Section 5.2 The Integers

What You Will Learn Integers Adding Integers Subtracting Integers Multiplying Integers Dividing Integers

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, …}

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

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.

Real Number Line –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 Positive integers extend to the right from zero, equally spaced Negative integers extend to the left from zero, using the same spacing Contains integers and all other real numbers that are not integers Line continues indefinitely in both directions

Inequality > is greater than < is less than –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 > is greater than < is less than On the number line, numbers increase from left to right 2 is to the left of 3 2 < 3 or 3 > 2 Symbol always points to the smaller number

Example 1: Writing an Inequality Insert either > or < in the shaded area between the paired numbers to make the statement correct. a) –7 8 < b) –7 –8 > d) –7 –4 < c) –7 0 <

Addition of Integers Represented geometrically using a number line Begin at 0 on the number line Represent the first addend by an arrow starting at 0 Draw the arrow to the right if the addend is positive Draw the arrow to the left if the addend is negative

Addition of Integers From the tip of the first arrow, draw a second arrow to represent the second addend Draw 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

Example 2: Adding Integers Evaluate using a number line. a) 3 + (–5) • • • –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 3 + (–5) = –2

Example 1: Adding Integers Evaluate using a number line. b) –1 + (–4) • • • –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 –1 + (–4) = –5

Example 1: Adding Integers Evaluate using a number line. c) –6 + 4 • • • –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 –6 + 4 = –2

Example 1: Adding Integers Evaluate using a number line. d) 3 + (–3) • • –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 3 + (–3) = 0

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.

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

Example 4: Subtracting: Adding the Inverse Evaluate. c) 7 – (–3) Solution 7 – (–3) = 7 + 3 = 10 d) 7 – 3 7 – 3 = 7 + (–3) = 4

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.

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

Example 6: Multiplying Integers Evaluate. a) 5 • 6 5 • 6 = 30 b) 5 • (–6) 5 • (–6)= –30 c) (–5) • 6 (–5) • 6 = –30 d) (–5) • (–6) (–5) • (–6) = 30

Division For any a, b, and c where b ≠ 0, means c • b = a.

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

Example 7: Dividing Integers Evaluate. = 7 = –7 = –7 = 7