n\):
\[ \omega_d = \omega_n \sqrt{1 - \zeta^2}. \]
The period of the underdamped response differs from the undamped response as well.
\[ \text{Undamped:} \quad \tau_n = \frac{2 \pi}{\omega_n} \]
\[ \text{Underdamped:} \quad \tau_d = \frac{2 \pi}{\omega_d} \]
Comparison of Viscous Damping Cases:
In the figure above, we can see that the critically-damped response results in the system returning to equilibrium the fastest. Also, we can see that the underdamped system amplitude is quite attenuated compared to the undamped case.


7.11: Learning to Read
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A huge milestone in middle childhood is learning to read and write. While the foundations of this were laid in infancy and early childhood, formal instruction on this process usually happens during the school-age years. There isn’t always complete agreement on how children are best taught to read. The following approaches to teaching reading are separated by their methodology, but today, models of reading strive for a balance between the two types of reading methods because they are both recognized as essential for learning to read.
60. Cognition and Instruction/Learning to Read by Wikibooks is licensed under CC BY-SA 3.0


13: Modern Understandings of Inheritance
The gene is the physical unit of inheritance, and genes are arranged in a linear order on chromosomes. The behaviors and interactions of chromosomes during meiosis explain, at a cellular level, the patterns of inheritance that we observe in populations. Genetic disorders involving alterations in chromosome number or structure may have dramatic effects and can prevent a fertilized egg from developing altogether.
-
- 13.0: Prelude to Modern Understandings of Inheritance
- The gene is the physical unit of inheritance, and genes are arranged in a linear order on chromosomes. The behaviors and interactions of chromosomes during meiosis explain, at a cellular level, the patterns of inheritance that we observe in populations. Genetic disorders involving alterations in chromosome number or structure may have dramatic effects and can prevent a fertilized egg from developing altogether.
-
- 13.1: Chromosomal Theory and Genetic Linkage
- The Chromosomal Theory of inheritance, proposed by Sutton and Boveri, states that chromosomes are the vehicles of genetic heredity. Neither Mendelian genetics nor gene linkage is perfectly accurate; instead, chromosome behavior involves segregation, independent assortment, and occasionally, linkage. Sturtevant devised a method to assess recombination frequency and infer the relative positions and distances of linked genes on a chromosome on the basis of the average number of crossovers.
-
- 13.2: Chromosomal Basis of Inherited Disorders
- The number, size, shape, and banding pattern of chromosomes make them easily identifiable in a karyogram and allows for the assessment of many chromosomal abnormalities. Disorders in chromosome number, or aneuploidies, are typically lethal to the embryo, although a few trisomic genotypes are viable. Because of X inactivation, aberrations in sex chromosomes typically have milder phenotypic effects. Aneuploidies also include instances in which segments of a chromosome are duplicated or deleted.
Thumbnail: Chromosomes are threadlike nuclear structures consisting of DNA and proteins that serve as the repositories for genetic information. The chromosomes depicted here were isolated from a fruit fly’s salivary gland, stained with dye, and visualized under a microscope. Akin to miniature bar codes, chromosomes absorb different dyes to produce characteristic banding patterns, which allows for their routine identification. (credit: modification of work by “LPLT”/Wikimedia Commons; scale-bar data from Matt Russell)


1.14.40: Laws of Thermodynamics
A remarkable feature of the subject called thermodynamics is the extent to which it is founded on four laws: Zeroeth, First, Second and Third. These laws summarize elegantly the results of experiments. Actually these are not laws in the sense of being laid down by government or by religious doctrine. Rather the laws are axioms . As McGlashan notes [1] each axiom is a ‘rule of the game’. These axioms refer to state variables such as temperature, pressure, energy and entropy. At this level the laws are not of immediate interest to chemists. However chemists have discovered how to ‘tell’ these axioms about chemical substances and chemical reactions.
The First Law invokes the concepts of energy and energy change. The law states that the energy of the universe is constant. In a realistic sense, at least for chemists, the law states that the energy of a chemical laboratory is constant. Then if the energy of system held in a reaction vessel increases, an equivalent amount of energy is lost from the rest of the laboratory. Then
\[\Delta \mathrm{U}(\text { system })+\Delta \mathrm{U}(\text { surroundings })=0 \nonumber \]
The Second Law of thermodynamics invokes the concepts of entropy and entropy change. In summary the law states that heat cannot flow spontaneously from low to high temperatures. The elegant studies carried out by James Prescott Joule (1818 -1889) were crucial to the development of thermodynamics [2].
Footnotes
[1] M. L. McGlashan, Chemical Thermodynamics, Academic Press, London 1979.
[2] L. Woodcock and L. Lue, Chem. Britain, 2001, August, p. 38.


10.9: Integer Exponents and Scientific Notation (Part 2)
Convert from Decimal Notation to Scientific Notation
Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on.
Consider the numbers 4000 and 0.004. We know that 4000 means 4 × 1000 and 0.004 means 4 × \(\dfrac{1}{1000}\). If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:
\[\begin{split} &4000 \qquad \qquad 0.004 \\ &4 \times 1000 \qquad 4 \times \dfrac{1}{1000} \\ &4 \times 10^{3} \qquad \; \; 4 \times \dfrac{1}{10^{3}} \\ &\qquad \qquad \quad \; \; \; 4 \times 10^{-3} \end{split}\]
When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation .
A number is expressed in scientific notation when it is of the form a × 10 n where a ≥ 1 and a < 10 and n is an integer.
It is customary in scientific notation to use × as the multiplication sign, even though we avoid using this sign elsewhere in algebra.
Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier.
If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.
In both cases, the decimal was moved 3 places to get the first factor, 4, by itself.
- The power of 10 is positive when the number is larger than 1: 4000 = 4 × 10 3 .
- The power of 10 is negative when the number is between 0 and 1: 0.004 = 4 × 10 −3 .
Write 37,000 in scientific notation.
Solution
| Step 1 : Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. | |
| Step 2 : Count the number of decimal places, n, that the decimal point was moved. |
3.70000 4 places |
| Step 3: Write the number as a product with a power of 10. | 3.7×10 4 |
|
If the original number is:
|
|
| Step 4 : Check.10 4 is 10,000 and 10,000 times 3.7 will be 37,000. | 37,000 = 3.7×10 4 |
Write in scientific notation: 96,000.
- Answer
-
9.6 × 10 4
Write in scientific notation: 48,300.
- Answer
-
4.83 × 10 4
Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
Step 2. Count the number of decimal places, n, that the decimal point was moved.
Step 3. Write the number as a product with a power of 10.
If the original number is:
- greater than 1, the power of 10 will be 10 n .
- between 0 and 1, the power of 10 will be 10 −n .
Step 4. Check.
Write in scientific notation: 0.0052.
