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Section 3

The Real Numbers

Fill in the blanks.The set of ___ numbers is $\{0,1,2,3,4,5, \ldots\}.$

Fill in the blanks.The set of ___ numbers is $\{1,2,3,4,5, \ldots\}.$

Fill in the blanks.The figure $\begin{array}{lllll}{+1} & {1} & {1} & {1} & {1} \\ \hline-2 & {-1} & {0} & {1} & {2}\end{array}$ is called a ______.

Fill in the blanks.The set of _____ numbers is $\{\ldots,-2,-1,0,1,2, \ldots\}.$

Fill in the blanks.Positive and negative numbers are called _____ numbers.

Fill in the blanks.GRAPH CANNOT COPY.

Fill in the blanks.The symbols $<$ and $>$ are ____ symbols.

Fill in the blanks.A ______ number is any number that can be expressed as a fraction with an integer numerator and a nonzero integer denominator.

Fill in the blanks.0.25 is called a ______ decimal and $0.333 \ldots$ is called a ______ decimal.

Fill in the blanks.An ______ . number cannot be expressed as a quotient of two integers.

Fill in the blanks.An irrational number is a nonterminating, nonrepeating ______ .

Fill in the blanks.The ______ ____ of a number is the distance on the number line between the number and $0 .$

Represent each situation using a signed number.a. A loss of $\$ 15$ millionb. A building foundation $\frac{5}{16}$ inch above grade

Show that each of the following numbers is a rational number by expressing it as a fraction with an integer numerator and a nonzero integer denominator: $6,-9,-\frac{7}{8}, 3 \frac{1}{2},-0.3,2.83.$

Give the opposite of each number.a. $20$ b. $-\frac{2}{3}$

What two numbers are a distance of 8 away from 5 on the number line?

What two numbers are a distance of 5 away from $-9$ on the number line?

Refer to the number line below. Use an inequality symbol, $<$ or $>,$ to make each statement true.a. $a \quad b$b. $b\quad a$c. $b \quad 0$ and $a$ $\quad 0$d. d. $|a| \quad|b|$GRAPH CANNOT COPY.

Fill in the blanks.$\sqrt{2}$ is read "the______ _____ of $2 "$

Fill in the blanks.$|-15|$ is read "the ______ _____ of $15 ".$

Fill in the blanks.

The symbol $=$ means ___ ______ _____ _____ .

The symbols $\{\quad \}$ are called ______.

The symbol $\pi$ is a letter from the ______ alphabet.

To find the decimal equivalent for $\frac{2}{3},$ we perform the following long division: SYMBOL CANNOT COPY.

$-\frac{4}{5}-\frac{}{5}=\frac{4}{ }$

Write each repeating decimal using an overbar.a. $0.666 \ldots$b. $0.24444 \ldots$c. $0.717171 . . .$d. $0.456456456 \dots$

Place check marks in the table to show the set or sets to which each number belongs. For example, the check shows that $\sqrt{2}$ is irrational. See Example $1 .$

TABLE CANNOT COPY.

Which numbers in the following set are natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers? $\left\{67, \frac{4}{13},-5.9,11 \frac{2}{3}, \sqrt{2}, 0,-3, \pi\right\}.$

Determine whether each statement is true or false. Every whole number is an integer.

Determine whether each statement is true or false. Every integer is a natural number.

Determine whether each statement is true or false. Every integer is a whole number.

Determine whether each statement is true or false. Every real number is either a rational number or an irrational number.

Determine whether each statement is true or false. Irrational numbers are real numbers.

Determine whether each statement is true or false. Every whole number is a rational number.

Determine whether each statement is true or false. Every rational number can be written as a fraction (ratio) of two integers.

Determine whether each statement is true or false. Every rational number is a whole number.

Use one of the symbols $<$ or $>$ to make each statement true. $$0 \quad\text{__}\quad-4 $$

Use one of the symbols $<$ or $>$ to make each statement true. $$0 \quad\text{__}\quad32 $$

Use one of the symbols $<$ or $>$ to make each statement true. $$917 \quad\text{__}\quad971 $$

Use one of the symbols $<$ or $>$ to make each statement true. $$898 \quad\text{__}\quad889 $$

Use one of the symbols $<$ or $>$ to make each statement true. $$-2\quad\text{__}\quad-3 $$

Use one of the symbols $<$ or $>$ to make each statement true. $$-5\quad\text{__}\quad-4 $$

Use one of the symbols $<$ or $>$ to make each statement true. $$-\frac{5}{8}\quad\text{__}\quad-\frac{3}{8} $$

Use one of the symbols $<$ or $>$ to make each statement true. $$-19 \frac{2}{3}\quad\text{__}\quad-19 \frac{1}{3} $$

Use one of the symbols $<$ or $>$ to make each statement true. $$\frac{2}{3}\quad\text{__}\quad\frac{3}{5} $$

Use one of the symbols $<$ or $>$ to make each statement true. $$\frac{3}{4}\quad\text{__}\quad\frac{5}{6} $$

Use one of the symbols $<$ or $>$ to make each statement true. $$-6.19\quad\text{__}\quad-5.8 $$

Use one of the symbols $<$ or $>$ to make each statement true. $$-2.27\quad\text{__}\quad-5.25 $$

Write each fraction as a decimal. If the result is a repeating decimal, use an overbar. $$\frac{5}{8}$$

