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Section A.13 The Real Number System

Subsection Subsets of the Real Numbers

The numbers associated with points on a number line are called the real numbers. The set of real numbers is denoted by \(\mathbb{R}\text{.}\) You are already familiar with several types, or subsets, of real numbers:

  • The set \(\mathbb{N} \) of natural, or counting numbers, as its name suggests, consists of the numbers \(1, 2, 3, 4 , \ldots ,\) where "\(\ldots\)" indicates that the list continues without end.

  • The set \(\mathbb{W} \) of whole numbers consists of the natural numbers and zero: \(0, 1, 2, 3 \ldots\text{.}\)

  • The set \(\mathbb{Z} \) of integers consists of the natural numbers, their negatives, and zero: \(\ldots , -3, -2, -1, 0, 1, 2, 3, \ldots\text{.}\)

All of these numbers are subsets of the rational numbers.

Subsection Rational Numbers

A number that can be expressed as the quotient of two integers \(\dfrac{a}{b} \) where \(b\ne 0\text{,}\) is called a rational number. The integers are rational numbers, and so are common fractions. Some examples of rational numbers are \(5, -2, 0, \dfrac{2}{9} , \sqrt{16},\) and \(\dfrac{-4}{17} \text{.}\) The set of rational numbers is denoted by \(\mathbb{Q} \text{.}\)

Every rational number has a decimal form that either terminates or repeats a pattern of digits. For example,

\begin{equation*} \frac{3}{4}=3\div 4=0.75, ~\text{a }\textbf{terminating decimal} \end{equation*}

and

\begin{equation*} \frac{2}{37}=9\div 37=0.243243243 \ldots \end{equation*}

where the pattern of digits \(243\) is repeated endlessly. We use the repeater bar notation to write a repeating decimal fraction:

\begin{equation*} \frac{9}{37}= 0.\overline{243} \end{equation*}

Subsection Irrational Numbers

Some real numbers cannot be written in the form \(\dfrac{a}{b} \) , where \(a\) and \(b\) are integers. For example, the number \(\sqrt{2} \) is not equal to any common fraction. Such numbers are called irrational numbers. Examples of irrational numbers are \(\sqrt{15}, \pi, \) and \(-\sqrt[3]{7} \text{.}\)

The decimal form of an irrational number never terminates, and its digits do not follow a repeating pattern, so it is impossible to write down an exact decimal equivalent for an irrational number. However, we can obtain decimal approximations correct to any desired degree of accuracy by rounding off. A graphing calculator gives the decimal representation of \(\pi\) as \(3.141592654\text{.}\) This is not the exact value of \(\pi\text{,}\) but for most calculations it is quite adequate.

Some \(n\)th roots are rational numbers and some are irrational numbers. For example,

\begin{equation*} \sqrt{49},~~~ \sqrt[3]{\frac{27}{8}}, ~~\text{ and } ~~81^{1/4} \end{equation*}

are rational numbers because they are equal to \(7, \dfrac{3}{2},\) and \(3\text{,}\) respectively. On the other hand,

\begin{equation*} \sqrt{5}, ~~~\sqrt[3]{54}, ~~\text{ and } ~~7^{1/5} \end{equation*}

are irrational numbers. We can use a calculator to obtain decimal approximations for each of these numbers:

\begin{equation*} \sqrt{5}\approx 2.236, ~~~\sqrt[3]{54}\approx 3.826, ~~~\text{ and }~~~ 7^{1/5}\approx 1.476 \end{equation*}

The subsets of the real numbers are related as shown in Figure A.113. Every natural number is also a whole number, every whole number is an integer, every integer is a rational number, and every rational number is real. Also, every real number is either rational or irrational.

real numbers
Figure A.113
Example A.114
  1. \(2\) is a natural number, a whole number, an integer, a rational number, and a real number.

  2. \(\sqrt{15}\) is an irrational number and a real number.

  3. The number \(\pi\text{,}\) whose decimal representation begins \(3.14159\ldots\) is irrational and real.

  4. \(3.14159\) is a rational and real number (which is close but not exactly equal to \(\pi\)).