Solution
| Move the decimal point to get 5.2, a number between 1 and 10. | |
| Count the number of decimal places the point was moved. | 3 places |
| Write as a product with a power of 10. | 5.2 × 10 −3 |
| Check your answer: | $$\begin{split} 5.2 &\times 10^{-3} \\ 5.2 &\times \dfrac{1}{10^{3}} \\ 5.2 &\times \dfrac{1}{1000} \\ 5.2 &\times 0.001 \\ 0.&0052 \end{split}$$ |
| 0.0052 = 5.2 × 10 −3 |
Write in scientific notation: 0.0078.
- Answer
-
7.8 × 10 -3
Write in scientific notation: 0.0129.
- Answer
-
1.29 × 10 -2
Convert Scientific Notation to Decimal Form
How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.
\[\begin{split} &9.12 \times 10^{4} \qquad \qquad 9.12 \times 10^{-4} \\ &9.12 \times 10,000 \qquad 9.12 \times 0.0001 \\ &91,200 \qquad \qquad \quad 0.000912 \end{split}\]
If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.
In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.
Convert to decimal form: 6.2 × 10 3 .
Solution
| Step 1 : Determine the exponent, n, on the factor 10. | 6.2 × 10 3 |
| Step 2 : Move the decimal point n places, adding zeros if needed. | |
|
6,200 |
| Step 3 : Check to see if your answer makes sense. | |
| 10 3 is 1000 and 1000 times 6.2 will be 6,200. | 6.2 × 10 3 = 6,200 |
Convert to decimal form: 1.3 × 10 3 .
- Answer
-
1,300
Convert to decimal form: 9.25 × 10 4 .
- Answer
-
92,500
Step 1. Determine the exponent, n, on the factor 10.
Step 2. Move the decimal n places, adding zeros if needed.
- If the exponent is positive, move the decimal point n places to the right.
- If the exponent is negative, move the decimal point |n| places to the left.
Step 3. Check.
Convert to decimal form: 8.9 × 10 −2 .
Solution
| Determine the exponent n, on the factor 10. | The exponent is −2. |
| Move the decimal point 2 places to the left. | |
| Add zeros as needed for placeholders. | 0.089 |
| 8.9 × 10 −2 = 0.089 | |
| The Check is left to you. |
Convert to decimal form: 1.2 × 10 −4 .
- Answer
-
0.00012
Convert to decimal form: 7.5 × 10 −2 .
- Answer
-
0.075
Multiply and Divide Using Scientific Notation
We use the Properties of Exponents to multiply and divide numbers in scientific notation.
Multiply. Write answers in decimal form: (4 × 10 5 )(2 × 10 −7 ).
Solution
| Use the Commutative Property to rearrange the factors. | 4 • 2 • 10 5 • 10 −7 |
| Multiply 4 by 2 and use the Product Property to multiply 10 5 by 10 −7 . | 8 × 10 −2 |
| Change to decimal form by moving the decimal two places left. | 0.08 |
Multiply. Write answers in decimal form: (3 × 10 6 )(2 × 10 −8 ).
- Answer
-
0.06
Multiply. Write answers in decimal form: (3 × 10 −2 )(3 × 10 −1 ).
- Answer
-
0.009
Divide. Write answers in decimal form: \(\dfrac{9 \times 10^{3}}{3 \times 10^{−2}}\).
Solution
| Separate the factors. | $$\dfrac{9}{3} \times \dfrac{10^{3}}{10^{-2}}$$ |
| Divide 9 by 3 and use the Quotient Property to divide 10 3 by 10 −2 . | 3 × 10 5 |
| Change to decimal form by moving the decimal five places right. | 300,000 |
Divide. Write answers in decimal form:\dfrac{8 \times 10^{4}}{2 \times 10^{-1}}.
- Answer
-
400,000
Divide. Write answers in decimal form:\dfrac{8 \times 10^{2}}{4 \times 10^{-2}}.
- Answer
-
20,000
Negative Exponents
Examples of Simplifying Expressions with Negative Exponents
Scientific Notation
Practice Makes Perfect
Use the Definition of a Negative Exponent
In the following exercises, simplify.
- 5 −3
- 8 −2
- 3 −4
- 2 −5
- 7 −1
- 10 −1
- 2 −3 + 2 −2
- 3 −2 + 3 −1
- 3 −1 + 4 −1
- 10 −1 + 2 −1
- 10 0 − 10 −1 + 10 −2
- 2 0 − 2 −1 + 2 −2
- (a) (−6) −2 (b) −6 −2
- (a) (−8) −2 (b) −8 −2
- (a) (−10) −4 (b) −10 −4
- (a) (−4) −6 (b) −4 −6
- (a) 5 • 2 −1 (b) (5 • 2) −1
- (a) 10 • 3 −1 (b) (10 • 3) −1
- (a) 4 • 10 −3 (b) (4 • 10) −3
- (a) 3 • 5 −2 (b) (3 • 5) −2
- n −4
- p −3
- c −10
- m −5
- (a) 4x −1 (b) (4x) −1 (c) (−4x) −1
- (a) 3q −1 (b) (3q) −1 (c) (−3q) −1
- (a) 6m −1 (b) (6m) −1 (c) (−6m) −1
- (a) 10k −1 (b) (10k) −1 (c) (−10k) −1
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
- p −4 • p 8
- r −2 • r 5
- n −10 • n 2
- q −8 • q 3
- k −3 • k −2
- z −6 • z −2
- a • a −4
- m • m −2
- p 5 • p −2 • p −4
- x 4 • x −2 • x −3
- a 3 b −3
- u 2 v −2
- (x 5 y −1 )(x −10 y −3 )
- (a 3 b −3 )(a −5 b −1 )
- (uv −2 )(u −5 v −4 )
- (pq −4 )(p −6 q −3 )
- (−2r −3 s 9 )(6r 4 s −5 )
- (−3p −5 q 8 )(7p 2 q −3 )
- (−6m −8 n −5 )(−9m 4 n 2 )
- (−8a −5 b −4 )(−4a 2 b 3 )
- (a 3 ) −3
- (q 10 ) −10
- (n 2 ) −1
- (x 4 ) −1
- (y −5 ) 4
- (p −3 ) 2
- (q −5 ) −2
- (m −2 ) −3
- (4y −3 ) 2
- (3q −5 ) 2
- (10p −2 ) −5
- (2n −3 ) −6
- u 9 u −2
- b 5 b −3
- x −6 x 4
- m 5 m −2
- q 3 q 12
- r 6 r 9
- n −4 n −10
- p −3 p −6
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
- 45,000
- 280,000
- 8,750,000
- 1,290,000
- 0.036
- 0.041
- 0.00000924
- 0.0000103
- The population of the United States on July 4, 2010 was almost 310,000,000.
- The population of the world on July 4, 2010 was more than 6,850,000,000.
- The average width of a human hair is 0.0018 centimeters.
- The probability of winning the 2010 Megamillions lottery is about 0.0000000057.
