1.1 Real Numbers

---

1.1.1 Learning Objectives

By the end of this section, you should be able to:

• classify natural numbers, integers, rational numbers, irrational numbers, and real numbers;
• identify rational and irrational numbers from their decimal representations;
• apply the commutative, associative, and distributive properties;
• perform operations with negatives and fractions;
• describe sets using inequalities and interval notation; and
• interpret absolute value as distance on the real number line.

---

1.1.2 The Real Number System

The real numbers are all the numbers represented by points on a number line. They contain several important subsets.

Set	Symbol	Description	Examples
Natural numbers	\(\mathbb{N}\)	Counting numbers	\(1,2,3,\ldots\)
Integers	\(\mathbb{Z}\)	Natural numbers, their negatives, and zero	\(\ldots,-2,-1,0,1,2,\ldots\)
Rational numbers	\(\mathbb{Q}\)	Numbers of the form \(\frac{m}{n}\), where \(m,n\in\mathbb{Z}\) and \(n\ne0\)	\(-5,\frac{3}{8},0.4\)
Irrational numbers		Real numbers that are not ratios of integers	\(\sqrt{2},\sqrt{7},\pi\)
Real numbers	\(\mathbb{R}\)	All rational and irrational numbers	Every number above

These sets are nested:

\[ \mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}. \]

For example, every integer \(k\) is rational because \(k=\frac{k}{1}\).

Definition 1.1 Decimal Representations

• A rational number has a decimal that terminates or eventually repeats.
• An irrational number has a decimal that neither terminates nor repeats in a fixed pattern.

For example,

\[ \frac{3}{8}=0.375, \qquad \frac{2}{11}=0.181818\ldots \]

are rational, while \(\sqrt{2}=1.414213\ldots\) is irrational.

Stopping a decimal after finitely many places gives an approximation. The notation

\[ \pi\approx3.142 \]

means that \(3.142\) is close to, but not exactly equal to, \(\pi\).

Example 1.1 Classifying Numbers

Classify \(12\), \(-3\), \(\frac59\), and \(\sqrt6\) using the smallest appropriate set.

Solution

• \(12\) is natural.
• \(-3\) is an integer.
• \(\frac59\) is rational.
• \(\sqrt6\) is irrational because \(6\) is not a perfect square.

Exercise 1.1 Classify each number using the smallest appropriate set:

\[ -9,\qquad 0,\qquad \frac{7}{12},\qquad \sqrt{49},\qquad \sqrt{10}. \]

Show answer

Answers: integer, integer, rational, natural, irrational.

Exercise 1.2 Classify \(0.125\), \(-14\), and \(\sqrt{13}\).

Show answer

Answer: rational, integer, irrational

---

1.1.3 Properties of Real Numbers

The properties of real numbers justify many familiar algebraic steps.

Property	General form	Meaning
Commutative property of addition	\(a+b=b+a\)	The order of addition may change.
Commutative property of multiplication	\(ab=ba\)	The order of multiplication may change.
Associative property of addition	\((a+b)+c=a+(b+c)\)	The grouping of addition may change.
Associative property of multiplication	\((ab)c=a(bc)\)	The grouping of multiplication may change.
Distributive property	\(a(b+c)=ab+ac\)	Multiplication distributes across addition.

Exercise 1.3 Name the property illustrated by \(4(x+y)=4x+4y\).

Show answer

Answer: distributive property

Zero is the additive identity, and \(1\) is the multiplicative identity:

\[ a+0=a,\qquad a\cdot1=a. \]

The additive inverse of \(a\) is \(-a\). The multiplicative inverse of a nonzero number \(a\) is \(\frac1a\):

\[ a+(-a)=0,\qquad a\cdot\frac1a=1. \]

Subtraction means adding an opposite, and division means multiplying by a reciprocal:

\[ a-b=a+(-b), \qquad \frac{a}{b}=a\cdot\frac1b\quad(b\ne0). \]

Division by zero is undefined.

The expression \(-a\) means “the opposite of \(a\).” It is not necessarily negative. If \(a=-5\), then \(-a=5\).