Write each fraction as a decimal. If the result is a repeating decimal, use an overbar. $$\frac{3}{32}$$

Write each fraction as a decimal. If the result is a repeating decimal, use an overbar. $$\frac{1}{30}$$

Write each fraction as a decimal. If the result is a repeating decimal, use an overbar. $$\frac{7}{9}$$

Write each fraction as a decimal. If the result is a repeating decimal, use an overbar. $$\frac{1}{60}$$

Write each fraction as a decimal. If the result is a repeating decimal, use an overbar. $$\frac{5}{11}$$

Write each fraction as a decimal. If the result is a repeating decimal, use an overbar. $$\frac{21}{50}$$

Write each fraction as a decimal. If the result is a repeating decimal, use an overbar. $$\frac{2}{125}$$

Graph each set of numbers on a number line.$$\left\{-\pi, 4.25,-1 \frac{1}{2},-0.333 \ldots, \sqrt{2},-\frac{35}{8}, 3\right\}$$

Graph each set of numbers on a number line.$$\left\{-2 \frac{1}{8}, \pi, 2.75,-\sqrt{2}, \frac{17}{4}, 0.666 \ldots,-3\right\}$$

Graph each set of numbers on a number line.The integers between $-5$ and 2

Graph each set of numbers on a number line.The whole numbers less than 4

Find each absolute value.$$|83|$$

Find each absolute value.$$|29|$$

Find each absolute value.$$\left|\frac{4}{3}\right|$$

Find each absolute value.$$\left|\frac{9}{16}\right|$$

Find each absolute value.$$|-11|$$

Find each absolute value.$$|-14|$$

Find each absolute value.$$|-6.1|$$

Find each absolute value.$$|-25.3|$$

Insert one of the symbols $>,<,$ or $=$ in the blank to make each statement true.

$$|3.4| \quad \text{_____}\quad-3$$

$$0.08 \quad \text{_____}\quad0.079$$

$$|-1.1| \quad \text{_____}\quad1.2$$

$$-5.5 \quad \text{_____}\quad-5 \frac{1}{2}$$

$|-\frac{15}{2}| {}\quad 7.5$

$$\left|-2 \frac{2}{3}\right| \quad \text{_____}\quad\frac{7}{3}$$

$$\frac{99}{100} \quad \text{_____}\quad0.99$$

$$|2| \quad \text{_____}\quad|-2|$$

$$0.3 \quad \text{_____}\quad0.333 \dots$$

$$-0.666 \dots \quad \text{_____}\quad-0.6$$

$$1 \quad \text{_____}\quad\left|-\frac{15}{16}\right|$$

$$\sqrt{2} \quad \text{_____}\quad \pi$$

Drafting. Which dimensions of the aluminum bracket shown below are natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers?GRAPH CANNOT COPY.

History. Refer to the time line shown below.a. What basic unit was used to scale the time line?b. What symbolism is used to represent zero?c. Which numbers could be thought of as positive and which as negative?d. Express the dates for the Maya civilization using positive and negative numbers.GRAPH CANNOT COPY.

iPhones. You can get a more accurate reading of an iPhone's signal strength by dialing $* 3001 \# 12345 \#^{*} .$ Field test mode is then activated and the standard signal strength bars (in the upper left corner of the display) are replaced by a negative number. The closer the negative number is to zero, the stronger the signal. Which iPhone shown below is receiving the strongest signal?

GRAPH CANNOT COPY.

Drafting. On an architect's scale, the edge marked 16 divides each inch into 16 equal parts. Find the decimal form for each fractional part of one inch that is highlighted on the scale.

Trade. Each year from 1994 through $2009,$ the United States imported more goods and services from Japan than it exported to Japan. This caused trade deficits, which are represented by negative numbers on the following graph.a. In which year was the deficit the worst? Express that deficit using a signed number.b. In which year was the deficit the smallest? Express that deficit using a signed number.GRAPH CANNOT COPY.

U.S. Budget. A budget deficit is a negative number that indicates the government spent more money than it took in that year. A budget surplus is a positive number that indicates the government took in more money than it spent that year.a. Refer to the graph in the next column that shows the U.S. Federal Budget Deficit/Surplus for the years 1980 through $2009 .$ For how many of those years was there a budget surplus?b. Consider the years in which there was a budget deficit. For how many of those years was it smaller than $\$ 300$ billion?

Explain the difference between a rational and an irrational number.

Can two different numbers have the same absolute value? Explain.

Explain how to find the decimal equivalent of a fraction.

What is a real number?

Pi Day (or Pi Approximation Day) is an unofficial holiday held to celebrate $\pi .$ Why do you think Pi Day is observed each year on March $14 ?$

Explain why $0.1 \overline{333}$ is not the simplest way to represent $0.1333 \ldots$

Simplify: $\frac{24}{54}$

Multiply: $\quad \frac{3}{4}\left(\frac{8}{5}\right)$

Divide: $5 \frac{2}{3} \div 2 \frac{5}{9}$

Add: $\quad \frac{3}{10}+\frac{2}{15}$

How many integers have an absolute value that is less than $1,000 ?$

Is $0.10100100010000 \ldots$ a repeating decimal? Explain.

Find a rational number between each pair of numbers.$$\frac{1}{8} \text { and } \frac{1}{9}$$

Find a rational number between each pair of numbers.$$1.7 \overline{1} \text { and } 1.7 \overline{2}$$