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
- 4.1 × 10 2
- 8.3 × 10 2
- 5.5 × 10 8
- 1.6 × 10 10
- 3.5 × 10 −2
- 2.8 × 10 −2
- 1.93 × 10−5
- 6.15 × 10−8
- In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was 2 × 10 4 .
- At the start of 2012, the US federal budget had a deficit of more than $1.5 × 10 13 .
- The concentration of carbon dioxide in the atmosphere is 3.9 × 10 −4 .
- The width of a proton is 1 × 10 −5 of the width of an atom.
Multiply and Divide Using Scientific Notation
In the following exercises, multiply or divide and write your answer in decimal form.
- (2 × 10 5 )(2 × 10 −9 )
- (3 × 10 2 )(1 × 10 −5 )
- (1.6 × 10 −2 )(5.2 × 10 −6 )
- (2.1 × 10 −4 )(3.5 × 10 −2 )
- \(\dfrac{6 \times 10^{4}}{3 \times 10^{−2}}\)
- \(\dfrac{8 \times 10^{6}}{4 \times 10^{−1}}\)
- \(\dfrac{7 \times 10^{-2}}{1 \times 10^{−8}}\)
- \(\dfrac{5 \times 10^{-3}}{1 \times 10^{−10}}\)
Everyday Math
-
Calories
In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by 1.5 trillion calories by the end of 2015.
- Write 1.5 trillion in decimal notation.
- Write 1.5 trillion in scientific notation.
-
Length of a year
The difference between the calendar year and the astronomical year is 0.000125 day.
- Write this number in scientific notation.
- How many years does it take for the difference to become 1 day?
- Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the probability of getting a particular 5-card hand from a deck of cards, Mario divided 1 by 2,598,960 and saw the answer 3.848 × 10 −7 . Write the number in decimal notation.
- Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the number of ways Barbara could make a collage with 6 of her 50 favorite photographs, she multiplied 50 • 49 • 48 • 47 • 46 • 45. Her calculator gave the answer 1.1441304 × 10 10 . Write the number in decimal notation.
Writing Exercises
- (a) Explain the meaning of the exponent in the expression 2 3 . (b) Explain the meaning of the exponent in the expression 2 −3 .
- When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) After looking at the checklist, do you think you are well prepared for the next section? Why or why not?
Contributors and Attributions
-
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at<EMAIL_ADDRESS>."


7.7E: Comparison of Female and Male Pelves
The female pelvis has evolved to its maximum width for childbirth and the male pelvis has been optimized for bipedal locomotion.
- Apply the diagnostic criteria to determine whether a pelvis is male or female
Key Points
- The pelvis is one of the most useful skeletal elements for differentiating between males and females.
- Female pelves are larger and wider than male pelves and have a rounder pelvic inlet.
- Male iliac crests are higher than females, causing their false pelves to look taller and narrower.
- The male sacrum is longer, narrower, straighter, and has a pronounced sacral promontory relative to the female sacrum.
- The angle between the inferior pubic rami is acute (70 degrees) in men, but obtuse (90–100 degrees) in women. Accordingly, the angle is called the subpubic angle in men and the pubic arch in women.
Key Terms
- pelvis : The large compound bone structure at the base of the spine that supports the legs. It consists of hip bone, sacrum, and coccyx.
Different Sex, Different Pelvis
Like the skull, the pelvis is highly useful for determining a skeleton’s biological sex. A wide pelvis is beneficial for child birth, however a narrow pelvis is beneficial for locomotion when walking upright. These conflicting demands are often termed the obstetrical dilemma.
The female pelvis has evolved to its maximum width for childbirth—a wider pelvis would make women unable to walk. In contrast, human male pelves are not constrained by the need to give birth and therefore are optimized for bipedal locomotion.
Diagnostic Criteria
There are several diagnostic criteria for differentiating a male from a female pelvis:
- The female pelvis is larger and broader than the male pelvis, which is taller (owing to a higher iliac crest), narrower, and more compact.
- The distance between the ischium bones is small in males. This causes the sides of the male pelvis to converge from the inlet to the outlet, whereas the sides of the female pelvis are wider apart. This results in the female inlet being large and oval in shape, while the male inlet is more heart shaped.
- The angle between the inferior pubic rami is acute (70 degrees) in men, but obtuse (90–100 degrees) in women. Accordingly, the angle is called the subpubic angle in men and pubic arch in women.
- The greater sciatic notch is wider in females.
- The ischial spines and tuberosities are heavier and project farther into the pelvic cavity in males.
- The male sacrum is long, narrow, straighter, and has a pronounced sacral promontory. The female sacrum is shorter, wider, more curved posteriorly, and has a less pronounced promontory.
- The acetabula are wider apart and face more medially in females than in males. This change in the angle of the femoral head gives the female gait its characteristic swinging of hips.


11.14: Reading- Products with Elastic and Inelastic Demand
Now that you’ve had some practice calculating the value of elasticity, let’s turn to some of the factors that play a role in whether a product is likely to have elastic or inelastic demand. The following factors can have an effect on elasticity:
- Substitutes : If it’s easy to choose a different product when prices change, the demand will be more elastic. If there are few or no alternatives, demand will be more inelastic.
- Absolute price : When a product is very expensive, even a small percentage change in price will make it prohibitively expensive to more buyers. If the price of a product is a tiny percentage of the buyer’s overall spending power, then a change in price will have less impact.
- Importance of use : In our previous example, we examined the elasticity of demand for cookies. A buyer may enjoy a cookie, but it doesn’t fulfill a critical need the way a snow shovel after a blizzard or a life-saving drug does. In general, the more important the product’s use, the more inelastic the demand will be.
- Competitive dynamics : Goods that are produced by a monopoly generally have inelastic demand, while products that exist in a competitive marketplace have elastic demand. This is because a competitive marketplace will create more options for the buyer.
With these considerations in mind, take a moment to see if you can figure out which of the following products have elastic demand and which have inelastic demand. It may be helpful to remember that when the buyer is in sensitive to price, demand is in elastic.
Gasoline (Generic Need)
The demand for gasoline generally is fairly inelastic . Car travel requires gasoline. The substitutes for car travel offer less convenience and control. Much car travel is necessary for people to move between activities and cannot be reduced to save money.
Gas from a Specific Station
The demand for gasoline from any single gas station, or chain of gas stations, is highly elastic . Buyers can choose between comparable products based on price. There are often many stations in a small geographic area that are equally convenient.
Traditional Textbooks
Generally an instructor assigns a textbook to the student, and the student who wants access to the learning materials must buy it, regardless of the price. Because the student can’t easily identify another textbook or resource that will ensure the same content and grade for the class, he has no substitutes and must buy the book at any price. Thus the demand is inelastic .
New Textbook Distribution Channels
Increasingly, students have new options to buy the same textbooks from different distribution channels at different price points. The introduction of new distribution channels is increasing options for buyers and having an impact on the price elasticity for publishers.