Subsection Properties of the Real Numbers

The real numbers have several useful properties governing the operations of addition and multiplication. If \(a\text{,}\) \(b\text{,}\) and \(c\) represent real numbers, then each of the following equations is true:

  • \(\begin{aligned}[t] \amp a + b = b + a \amp\amp\hphantom{blankblank}\blert{\text{Commutative properties}}\\ \amp ab = ba \end{aligned}\)

  • \(\begin{aligned}[t] \amp (a + b)+c = a+(b+c) \amp\amp\blert{\text{Associative properties}}\\ \amp (ab) = a(bc) \end{aligned}\)

  • \(\begin{aligned}[t] \amp a(b+c) = ab+ac \amp\amp\hphantom{blank0}\blert{\text{Distributive property}} \end{aligned}\)

  • \(\begin{aligned}[t] \amp a + 0 = a \amp\amp\hphantom{blankblank0000}\blert{\text{Identity properties}}\\ \amp a\cdot 1 = a \end{aligned}\)

These properties do not mention subtraction or division. But we can define subtraction and division in terms of addition and multiplication. For example, we can define the difference \(a-b\) as follows:

\begin{equation*} a - b = a + (-b) \end{equation*}

where \(-b\text{,}\) the additive inverse (or opposite) of \(b\text{,}\) is the number that satisfies

\begin{equation*} b + (-b) = 0 \end{equation*}

Similarly, we can define the quotient \(\dfrac{a}{b}\text{:}\)

\begin{equation*} \frac{a}{b}=a\left(\frac{1}{b} \right) \hphantom{blank} (b\ne 0) \end{equation*}

where \(\dfrac{1}{b} \text{,}\) the multiplicative inverse (or reciprocal) of \(b\text{,}\) is the number that satisfies

\begin{equation*} b\cdot\frac{1}{b}=1 \hphantom{blank} (b\ne 0) \end{equation*}

Division by zero is not defined.

Example A.115

Use the commutative and associative laws to simplify the computations.

  1. \(24 + 18 + 6\)

  2. \(4 \cdot 27\cdot 25\)

Solution
  1. Apply the commutative law of addition.

    \begin{align*} 24 + 18 + 6 \amp = (24 + 6) + 18\\ \amp = 30 + 18 = 48 \end{align*}
  2. Apply the commutative law of multiplication.

    \begin{align*} 4 \cdot 27\cdot 25 \amp = (4\cdot 25)\cdot 27\\ \amp = 100 \cdot 27 = 2700 \end{align*}

Subsection Order Properties of the Real Numbers

Real numbers obey properties about order, that is, properties about inequalities. The familiar inequality symbols, \(\lt\) and \(\gt\text{,}\) have the following properties:

  • If \(a\) and \(b\) are any real numbers, then one of three things is true:

    \begin{equation*} a \lt b, ~~\text{ or }~~ a\gt b, ~~\text{ or }~~ a = b \end{equation*}
  • (Transitive property) For real numbers \(a\text{,}\) \(b\text{,}\) and \(c\text{,}\)

    \begin{equation*} \text{if } a \lt b ~\text{ and }~ b \lt c, ~\text{ then }~ a \lt c \end{equation*}

We also have three properties that are useful for solving inequalities:

  • If \(a\lt b\text{,}\) then \(a+c\lt b+c\text{.}\)

  • If \(a\lt b\) and \(c\gt 0\text{,}\) then \(ac \lt bc\text{.}\)

  • If \(a\lt b\) and \(c\lt 0\text{,}\) then \(ac\gt bc\text{.}\)

Example A.116
  1. If \(x\lt y\) and \(y\lt -2\text{,}\) then \(x\lt -2\)

  2. \(\pi \lt 3.1416\text{,}\) so \(10\pi\lt 31.416\text{.}\)

  3. \(\dfrac{1}{3}\gt 0.33\text{,}\) so \(-\dfrac{1}{3}\lt -0.33.\)

Subsection Section Summary

Subsubsection Vocabulary

Look up the definitions of new terms in the Glossary.

