Section 5.2 The Integers

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 5.2

The Integers


What You Will Learn

IntegersAdding IntegersSubtracting IntegersMultiplying IntegersDividing Integers

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Number Theory

The study of numbers and their properties.The numbers we use to count are called counting numbers, or natural numbers, denoted by N.

N = {1, 2, 3, 4, 5, …}

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Whole Numbers

The set of whole numbers contains the set of natural numbers and the number 0.Whole numbers = {0, 1, 2, 3, 4,…}

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Integers

The set of integers consists of 0, the natural numbers, and the negative natural numbers. Integers= {…, –4, –3, –2, –1, 0, 1, 2, 3, 4,…}On a number line, the positive numbers extend to the right from zero; the negative numbers extend to the left from zero.

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Real Number Line

Positive integers extend to the right from zero, equally spacedNegative integers extend to the left from zero, using the same spacingContains integers and all other real numbers that are not integersLine continues indefinitely in both directions

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

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Inequality

> is greater than< is less thanOn the number line, numbers increase from left to right2 is to the left of 32 < 3 or 3 > 2Symbol always points to the smaller number

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6

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Example 1: Writing an InequalityInsert either > or < in the shaded area between the paired numbers to make the statement correct.

a) –7 8< b) –7 –8>

c) –7 0< d) –7 –4<

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Addition of IntegersRepresented geometrically using a number lineBegin at 0 on the number line Represent the first addend by an arrow starting at 0Draw the arrow to the right if the addend is positiveDraw the arrow to the left if the addend is negative

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Addition of IntegersFrom the tip of the first arrow, draw a second arrow to represent the second addendDraw the second arrow to the right (positive addend) or left (negative addend)Sum of the two integers is found at the tip of the second arrow

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Example 2: Adding IntegersEvaluate using a number line.

a) 3 + (–5)

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6• ••

3 + (–5) = –2

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b) –1 + (–4)

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6•••

–1 + (–4) = –5

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c) –6 + 4

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6•• •

–6 + 4 = –2

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d) 3 + (–3)

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6• •

3 + (–3) = 0

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Subtraction of Integers

Any subtraction problem can be rewritten as an addition problem.

a – b = a + (–b)

The rule for subtraction indicates that to subtract b from a, add the additive inverse of b to a.

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Example 4: Subtracting: Adding the InverseEvaluate.a) –7 – 3Solution–7 – 3 = –7 + (–3) = –10

b) –7 – (–3)Solution–7 – (–3) = –7 + 3 = –4

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Example 4: Subtracting: Adding the InverseEvaluate.c) 7 – (–3)Solution7 – (–3) = 7 + 3 = 10

d) 7 – 3Solution7 – 3 = 7 + (–3) = 4

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Try This

-6 – 2 6 – (8 + 6)

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Multiplication Property of Zero

a • 0 = 0 • a = 0

The multiplication property of zero is important in our discussion of multiplication of integers.It indicates that the product of 0 and any number is 0.

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Rules for MultiplicationThe product of two numbers with like signs (positive × positive ornegative × negative) is a positive number.

The product of two numbers with unlike signs (positive × negative or negative × positive) is a negative number.

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Example 6: Multiplying IntegersEvaluate.

a) 5 • 65 • 6 = 30

b) 5 • (–6)5 • (–6)= –30

c) (–5) • 6(–5) • 6 = –30

d) (–5) • (–6)(–5) • (–6) =

30

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Try this

(5)(-2) (-3)(2)(-4)

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Division

For any a, b, and c where b ≠ 0,

means c • b = a.

a

bc

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Rules for DivisionThe quotient of two numbers with like signs (positive ÷ positive ornegative ÷ negative) is a positive number.

The quotient of two numbers with unlike signs (positive ÷ negative or negative ÷ positive) is a negative number.

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Example 7: Dividing IntegersEvaluate.

= 7 a)

63

9

= –7 b)

63

9

= –7 c)

63

9

= 7 d)

63

9

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Homework

P. 227 # 6 – 69 (x3)

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Section 5.2 The Integers

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