Specialty Coffee Drinks
Many coffee shops have developed branded drinks and specialized experiences in order to reduce substitutes and build customer loyalty. While black coffee is available almost universally, there are few substitutes for a Starbucks Java Chip Frappuccino. Demand for such products is more inelastic .
Black Coffee
Coffee is generally widely available at a level of quality that meets the needs of most buyers. The combination of a low price, relative to the buyer’s spending power, and the fact that the product is sold by many different suppliers in a competitive market make the demand highly elastic .
Concert Tickets
Only Taylor Swift can offer a Taylor Swift concert. She holds a monopoly on the creation and delivery of that experience. There is no substitute, and loyal fans are willing to pay for the experience. Because it is a scarce resource and the delivery is tightly controlled by a single provider, access to concerts has inelastic demand.
Airline Tickets
Airline tickets are sold in a fiercely competitive market. Buyers can easily compare prices, and buyers experience the services provided by competitors as being very similar. Buyers can often choose not to travel it the cost is too high, or to substitute travel by car or train. This makes the demand elastic .
Medical Procedures
Essential medical procedures have inelastic demand. The patient will pay what she can or what she must. In general, products that significantly affect health and well-being have inelastic demand.
Soft Drinks
Soft drinks and many other nonessential items have highly elastic demand. There is competition among every brand and type of soda, and there are many substitutes for the entire category of soft drinks.
Contributors and Attributions
- Products with Elastic and Inelastic Demand. Provided by : Lumen Learning. License : CC BY: Attribution
- Coffee Beans. Authored by : Mark Daynes. Located at : https://unsplash.com/photos/QKylkd3HS6o . License : CC0: No Rights Reserved


7.8: Suggested Readings
Benítez-López, A., L. Santini, A.M. Schipper, et al. 2019. Intact but empty forests? Patterns of hunting-induced mammal defaunation in the tropics. PLoS ONE 17: e3000247. https://doi.org/10.1371/journal.pbio.3000247 African mammals are undergoing catastrophic declines due to unregulated hunting.
Bouwman, H., I.M. Viljoen, L.P. Quinn, et al. 2008. Halogenated pollutants in terrestrial and aquatic bird eggs: Converging patterns of pollutant profiles and impacts and risks from high level. Environmental Research 126: 240–52. https://doi.org/10.1016/j.envres.2013.06.003 DDT threaten African biodiversity and people even today
Campbell, L., P. Verburg, D.G. Dixon, et al. 2008. Mercury biomagnification in the food web of Lake Tanganyika (Tanzania, East Africa). Science of The Total Environment 402: 184–91. https://doi.org/10.1016/j.scitotenv.2008.04.017 Biomagnification of heavy metals are making some fish unsafe to eat.
Galloway, T.S., and C.N. Lewis. 2016. Marine microplastics spell big problems for future generations. Proceedings of the National Academy of Sciences 113: 2331–33. https://doi.org/10.1073/pnas.1600715113 The threat of plastic pollution to ocean life.
Gottdenker, N.L., D.G. Streicker, C.L. Faust, et al. 2014. Anthropogenic land use change and infectious diseases: A review of the evidence. EcoHealth 11: 619632. https://doi.org/10.1007/s10393-014-0941-z Linking environmental degradation to disease risk in humans and wildlife.
le Maitre, D.C., G.G. Forsyth, S. Dzikiti, et al. 2016. Estimates of the impacts of invasive alien plants on water flows in South Africa. Water SA 42: 659–72. http://dx.doi.org/10.4314/wsa.v42i4.17 The impact of invasive plants on water cycles
Maxwell, S.L., R.A. Fuller, T.M. Brooks, et al. 2016. The ravages of guns, nets and bulldozers. Nature 536: 143–45. https://doi.org/10.1038/536143a Biodiversity faces many anthropogenic threats.
Pringle, R.M. 2005. The origins of the Nile perch in Lake Victoria. BioScience 55: 780–87. https://doi.org/10.1641/0006-3568(2005)055 [0780:TOOTNP]2.0.CO;2 The devastating impact of invasive fish in Lake Victoria.
Stauffer, J.R., H. Madsen, K. McKaye, et al. 2006. Schistosomiasis in Lake Malawi: Relationship of fish and intermediate host density to prevalence of human infection. EcoHealth 3: 22-27. https://doi.org/10.1007/s10393-005-0007-3 Trophic cascades can even affect human health.


3.2.1: Sarojini Naidu (1879-1949)
The Golden Threshold Indian Postcolonialism Sarojini Naidu was born Sarojini Chattapadhyay. She received her education from the University of Madras at King's College, London. She would go on to study at Girton College, Cambridge. During her time in England, she became familiar with the suffragist movement, and she continued her political interests in India. She became the first women to be president of the Indian National Congress and appointed Indian state governor. She was a political activist, feminist, and poet. Her writing was very influential an attracted many leading intellectuals to her salon in Bombay. She went on to become a fellow of the Royal Society of Literature in 1914. The Golden Threshold (1905) was the first of three volumes of poetry she published. Due to the musical quality of her work, which critics have compared to Walt Whitman and the Song of Solomon, she has been given the title the Nightingale of India. Consider while reading:
- What magical elements are present in Naidu's work? What is the effect of those elements in the poetry?
- How is Naidu's work similar to Walt Whitman's work?
- Describe the musical qualities you find in Naidu's work.
- How does the prosody of the poem reinforce her meaning?
Written by Laura Ng


1.6: Evidence-Based Practice and Nutrition
By the end of this section, you should be able to:
- 1.5.1 Apply EBP to client nutrition education and care.
- 1.5.2 Use EBP to evaluate trending nutritional views.
- 1.5.3 Use EBP to evaluate trending social media influencers.
Applying Evidence-Based Practice
Health information is readily available via the Internet. Unfortunately, only some of this information is vetted for appropriateness and validity. Nurses should steer clients toward reliable sources of evidence-based information.
EBP is fundamental to nursing practice—whether collaborating with an interdisciplinary team or working individually with clients directly. Throughout the remainder of this educational resource, the nurse will have opportunities to apply EBP to various nutritionally based client scenarios. Working through these scenarios, via unfolding client case studies, will require the nursing student to apply concepts of clinical judgment (introduced from the NCJMM) to foster health and well-being to clients across all body systems and the lifespan. Unfolding case studies will build on prior knowledge and scenarios. When using EBP to navigate the case studies, an understanding of nutrition and body system disorders will be used to develop the best plan of care for the client.
Research on nutrition has traditionally focused on biological relationships between diet and health. Although this is important, researchers must consider other factors. SDOH, health disparities, and health inequities all affect clients’ nutritional intake and need to be addressed in research. Doing this will help researchers shift their nutritional recommendations from a “one size fits all approach” to evidence-based recommendations that can be utilized by diverse populations (Mattes et al., 2022).