Useful rules for negatives include

\[ -(-a)=a,\qquad (-a)b=-(ab),\qquad (-a)(-b)=ab, \]

and

\[ -(a+b)=-a-b. \]

Example 1.2 Applying Real Number Properties

Simplify:

\[ 5(2x-3)+4(x+1). \]

Solution

\[ 5(2x-3)+4(x+1) =10x-15+4x+4 =\boxed{14x-11}. \]

Exercise 1.4 Simplify \(-[3x-(2x-5)]\).

Show answer

Answer: \(-x-5\)

Exercise 1.5 Simplify:

\[ 3(2x-5)-2(x+4). \]

Show answer

Answer: \(4x-23\)

---

1.1.4 Fractions

For nonzero denominators,

\[ \frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}, \qquad \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}, \]

\[ \frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}, \qquad \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}. \]

When adding fractions, a least common denominator often makes the arithmetic easier. Cancellation is valid only for factors of the entire numerator and denominator.

Example 1.3 Adding Fractions

Evaluate:

\[ \frac5{18}+\frac7{24}. \]

Solution

The least common denominator is \(72\):

\[ \frac5{18}+\frac7{24} =\frac{20}{72}+\frac{21}{72} =\boxed{\frac{41}{72}}. \]

Exercise 1.6 Evaluate \(\frac3{10}+\frac5{12}\).

Show answer

Answer: \(\frac{43}{60}\)

---

1.1.5 The Real Number Line

On a real number line, values increase from left to right. The statement \(a<b\) means that \(a\) lies to the left of \(b\). Positive numbers lie to the right of \(0\), and negative numbers lie to the left.

---

1.1.6 Sets and Intervals

Definition 1.2 A set is a collection of objects called elements.

We write \(x\in S\) when \(x\) belongs to \(S\), and \(x\notin S\) when it does not.

A set may be listed:

\[ A=\{1,2,3,4\}, \]

or written in set-builder notation:

\[ A=\{x\in\mathbb{Z}\mid 1\le x\le4\}. \]

Definition 1.3 For sets \(S\) and \(T\):

• \(S\cup T\), the union, contains elements in either set or in both.
• \(S\cap T\), the intersection, contains only elements common to both sets.
• \(\varnothing\), the empty set, contains no elements.

Example 1.4 Union and Intersection

Let

\[ S=\{1,3,5,7\}, \qquad T=\{3,4,5,6\}. \]

Then

\[ S\cup T=\{1,3,4,5,6,7\}, \qquad S\cap T=\{3,5\}. \]

Intervals describe continuous portions of the real line.

Interval	Inequality	Endpoints
\((a,b)\)	\(a<x<b\)	Neither included
\([a,b]\)	\(a\le x\le b\)	Both included
\([a,b)\)	\(a\le x<b\)	Left included
\((a,b]\)	\(a<x\le b\)	Right included
\((a,\infty)\)	\(x>a\)	Extends right
\((-\infty,b]\)	\(x\le b\)	Extends left

Infinity is not a real number, so an infinite endpoint always uses a parenthesis.

Exercise 1.7 Write \(-2\le x<5\) in interval notation.

Show answer

Answer: \([-2,5)\)

Exercise 1.8 Write \(x>-4\) in interval notation.

Show answer

Answer: \((-4,\infty)\)

Example 1.5 Intersecting Intervals

Find

\[ (-3,4]\cap[1,7). \]

Solution

The values common to both intervals run from \(1\) through \(4\), including both endpoints:

\[ \boxed{[1,4]}. \]

Exercise 1.9 Find \([-2,6)\cap(3,9]\).

Show answer

Answer: \((3,6)\)

---

1.1.7 Absolute Value and Distance

The absolute value of \(a\), written \(|a|\), is the distance from \(a\) to \(0\):

\[ |a|= \begin{cases} a, & a\ge0,\\ -a, & a<0. \end{cases} \]

Thus \(|a|\ge0\) for every real number \(a\). Important properties include

\[ |-a|=|a|,\qquad |ab|=|a||b|, \qquad \left|\frac{a}{b}\right|=\frac{|a|}{|b|}\quad(b\ne0). \]

The distance between \(a\) and \(b\) is

\[ d(a,b)=|b-a|. \]

Example 1.6 Distance

Find the distance between \(-7\) and \(5\).

