Integers and Introduction to Integers

1 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 2 Integers and Introduction to Integers Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Introduction to Integers Martin-Gay, Prealgebra, 6ed 3 3 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Numbers greater than 0 are called positive numbers. Numbers less than 0 are called negative numbers. negative numbers zero positive numbers 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 Positive and Negative Numbers Martin-Gay, Prealgebra, 6ed 4 4 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Some signed numbers are Integers . negative numbers zero positive numbers 6 5 4 3 2 1 0 1 2 3 4 5 6 The Integers are { .., 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, ..} Integers Martin-Gay, Prealgebra, 6ed 5 5 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. 3 indicates negative three. 3 and + 3 both indicate positive three.

2 The number 0 is neither positive nor negative. negative numbers zero positive numbers 6 5 4 3 2 1 0 1 2 3 4 5 6 Negative and Positive Numbers Martin-Gay, Prealgebra, 6ed 6 6 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. We compare Integers just as we compare whole numbers. For any two numbers graphed on a number line, the number to the right is the greater number and the number to the left is the smaller number. < means is less than > means is greater than Comparing Integers Martin-Gay, Prealgebra, 6ed 7 7 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. The graph of 5 is to the left of 3, so 5 is less than 3, written as 5 < 3 . We can also write 3 > 5. Since 3 is to the right of 5, 3 is greater than 5. 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 6 5 4 3 2 1 0 1 2 3 4 5 6 Graphs of Integers Martin-Gay, Prealgebra, 6ed 8 8 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. The absolute value of a number is the number s distance from 0 on the number line.

3 The symbol for absolute value is | |. 2is 2 because 2 is 2 units from 0. 6 5 4 3 2 1 0 1 2 3 4 5 6 is 2 because 2 is 2 units from 0. 2 6 5 4 3 2 1 0 1 2 3 4 5 6 Absolute Value Martin-Gay, Prealgebra, 6ed 9 9 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Since the absolute value of a number is that number s distance from 0, the absolute value of a number is always 0 or positive. It is never negative. 0 = 06 = 6 zero a positive number Helpful Hint Martin-Gay, Prealgebra, 6ed 10 10 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Two numbers that are the same distance from 0 on the number line but are on the opposite sides of 0 are called opposites. 5 units 5 units 5 and 5 are opposites. 6 5 4 3 2 1 0 1 2 3 4 5 6 Opposite Numbers Martin-Gay, Prealgebra, 6ed 11 11 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. 5 is the opposite of 5 and 5 is the opposite of 5. The opposite of 4 is 4 is written as (4) = 4 The opposite of 4 is 4 is written as ( 4) = 4 ( 4) = 4 If a is a number, then ( a) = a.

4 Opposite Numbers Martin-Gay, Prealgebra, 6ed 12 12 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Remember that 0 is neither positive nor negative. Therefore, the opposite of 0 is 0. Helpful Hint Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Adding Integers Martin-Gay, Prealgebra, 6ed 14 14 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 Adding Two Numbers with the Same Sign 2 + 3 = 2 Start End 2 + ( 3) = 2 3 Start End 3 6 5 4 3 2 1 0 1 2 3 4 5 6 6 5 4 3 2 1 0 1 2 3 4 5 6 5 5 Martin-Gay, Prealgebra, 6ed 15 15 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Adding Two Numbers with the Same Sign Step 1: Add their absolute values. Step 2: Use their common sign as the sign of the sum. Examples: 3 + ( 5) = 8 5 + 2 = 7 Martin-Gay, Prealgebra, 6ed 16 16 Copyright 2011 Pearson Education, Inc.

5 Publishing as Prentice Hall. Adding Two Numbers with Different Signs 2 + ( 3) = 2 3 2 + 3 = 2 3 Start End Start End 6 5 4 3 2 1 0 1 2 3 4 5 6 6 5 4 3 2 1 0 1 2 3 4 5 6 1 1 Martin-Gay, Prealgebra, 6ed 17 17 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Step 1: Find the larger absolute value minus the smaller absolute value. Step 2: Use the sign of the number with the larger absolute value as the sign of the sum. Examples: 4 + 5 = 1 6 + ( 8) = 2 Adding Two Numbers with Different Signs Martin-Gay, Prealgebra, 6ed 18 18 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. If a is a number, then a is its opposite. a + ( a) = 0 a + a = 0 The sum of a number and its opposite is 0. Helpful Hint Martin-Gay, Prealgebra, 6ed 19 19 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Don t forget that addition is commutative and associative. In other words, numbers may be added in any order.