Evaluating Trending Nutritional Views
As mentioned earlier, an overwhelming amount of information is available to the consumer through the Internet. Consumers use the Internet to search health-related issues, find health care providers, and explore available resources. New (and sometimes not so new) nutrition trends are always appearing. Table 1.7 lists some of the more prevalent trends in recent years. Encourage clients to use the approaches previously described in Nutrition and Population Health to review and discuss with their health care provider any trends that they are considering.
| Trend | Description |
|---|---|
| South Beach Diet | Consume nutrient-rich carbohydrates and fats |
| Raw foods | Eat unprocessed/organic foods at a certain temperature |
| Fasting/cleanse | Consume liquids designed to eliminate toxins |
| Organic | Consume food certified as grown without chemical pesticides |
| NonGMO | Consume food certified as having no genetic modifications |
| Vegan/vegetarian | Eliminate animal-source protein/products from the diet either completely or selectively depending on specific type of vegetarian diet |
| Gluten free | Minimize or eliminate the ingestion of gluten found in wheat, barley, and rye |
| Paleo | Eliminate modern processed foods to focus on vegetables, fruits, nuts, eggs, fish, and lean meats |
| Atkins/Keto | Consume high-protein, low-carb foods |
| Calorie counting | Track food intake using calories; lose weight by eating fewer calories than required |
| Whole 30 | Avoid inflammatory foods for better gut health |
| Probiotics | Eat foods containing probiotics to enhance gut and digestive health |
| Weight Watchers | Track food intake using point system; lose weight by eating a certain level of points |
| Diet apps | MyFitnessPal, for example, to quantify exercise and macronutrients |
| Mediterranean | Focus on fruit, vegetables, beans, nuts, lean meats, fish, and olive oil |
| Intermittent fasting | Focus on when to eat, not necessarily what to eat |
Evaluating Social Media Sources for Nutrition Information
Over 4 billion social media users view information on social media sites. Social media can provide 24/7 support from other users or support groups, instant communication, and cost-effective health care options. Social media and other Internet sites also pose security risks, provide false information, promote inappropriate sharing of research findings, and promote self-diagnosis without the benefit of an assessment from a trained, experienced health care provider. A recent study of nutritional information on Instagram found the quality of the content to be extremely low (Kabata et al, 2022). When looking at social media posts, consumers should consider (Johns Hopkins Sheridan Libraries, 2023):
- The poster—Is this person qualified to post on this topic?
- Network—Do other credible people or organizations follow this account?
- Content—Can other sources validate the information?
- Contextual updates—Is this material they usually post about?
- Age—Is this an established account, or recently created?
- Reliability—Is the information source reliable?
As the American Academy of Nutrition and Dietetics noted, EBP uses scientific evidence rather than anecdotal evidence. This fundamental difference informs the nurse who evaluates trends in nutrition, especially those found on social media, and makes trusted recommendations to clients.
Benefits of Social Media Sources
Several sources channel the benefits of using social media to influence nutrition and nutritional education positively. Food Hero , an online nutrition resource provided by Oregon State University through funding from the USDA, provides an excellent format for social media sources to use. From the inclusion of several cultural resources to videos on demand and gardening tips, their site provides information in a dynamic format. Food Hero is found on Snapchat, YouTube, Twitter, Facebook, and Pinterest and serves as a great example for best practices on social media. Several other examples can be found by following the links in Table 1.8.
| Website | URL |
|---|---|
| Clem & Thyme Nutrition | https://clemandthyme.com/# |
| kaelyrd | https://www.kaelyrd.com/about |
| Gathered Nutrition | https://gatherednutrition.com/ |
| no food rules | https://colleenchristensennutrition.com/course/ |
| Christy Harrison | https://christyharrison.com/foodpsych |
Pitfalls of Social Media Sources
As mentioned earlier, the Internet is full of conflicting information. The nurse should encourage clients to evaluate nutrition information on social media sites in the same way information from websites and the media is scrutinized. Much of the nutrition advice on social media is based on anecdotal findings and experiences rather than scientific inquiry. Non-scientific social media content can promote eating disorders, distorted views of body image, and an unrealistic promotion of diet culture, all of which negatively impact nutrition and wellness.


1.7: Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
One simple graph, the stem-and-leaf graph or stemplot , comes from the field of exploratory data analysis. It is a good choice when the data sets are small. To create the plot, divide each observation of data into a stem and a leaf. The leaf consists of a final significant digit. For example, 23 has stem two and leaf three. The number 432 has stem 43 and leaf two. Likewise, the number 5,432 has stem 543 and leaf two. The decimal 9.3 has stem nine and leaf three. Write the stems in a vertical line from smallest to largest. Draw a vertical line to the right of the stems. Then write the leaves in increasing order next to their corresponding stem.
Creating a stem-and-leaf plot involves organizing numerical data into a visual format that shows the distribution and helps identify patterns. Here are the steps to create a stem-and-leaf plot:
Step 1. Organize the Data:
- Gather the data you want to represent in the stem-and-leaf plot.
- Arrange the data in ascending order for easier plotting.
Step 2. Determine the Stem and Leaf:
- Decide on the place value for the stem and the leaf. Typically, the stem represents the leading digits, and the leaf represents the trailing digits.
- For example, for numbers in the tens (e.g., 43, 56, 78), the stem could be the tens place (4, 5, 7) and the leaf could be the ones place (3, 6, 8).
Step 3. Create the Stem Column:
- List the stems in a vertical column from the smallest to the largest.
- Write the stems only once, even if they appear multiple times in the data.
Step 4. Add the Leaves:
- For each data point, write the leaf next to its corresponding stem.
- Arrange the leaves in ascending order next to each stem.
Step 5. Title and Label the Plot:
- Provide a title that describes the data being represented.
- Label the stem and leaf columns to make the plot easy to understand.
Consider the following data set:
\[ 52, 63, 55, 32, 67, 35, 36, 41, 46, 48, 57, 59, 63, 64, 65, 42, 44 \]
Step-by-Step Process:
1. Organize the Data:
- Sorted data: \[ 32, 35, 36, 41, 42, 44, 46, 48, 52, 55, 57, 59, 63, 64, 65, 67 \]
2. Determine the Stem and Leaf:
- Stem: Tens place (3, 4, 5, 6)
- Leaf: Ones place
3. Create the Stem Column:
```
Stem | Leaf
------------
3 |
4 |
5 |
6 |
```
4. Add the Leaves:
```
Stem | Leaf
------------
3 | 2 5 6
4 | 1 2 4 6 8
5 | 2 5 7 9
6 | 3 4 5 7
```
5. Title and Label the Plot:
```
Distribution of Data
Stem | Leaf
------------
3 | 2 5 6
4 | 1 2 4 6 8
5 | 2 5 7 9
6 | 3 4 5 7
```
By following these steps, you can create a stem-and-leaf plot that visually represents your data set.