Solution

\[ d(-7,5)=|5-(-7)|=|12|=\boxed{12}. \]

Exercise 1.10 Find the distance between \(-4.5\) and \(2\).

Show answer

Answer: \(6.5\)

---

1.1.8 Conceptual Takeaways

• Real numbers consist of rational and irrational numbers.
• Natural numbers, integers, and rational numbers form nested subsets.
• Terminating and repeating decimals are rational.
• Real-number properties justify the steps used to rewrite algebraic expressions.
• Parentheses exclude finite endpoints, while brackets include them.
• Union combines sets; intersection keeps their common elements.
• Absolute value measures distance and is never negative.

---

1.1.9 Skills You Should Be Able to Do

You should be able to:

• classify real numbers;
• distinguish rational and irrational decimal representations;
• apply the commutative, associative, and distributive properties;
• simplify expressions involving negative signs and fractions;
• compare and order real numbers;
• find unions and intersections;
• convert among inequalities, interval notation, and number-line graphs;
• evaluate absolute values; and
• calculate distances on the real number line.

---

1.1.10 Practice Problems with Solutions

The problems progress from basic skills to multi-step reasoning.

1. Classify each number using the smallest appropriate set: \[ 8,\qquad -11,\qquad \frac4{15},\qquad \sqrt{81},\qquad \sqrt{15}. \]

   Show Solution

   \[ 8\in\mathbb{N},\quad -11\in\mathbb{Z},\quad \frac4{15}\in\mathbb{Q}. \]

   Since \(\sqrt{81}=9\), it is natural. Since \(15\) is not a perfect square, \(\sqrt{15}\) is irrational.

2. State the property:

   1. \(9+u=u+9\)
      
   2. \((2x)y=2(xy)\)
      
   3. \(5(a-b)=5a-5b\)

   Show Solution

   1. Commutative property of addition
      
   2. Associative property of multiplication
      
   3. Distributive property

3. Simplify: \[ -2(3x-4)+5(x+1). \]

   Show Solution

   \[ -2(3x-4)+5(x+1) =-6x+8+5x+5 =\boxed{-x+13}. \]

4. Evaluate: \[ \frac7{20}-\frac5{18}. \]

   Show Solution

   \[ \frac7{20}-\frac5{18} =\frac{63}{180}-\frac{50}{180} =\boxed{\frac{13}{180}}. \]

5. If \(A=\{1,2,4,8\}\) and \(B=\{2,3,5,8\}\), find \(A\cup B\) and \(A\cap B\).

   Show Solution

   \[ A\cup B=\{1,2,3,4,5,8\}, \qquad A\cap B=\{2,8\}. \]

6. Write in interval notation:

   1. \(-3<x\le6\)
      
   2. \(x\ge2\)
      
   3. \(x<-1\)

   Show Solution

   1. \((-3,6]\)
      
   2. \([2,\infty)\)
      
   3. \((-\infty,-1)\)

7. Find \([-5,2)\cap(-1,6]\).

   Show Solution

   \[ \boxed{(-1,2)}. \]

8. Find \((-\infty,3]\cup(1,7)\).

   Show Solution

   Together, the intervals contain every number less than \(7\):

   \[ \boxed{(-\infty,7)}. \]

9. Find the distance between \(-\frac52\) and \(\frac74\).

   Show Solution

   \[ \left|\frac74-\left(-\frac52\right)\right| =\left|\frac74+\frac{10}{4}\right| =\boxed{\frac{17}{4}}. \]

10. A point \(x\) is exactly \(6\) units from \(-2\). Write an absolute-value equation and find \(x\).

    Show Solution

    \[ |x-(-2)|=6 \quad\Longrightarrow\quad |x+2|=6. \]

    Therefore \(x+2=6\) or \(x+2=-6\), giving

    \[ \boxed{x=4\text{ or }x=-8}. \]

---