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Math 40 Prealgebra Section 2.1 – Introduction to Integers 2.1 Introduction to Integers Recall The Set of Natural Numbers 1, 2, 3, 4, 5, 6, ... The Set of Whole Numbers 0, 1, 2, 3, 4, 5, 6, ... The set of whole numbers includes zero while the set of natural numbers does not. One way to help you remember the difference between these two sets is to think of the letter o in the word “whole” as a zero. The Set of Integers ..., 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, ... The set of integers is the set of whole numbers combined with their opposites. There are no fractions or decimal numbers in the set of integers. Number Line Zero is also known as the origin. Zero is neither positive nor negative. Integers greater than zero are called positive integers. Integers less than zero are called negative integers. 1 2015 Carreon Math 40 Prealgebra Section 2.1 – Introduction to Integers Notation Positive integers can be written in two ways. 1) Write a positive sign in front of the integer: 5 “positive 5” 2) Do not write any sign: 5 is assumed to be “positive five” Negative integers are written with a negative sign in front of the integer and can be written in two ways. 1) Write a raised negative sign: —4 “negative 4” or “the opposite of 4” 2) Write the negative sign in the middle : 4 Graphing Integers Integers are sometimes represented as points on a number line. Example 1: Graph 3 and 3 on the number line below. Solution: To graph 3 on a number line, we draw a closed circle at its location on the number line. To graph 3 on a number line, we draw a closed circle at its location on the number line. You Try It 1: Graph 7, 5, 4, 0, and 8 on the number line below. 2 2015 Carreon Math 40 Prealgebra Section 2.1 – Introduction to Integers Comparing Integers Notation = means “is equal to” > means “greater than” < means “less than” Consider the graph of 5 and 8 : 5 is to the left of 8 so we say “ 5 is less than 8 ” and write 5 8 . 8 is to the right of 5 so we say “ 8 is greater than 5 ” and write 8 5 . Now consider the graph of 7 and 2 : 7 is to the left of 2 so we say “ 7 is less than 2 ” and write 7 2 . 2 is to the right of 7 so we say “ 2 is greater than 7 ” and write 2 7 . Example 2: Write < or > between each pair of integers to make a true statement. a) 0 b) 7 9 6 c) 3 Solution: a) 0 is to the left of 9 on the number line, so use a less than symbol: 0 8 b) 7 is to the left of 6 on the number line, so use a less than symbol: 7 9 c) 3 is to the right of 8 on the number line, so use a greater than symbol: 3 6 8 You Try It 2: Write < or > between each pair of integers to make a true statement. a) 6 9 b) 0 2 c) 1 4 3 2015 Carreon Math 40 Prealgebra Section 2.1 – Introduction to Integers Absolute Value Absolute Value of an Integer The absolute value of an integer is its distance from 0 on the number line. Absolute value is indicated by enclosing the integer (or expression) using two vertical bars “the absolute value of 4 ” is written as 4 Example 3: Simplify each expression. a) 4 b) 7 c) 0 Solution: Each integer is enclosed in vertical bars. This means we find the absolute value of each integer. a) The integer 4 is 4 units from 0 on the number line so 4 4 . 4 units b) The integer 7 is 7 units from 0 on the number line so 7 7 . 7 units c) The integer 0 is 0 units from 0 on the number line so 0 0 . 0 units Note: Distance is never a negative quantity. This means that the absolute value of an integer can only be positive or zero. 4 2015 Carreon Math 40 Prealgebra Section 2.1 – Introduction to Integers You Try It 3: Simplify each expression. a) 1 b) 16 c) 20 Translating from English to Math Let’s go back and look at the graph of Example 1: Notice that 3 and 3 are on opposite sides of the origin. We say that 3 and 3 are opposites. Translating from English to Math There are two ways to verbally indicate that a negative sign is in front of the integer (or expression): 1) “the opposite of” 2) “negative” Example 4: Translate the following English phrases into a numerical expression and simplify if possible. a) the opposite of 5 b) the opposite of 4 c) negative seventeen d) the opposite of negative twelve Solution: English a) the opposite of 5 b) the opposite of 4 c) negative seventeen Expression 5 4 Math grammar: Use parentheses to avoid two symbols in a row. 17 12 d) the opposite of negative twelve Math grammar: Use parentheses to avoid two symbols in a row. Simplify 5 (already simplified) 4 The opposite of negative 4 is positive 4. 17 (already simplified) 12 The opposite of negative 12 is positive 12. 5 2015 Carreon Math 40 Prealgebra Section 2.1 – Introduction to Integers You Try It 4: Translate the following English phrases into a numerical expression and simplify if possible. Complete the table below. English Expression Simplify a) the opposite of 15 b) the opposite of 20 c) the opposite of negative one Example 5: Translate the following English phrases into a numerical expression and simplify if possible. a) the absolute value of 9 b) the opposite of the absolute value of 9 c) the opposite of the absolute value of 9 d) the opposite of 9 Solution: English a) the absolute value of 9 Expression Simplify 9 9 9 b) the opposite of the absolute value of nine 9 Find the absolute value of 9 first, then take the opposite. c) the opposite of the absolute value of 9 9 Find the absolute value of negative 9 first, then take the opposite. 9 9 d) the opposite of negative nine Math grammar: Use parentheses to avoid two symbols in a row. 9 The opposite of negative 12 is positive 12. You Try It 5: Translate the following English phrases into a numerical expression and simplify if possible. Complete the table below. English a) the opposite of the absolute value of fourteen Expression Simplify b) the opposite of 14 6 2015 Carreon