For Susan Dean's spring pre-calculus class, scores for the first exam were as follows (smallest to largest):
33; 42; 49; 49; 53; 55; 55; 61; 63; 67; 68; 68; 69; 69; 72; 73; 74; 78; 80; 83; 88; 88; 88; 90; 92; 94; 94; 94; 94; 96; 100
| Stem | Leaf |
|---|---|
| 3 | 3 |
| 4 | 2 9 9 |
| 5 | 3 5 5 |
| 6 | 1 3 7 8 8 9 9 |
| 7 | 2 3 4 8 |
| 8 | 0 3 8 8 8 |
| 9 | 0 2 4 4 4 4 6 |
| 10 | 0 |
The stemplot shows that most scores fell in the 60s, 70s, 80s, and 90s. Eight out of the 31 scores or approximately 26% \(\left(\frac{8}{31}\right)\) were in the 90s or 100, a fairly high number of As.
For the Park City basketball team, scores for the last 30 games were as follows (smallest to largest):
32; 32; 33; 34; 38; 40; 42; 42; 43; 44; 46; 47; 47; 48; 48; 48; 49; 50; 50; 51; 52; 52; 52; 53; 54; 56; 57; 57; 60; 61
Construct a stem plot for the data.
- Answer
-
Stem Leaf 3 2 2 3 4 8 4 0 2 2 3 4 6 7 7 8 8 8 9 5 0 0 1 2 2 2 3 4 6 7 7 6 0 1
The stemplot is a quick way to graph data and gives an exact picture of the data. You want to look for an overall pattern and any outliers. An outlier is an observation of data that does not fit the rest of the data. It is sometimes called an extreme value. When you graph an outlier, it will appear not to fit the pattern of the graph. Some outliers are due to mistakes (for example, writing down 50 instead of 500) while others may indicate that something unusual is happening. It takes some background information to explain outliers, so we will cover them in more detail later.
The data are the distances (in kilometers) from a home to local supermarkets. Create a stemplot using the data:
1.1; 1.5; 2.3; 2.5; 2.7; 3.2; 3.3; 3.3; 3.5; 3.8; 4.0; 4.2; 4.5; 4.5; 4.7; 4.8; 5.5; 5.6; 6.5; 6.7; 12.3
Do the data seem to have any concentration of values?
HINT: The leaves are to the right of the decimal.
Answer
The value 12.3 may be an outlier. Values appear to concentrate at three and four kilometers.
| Stem | Leaf |
|---|---|
| 1 | 1 5 |
| 2 | 3 5 7 |
| 3 | 2 3 3 5 8 |
| 4 | 0 2 5 5 7 8 |
| 5 | 5 6 |
| 6 | 5 7 |
| 7 | |
| 8 | |
| 9 | |
| 10 | |
| 11 | |
| 12 | 3 |
The following data show the distances (in miles) from the homes of off-campus statistics students to the college. Create a stem plot using the data and identify any outliers:
0.5; 0.7; 1.1; 1.2; 1.2; 1.3; 1.3; 1.5; 1.5; 1.7; 1.7; 1.8; 1.9; 2.0; 2.2; 2.5; 2.6; 2.8; 2.8; 2.8; 3.5; 3.8; 4.4; 4.8; 4.9; 5.2; 5.5; 5.7; 5.8; 8.0
- Answer
-
Stem Leaf 0 5 7 1 1 2 2 3 3 5 5 7 7 8 9 2 0 2 5 6 8 8 8 3 5 8 4 4 8 9 5 2 5 7 8 6 7 8 0 The value 8.0 may be an outlier. Values appear to concentrate at one and two miles.
A side-by-side stem-and-leaf plot allows a comparison of the two data sets in two columns. In a side-by-side stem-and-leaf plot, two sets of leaves share the same stem. The leaves are to the left and the right of the stems. Tables \(\PageIndex{1}\) and \(\PageIndex{2}\) show the ages of presidents at their inauguration and at their death. Construct a side-by-side stem-and-leaf plot using this data.
| President | Ageat Inauguration | President | Age | President | Age |
|---|---|---|---|---|---|
| Pierce | 48 | Harding | 55 | Obama | 47 |
| Polk | 49 | T. Roosevelt | 42 | G.H.W. Bush | 64 |
| Fillmore | 50 | Wilson | 56 | G. W. Bush | 54 |
| Tyler | 51 | McKinley | 54 | Reagan | 69 |
| Van Buren | 54 | B. Harrison | 55 | Ford | 61 |
| Washington | 57 | Lincoln | 52 | Hoover | 54 |
| Jefferson | 57 | Grant | 46 | Truman | 60 |
| Madison | 57 | Hayes | 54 | Eisenhower | 62 |
| J. Q. Adams | 57 | Arthur | 51 | L. Johnson | 55 |
| Monroe | 58 | Garfield | 49 | Kennedy | 43 |
| J. Adams | 61 | A. Johnson | 56 | F. Roosevelt | 51 |
| Jackson | 61 | Cleveland | 47 | Nixon | 56 |
| Taylor | 64 | Taft | 51 | Clinton | 47 |
| Buchanan | 65 | Coolidge | 51 | Trump | 70 |
| W. H. Harrison | 68 | Cleveland | 55 | Carter | 52 |
| President | Age | President | Age | President | Age |
|---|---|---|---|---|---|
| Washington | 67 | Lincoln | 56 | Hoover | 90 |
| J. Adams | 90 | A. Johnson | 66 | F. Roosevelt | 63 |
| Jefferson | 83 | Grant | 63 | Truman | 88 |
| Madison | 85 | Hayes | 70 | Eisenhower | 78 |
| Monroe | 73 | Garfield | 49 | Kennedy | 46 |
| J. Q. Adams | 80 | Arthur | 56 | L. Johnson | 64 |
| Jackson | 78 | Cleveland | 71 | Nixon | 81 |
| Van Buren | 79 | B. Harrison | 67 | Ford | 93 |
| W. H. Harrison | 68 | Cleveland | 71 | Reagan | 93 |
| Tyler | 71 | McKinley | 58 | ||
| Polk | 53 | T. Roosevelt | 60 | ||
| Taylor | 65 | Taft | 72 | ||
| Fillmore | 74 | Wilson | 67 | ||
| Pierce | 64 | Harding | 57 | ||
| Buchanan | 77 | Coolidge | 60 |
Answer
| Ages at Inauguration | Ages at Death | |
|---|---|---|
| 9 9 8 7 7 7 6 3 2 | 4 | 6 9 |
| 8 7 7 7 7 6 6 6 5 5 5 5 4 4 4 4 4 2 1 1 1 1 1 0 | 5 | 3 6 6 7 7 8 |
| 9 5 4 4 2 1 1 1 0 | 6 | 0 0 3 3 4 4 5 6 7 7 7 8 |
| 7 | 0 0 1 1 1 3 4 7 8 8 9 | |
| 8 | 0 1 3 5 8 | |
| 9 | 0 0 3 3 |
The table shows the number of wins and losses the Atlanta Hawks have had in 42 seasons. Create a side-by-side stem-and-leaf plot of these wins and losses.