6 Helpful Hint Martin-Gay, Prealgebra, 6ed 20 20 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Evaluate x + y for x = 5 and y = 9. x + y = ( ) + ( ) Replace x with 5 and y with 9 in x + y. 5 9 = 4 Evaluating Algebraic Expressions Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Subtracting Integers Martin-Gay, Prealgebra, 6ed 22 22 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. To subtract Integers , rewrite the subtraction problem as an addition problem. Study the examples below. 9 5 = 4 9 + ( 5) = 4 equal 4, we can say 9 5 = 9 + ( 5) = 4 Since both expressions Subtracting Integers Martin-Gay, Prealgebra, 6ed 23 23 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Subtracting Two Numbers If a and b are numbers, then a b = a + ( b). To subtract two numbers, add the first number to the opposite (called additive inverse) of the second number. Martin-Gay, Prealgebra, 6ed 24 24 Copyright 2011 Pearson Education, Inc.

7 Publishing as Prentice Hall. subtraction = first number + opposite of second number 7 4 = 7 + ( 4) = 3 5 3 = 5 + ( 3) = 8 3 ( 6) = 3 + 6 = 9 8 ( 2) = 8 + 2 = 6 Subtracting Two Numbers Martin-Gay, Prealgebra, 6ed 25 25 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. If a problem involves adding or subtracting more than two Integers , rewrite differences as sums and add. By applying the associative and commutative properties, add the numbers in any order. 9 3 + ( 5) ( 7) = 9 + ( 3) + ( 5) + 7 6 + ( 5) + 7 1 + 7 8 Adding and Subtracting Integers Martin-Gay, Prealgebra, 6ed 26 26 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Evaluate x y for x = 6 and y = 8. x y Replace x with 6 and y with 8 in x y. = ( ) ( ) 6 8 = 14 = ( ) + ( ) 6 8 Evaluating Algebraic Expressions Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiplying and Dividing Integers Martin-Gay, Prealgebra, 6ed 28 28 Copyright 2011 Pearson Education, Inc.

8 Publishing as Prentice Hall. Consider the following pattern of products. 3 5 = 15 2 5 = 10 1 5 = 5 0 5 = 0 This pattern continues as follows. 1 5 = - 5 2 5 = - 10 3 5 = - 15 This suggests that the product of a negative number and a positive number is a negative number. First factor decreases by 1 each time. Product decreases by 5 each time. Multiplying Integers Martin-Gay, Prealgebra, 6ed 29 29 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. 2 ( 5) = 10 0 ( 5) = 0 This pattern continues as follows. 1 ( 5) = 5 2 ( 5) = 10 3 ( 5) = 15 This suggests that the product of two negative numbers is a positive number. Product increases by 5 each time. 1 ( 5) = 5 Observe the following pattern. Multiplying Integers Martin-Gay, Prealgebra, 6ed 30 30 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiplying Integers The product of two numbers having the same sign is a positive number. 2 ( 4) = 8 2 4 = 8 2 ( 4) = 8 2 4 = 8 The product of two numbers having different signs is a negative number.

9 Martin-Gay, Prealgebra, 6ed 31 31 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiplying Integers Product of Like Signs ( + )( + ) = + ( )( + ) = ( + )( ) = Product of Different Signs ( )( ) = + Martin-Gay, Prealgebra, 6ed 32 32 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. If we let ( ) represent a negative number and ( + ) represent a positive number, then ( ) ( ) = ( + ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) ( ) = ( + ) ( ) ( ) ( ) ( ) ( ) = ( ) The product of an even number of negative numbers is a positive result. The product of an odd number of negative numbers is a negative result. Helpful Hint Martin-Gay, Prealgebra, 6ed 33 33 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Division of Integers is related to multiplication of Integers . 3 3 2 6 = = because 6 2 = = 3 3 2 6 because 6 2 3 3 ( 2) 6 = = because 6 2 = 3 6 because = 3 ( 2) 6 2 Dividing Integers Martin-Gay, Prealgebra, 6ed 34 34 Copyright 2011 Pearson Education, Inc.

10 Publishing as Prentice Hall. Dividing Integers The quotient of two numbers having the same sign is a positive number. 12 ( 4 ) = 3 12 4 = 3 12 4 = 3 12 ( 4) = 3 The quotient of two numbers having different signs is a negative number. Martin-Gay, Prealgebra, 6ed 35 35 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Dividing Numbers Quotient of Like Signs ()() ()() Quotient of Different Signs ()() ()() Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Order of Operations Martin-Gay, Prealgebra, 6ed 37 37 Copyright 2011 Pearson Education, Inc. Publishing as Prentice Hall. Order of Operations 1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars, starting with the innermost set. 2. Evaluate any expressions with exponents. 3. Multiply or divide in order from left to right. 4. Add or subtract in order from left to right. Martin-Gay, Prealgebra, 6ed 38 38 Copyright 2011 Pearson Education, Inc.