  • Real number

  • Multiplicative inverse

  • Additive inverse

  • Distributive property

  • Whole number

  • Natural number

  • Reciprocal

  • Opposite

  • Irrational number

  • Integers

  • Counting number

  • Transitive property

  • Commutative property

  • Terminating decimal

  • Rational number

  • Identity property

  • Associative property

  • Repeater bar

Subsubsection SKILLS

Practice each skill in the exercises listed.

  1. Identify types of numbers: #1–12

  2. Write the decimal form of a fraction: #13–20

  3. Use the properties governing arithmetic operations: #21–40

  4. Use the properties of order: #41–46

Subsection Exercises A.13

For Problems 1-12, name the subsets of the real numbers to which the number belongs

1

\(-\dfrac{5}{8} \)

2

\(137\)

3

\(\sqrt{8} \)

4

\(2.71828\ldots\)

5

\(-36 \)

6

\(\sqrt{49} \)

7

\(0 \)

8

\(0.0\overline{357} \)

9

\(13\overline{289} \)

10

\(\sqrt{\dfrac{4}{9}} \)

11

\(2\pi \)

12

\(\dfrac{13}{7} \)

For Problems 13-20, write the rational number in decimal form. Does the decimal terminate or does it repeat a pattern?

13

\(\dfrac{3}{8}\)

14

\(\dfrac{5}{6}\)

15

\(\dfrac{2}{7}\)

16

\(\dfrac{43}{11}\)

17

\(\dfrac{7}{16}\)

18

\(\dfrac{5}{12}\)

19

\(\dfrac{11}{13}\)

20

\(\dfrac{25}{6}\)

For Problems 21-30, fill in the blank according to the indicated property.

21

Commutative property

\(7+10=10+ \underline{\hspace{2.727272727272727em}} \)

22

Associative property

\((6\cdot 4)\cdot 3 = 6\cdot (4\cdot \underline{\hspace{2.727272727272727em}} )\)

23

Associative property

\((3+6)+9= \underline{\hspace{2.727272727272727em}}+(6+9) \)

24

Commutative property

\((8\cdot 12) = \underline{\hspace{2.727272727272727em}}\cdot 8 \)

25

Commutative property

\(36\cdot 147= \underline{\hspace{2.727272727272727em}}\cdot 36 \)

26

Commutative property

\(13+87=87+ \underline{\hspace{2.727272727272727em}} \)

27

Associative property

\((17\cdot 2)\cdot 5 =17\cdot (\underline{\hspace{2.727272727272727em}}\cdot \underline{\hspace{2.727272727272727em}}) \)

28

Associative property

\((44+12)+8=44+ (\underline{\hspace{1.818181818181818em}}+\underline{\hspace{1.818181818181818em}}) \)

29

Commutative property

\((5+9)+4=(9+\underline{\hspace{2.727272727272727em}})+4 \)

30

Commutative property

\((8\cdot 9)\cdot 3= (9\cdot \underline{\hspace{2.727272727272727em}})\cdot 3 \)

For Problems 31-40, use the commutative and associative properties to compute mentally.

31

\(47+28+3\)

32

\(12+147+8\)

33

\(26+37+3+4\)

34

\(55+32+5+8\)

35

\(2\cdot 7\cdot 5\)

36

\(15\cdot 6\cdot 2\)

37

\(50\cdot 13\cdot 2\)

38

\(4\cdot 26\cdot 25\)

39

\(4\cdot 6\cdot 5\cdot 5\)

40

\(8\cdot 8\cdot 5\cdot 5\)

For Problems 41-46, fill in the blank with the correct symbol: <, >, or \(=\text{.}\)

41

\(-0.667 \underline{\hspace{2.727272727272727em}} ~{-\dfrac{2}{3}}\)

42

\(\sqrt{2} \underline{\hspace{2.727272727272727em}} ~ 1.4\)

43

If \(x\gt 8\text{,}\) then \(x-7 \underline{\hspace{1.818181818181818em}} ~1.\)

44

If \(x\lt {-6}\text{,}\) then \(x-6 \underline{\hspace{1.818181818181818em}} ~{-12}.\)

45

If \(x\gt -2\text{,}\) then \(-9x \underline{\hspace{1.818181818181818em}} ~18.\)

46

If \(x\lt -4\text{,}\) then \(3x \underline{\hspace{1.818181818181818em}} ~{-12}.\)