| Losses | Wins | Year | Losses | Wins | Year |
|---|---|---|---|---|---|
| 34 | 48 | 1968–1969 | 41 | 41 | 1989–1990 |
| 34 | 48 | 1969–1970 | 39 | 43 | 1990–1991 |
| 46 | 36 | 1970–1971 | 44 | 38 | 1991–1992 |
| 46 | 36 | 1971–1972 | 39 | 43 | 1992–1993 |
| 36 | 46 | 1972–1973 | 25 | 57 | 1993–1994 |
| 47 | 35 | 1973–1974 | 40 | 42 | 1994–1995 |
| 51 | 31 | 1974–1975 | 36 | 46 | 1995–1996 |
| 53 | 29 | 1975–1976 | 26 | 56 | 1996–1997 |
| 51 | 31 | 1976–1977 | 32 | 50 | 1997–1998 |
| 41 | 41 | 1977–1978 | 19 | 31 | 1998–1999 |
| 36 | 46 | 1978–1979 | 54 | 28 | 1999–2000 |
| 32 | 50 | 1979–1980 | 57 | 25 | 2000–2001 |
| 51 | 31 | 1980–1981 | 49 | 33 | 2001–2002 |
| 40 | 42 | 1981–1982 | 47 | 35 | 2002–2003 |
| 39 | 43 | 1982–1983 | 54 | 28 | 2003–2004 |
| 42 | 40 | 1983–1984 | 69 | 13 | 2004–2005 |
| 48 | 34 | 1984–1985 | 56 | 26 | 2005–2006 |
| 32 | 50 | 1985–1986 | 52 | 30 | 2006–2007 |
| 25 | 57 | 1986–1987 | 45 | 37 | 2007–2008 |
| 32 | 50 | 1987–1988 | 35 | 47 | 2008–2009 |
| 30 | 52 | 1988–1989 | 29 | 53 | 2009–2010 |
- Answer
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Table \(\PageIndex{3}\): Atlanta Hawks Wins and Losses Number of Wins Number of Losses 3 1 9 9 8 8 6 5 2 5 5 9 8 7 6 6 5 5 4 3 1 1 1 1 0 3 0 2 2 2 2 4 4 5 6 6 6 9 9 9 8 8 7 6 6 6 3 3 3 2 2 1 1 0 4 0 0 1 1 2 4 5 6 6 7 7 8 9 7 7 6 3 2 0 0 0 0 5 1 1 1 2 3 4 4 6 7 6 9
Another type of graph that is useful for specific data values is a line graph . In the particular line graph shown in Example, the x -axis (horizontal axis) consists of data values and the y -axis (vertical axis) consists of frequency points . The frequency points are connected using line segments.
In a survey, 40 mothers were asked how many times per week a teenager must be reminded to do his or her chores. The results are shown in Table and in Figure.
| Number of times teenager is reminded | Frequency |
|---|---|
| 0 | 2 |
| 1 | 5 |
| 2 | 8 |
| 3 | 14 |
| 4 | 7 |
| 5 | 4 |
Answer
In a survey, 40 people were asked how many times per year they had their car in the shop for repairs. The results are shown in Table. Construct a line graph.
| Number of times in shop | Frequency |
|---|---|
| 0 | 7 |
| 1 | 10 |
| 2 | 14 |
| 3 | 9 |
Answer
Bar graphs consist of bars that are separated from each other. The bars can be rectangles or they can be rectangular boxes (used in three-dimensional plots), and they can be vertical or horizontal. The bar graph shown in Example \(\PageIndex{9}\) has age groups represented on the x -axis and proportions on the y -axis .
By the end of 2011, Facebook had over 146 million users in the United States. Table shows three age groups, the number of users in each age group, and the proportion (%) of users in each age group. Construct a bar graph using this data.
| Age groups | Number of Facebook users | Proportion (%) of Facebook users |
|---|---|---|
| 13–25 | 65,082,280 | 45% |
| 26–44 | 53,300,200 | 36% |
| 45–64 | 27,885,100 | 19% |
Answer
The population in Park City is made up of children, working-age adults, and retirees. Table shows the three age groups, the number of people in the town from each age group, and the proportion (%) of people in each age group. Construct a bar graph showing the proportions.
| Age groups | Number of people | Proportion of population |
|---|---|---|
| Children | 67,059 | 19% |
| Working-age adults | 152,198 | 43% |
| Retirees | 131,662 | 38% |
Answer
The columns in Table contain: the race or ethnicity of students in U.S. Public Schools for the class of 2011, percentages for the Advanced Placement examine population for that class, and percentages for the overall student population. Create a bar graph with the student race or ethnicity (qualitative data) on the x -axis, and the Advanced Placement examinee population percentages on the y -axis.
| Race/Ethnicity | AP Examinee Population | Overall Student Population |
|---|---|---|
| 1 = Asian, Asian American or Pacific Islander | 10.3% | 5.7% |
| 2 = Black or African American | 9.0% | 14.7% |
| 3 = Hispanic or Latino | 17.0% | 17.6% |
| 4 = American Indian or Alaska Native | 0.6% | 1.1% |
| 5 = White | 57.1% | 59.2% |
| 6 = Not reported/other | 6.0% | 1.7% |
Solution
Park city is broken down into six voting districts. The table shows the percent of the total registered voter population that lives in each district as well as the percent total of the entire population that lives in each district. Construct a bar graph that shows the registered voter population by district.
| District | Registered voter population | Overall city population |
|---|---|---|
| 1 | 15.5% | 19.4% |
| 2 | 12.2% | 15.6% |
| 3 | 9.8% | 9.0% |
| 4 | 17.4% | 18.5% |
| 5 | 22.8% | 20.7% |
| 6 | 22.3% | 16.8% |
Answer
Summary
A stem-and-leaf plot is a way to plot data and look at the distribution. In a stem-and-leaf plot, all data values within a class are visible. The advantage in a stem-and-leaf plot is that all values are listed, unlike a histogram, which gives classes of data values. A line graph is often used to represent a set of data values in which a quantity varies with time. These graphs are useful for finding trends. That is, finding a general pattern in data sets including temperature, sales, employment, company profit or cost over a period of time. A bar graph is a chart that uses either horizontal or vertical bars to show comparisons among categories. One axis of the chart shows the specific categories being compared, and the other axis represents a discrete value. Some bar graphs present bars clustered in groups of more than one (grouped bar graphs), and others show the bars divided into subparts to show cumulative effect (stacked bar graphs). Bar graphs are especially useful when categorical data is being used.
References
- Burbary, Ken. Facebook Demographics Revisited – 2001 Statistics, 2011. Available online at www.kenburbary.com/2011/03/fa...-statistics-2/ (accessed August 21, 2013).
- “9th Annual AP Report to the Nation.” CollegeBoard, 2013. Available online at http://apreport.collegeboard.org/goa...omoting-equity (accessed September 13, 2013).
- “Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013).


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5: Series And Parallel Circuits
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5.1: What are “Series” and “Parallel” Circuits?
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Circuits consisting of just one battery and one load resistance are very simple to analyze, but they are not often found in practical applications. Usually, we find circuits where more than two components are connected together.
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5.2: Simple Series Circuits
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5.3: Simple Parallel Circuits
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5.4: Conductance
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5.5: Power Calculations
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An interesting rule for total power versus individual power is that it is additive for any configuration of circuit: series, parallel, series/parallel, or otherwise. Power is a measure of rate of work, and since power dissipated must equal the total power applied by the source(s) (as per the Law of Conservation of Energy in physics), circuit configuration has no effect on the mathematics.
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5.6: Correct use of Ohm’s Law
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5.7: Component Failure Analysis
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5.8: Building Simple Resistor Circuits
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In the course of learning about electricity, you will want to construct your own circuits using resistors and batteries. Some options are available in this matter of circuit assembly, some easier than others. In this section, I will explore a couple of fabrication techniques that will not only help you build the circuits shown in this chapter, but also more advanced circuits.


4.5: Oxidation-reduction reactions
What are Oxidation and Reduction?
The oxidation-reduction is defined in three ways:
- O xidation i s the l oss of electrons and r eduction i s the g ain of electrons. The word OIL RIG helps in remembering this definition.
- The addition of oxygen is oxidation, and the removal of oxygen is a reduction.
- The removal of hydrogen is oxidation, and the addition of hydrogen is a reduction.
The oxidation-reduction or in short redox reaction is one of the most common types of chemical reactions happening in and around us. For example, rusting of metals, photosynthesis, digestion of food, and combustion of fuels are redox reactions.
Oxidation-half and reduction-half reactions
The oxidation-reduction reactions happen in a pair because one thing loses electrons and is oxidized; the other thing gains the electrons and is reduced. That is why it is commonly called redox reaction, where red- represents reduction and –ox represents oxidation.
The green patina on the statue of liberty, shown in Fig. 4.5.1, is a result of the oxidation of copper:
\[\ce{2Cu(s) + O2(g) -> CuO(s)}\nonumber\]
The transfer of electrons becomes apparent when the reaction is split into the oxidation-half and reduction-half. The oxidation-half is:
\[\ce{Cu -> Cu^2+ + 2e^-}\nonumber\]
copper lost electrtons, so copper is oxides. The reduction half is:
\[\ce{O2 + 4e^- -> 2O^2-}\nonumber\]
Oxygen gained electrons, so oxygen is reduced.
- The substance that oxidizes another substance is an oxidizing agent . Oxygen is an oxidizing agent in the above reaction, as it oxidizes copper.
- The substance that reduces another substance is a reducing agent . Copper is a reducing agent in the above reaction, as it reduces oxygen.
The electrons lost in an oxidation-half must be equal to the electrons gained by the accompanying reduction-half. Multiplying the oxidation-half with 2 makes the electrons lost equal to the electrons gained in the reduction half in the above reaction. Then adding the oxidation and the reduction half gives the overall reaction:
\[\ce{2Cu -> 2Cu^2+ + \cancel{4e^-}}\nonumber\]
\[\ce{O2 + \cancel{4e^-} -> 2O^2-}\nonumber\]
\[\text {Overall reaction: } \ce{2Cu(s) + O2(g) -> CuO(s)}\nonumber\]
The chemical equations can be manipulated like algebraic equations, i.e., they can be multiplied or divided by a constant, added, and subtracted, as demonstrated in the example of the copper redox-half reactions above. Note that electrons on one side of the equation canceled the electrons on the other side of the equation during the addition operation.
A silver color zinc strip dipped in copper nitrate solution becomes coated with a layer of reddish color copper, as shown in Fig. 4.5.2. The molecular equation of the reaction, that shows formula units of compounds in the reactants and products, is:
\[\ce{Zn(s) + Cu(NO3)2(aq) -> Cu(s) + Zn(NO3)2(aq)}\nonumber\]
The complete ionic equation for the reaction is obtained by showing the dissolved ionic compounds as ions, e.g.:
\[\ce{Zn(s) + Cu^2+ (aq) + 2NO3^{-} (aq) -> Cu(s) + Zn^2+ (aq) + 2NO3^- (aq)}\nonumber\]
The oxidation-half of the reaction is:
\[\ce{Zn(s) -> Zn^2+ (aq) + 2e^-}\nonumber\]
The reduction-half of the reaction is:
\[\ce{Cu^2+ (aq) + 2e{-} -> Cu(s)}\nonumber\]
The net ionic equation that is the addition of the oxidation-half and the reduction-half is:
\[\ce{Zn(s) + Cu^2+ (aq) -> Cu(s) + Zn^2+ (aq)}\nonumber\]
Note that NO 3 - was on both sides of the complete ionic equation and has been canceled out in the net ionic equation:
\[\ce{Zn(s) + Cu^2+ (aq) + \cancel{2NO3^{-} (aq)} -> Cu(s) + Zn^2+ (aq) + \cancel{2NO3^- (aq)}}\nonumber\]
Ions that do not participate in the chemical reaction are called spectator ions, and they appear on both sides of the molecular equation, like NO 3 - in this case.
Biological oxidation and reduction reactions
In the redox reactions involving metal species, the transfer of electrons is usually evident through the oxidation-half and the reduction-half reactions, as in the above examples. In the organic and biochemical redox reactions, the transfer of electrons is usually not so obvious, but the transfer of hydrogen or oxygen is usually apparent. For example, the metabolism of methanol \(\ce{H3C-OH}\) starts with oxidation through the loss of hydrogen:
\[\ce{H3C-OH -> H2C=O + 2H}\nonumber\]
Metanol is oxidized to formaldehyde \(\ce{H2C=O}\). Formaldehyde oxidizes further by gaining oxygen:
\[\ce{H2C=O -> HCOOH}\nonumber\]
Finally, the formic acid (HCOOH) is oxidized by gaining oxygen and forming carbon dioxide and water:
\[\ce{2HCOOH -> 2O=C=O + 2H2O}\nonumber\]
The increase in the C-O bond from a single bond in methanol to two double bonds (four C-O bonds) in carbon dioxide is a clear indication of the oxidation of the carbon.
Removal of hydrogen is also oxidation, e.g., hydroxymalonate is oxidized to oxomalonate by an enzyme hydroxymalonate dehydrogenase:
The reverse of these, i.e., removal of oxygen and addition of hydrogen is reduction.