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ch05MGF1106

ch05MGF1106 - Chapter 5 Number Theory and the Real Number...

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Slide 5 - 1 Chapter 5 Number Theory and the Real Number System

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Slide 5 - 2 5.2 The Integers
Slide 5 - 3 Number Theory (from Chapter 5.1) The study of numbers and their properties. The numbers we use to count are called natural numbers, , or counting numbers. ¥ = {1,2,3,4,5,...} ¥

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Slide 5 - 4 Whole Numbers The set of whole numbers contains the set of natural numbers and the number 0. Whole numbers = {0,1,2,3,4,…}
Slide 5 - 5 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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Slide 5 - 6 Inequality A number line can be used to determine the greater (or lesser) of two integers. Numbers on a number line increase from left to right. We us inequality symbols to indicate the greater (or lesser) of two integers. < “less than” > “greater than”
Slide 5 - 7 Writing an Inequality Insert either > or < in the box between the paired numbers to make the statement correct. a) - 3 - 1 b) - 9 - 7 - 3 < - 1 - 9 < - 7 c) 0 - 4 d) 6 8 0 > - 4 6 < 8

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Slide 5 - 8 Addition of integers Add the following integers. 3 + -6 = -3 + -7 = -7 + 3 = 8 + (-12) =
Slide 5 - 9 Addition of integers Add the following integers. 3 + -6 = -3 -3 + -7 = -10 -7 + 3 = -4 8 + (-12) = -4

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Slide 5 - 10 Subtraction of Integers a b = a + ( - b ) Evaluate: a) –7 – 3 = –7 + (–3) = –10 b) –7 – (–3) = –7 + 3 = –4
Slide 5 - 11 Subtraction of integers Rule: a – b = a + (-b) & -(-a) = a 1. -6 – 8 = 2. 6 – (-8) = 3. 6 – 8 = 4. -6 – (-8) =

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Slide 5 - 12 Subtraction of integers Rule: a – b = a + (-b) & -(-a) = a 1. -6 – 8 = -14 2. 6 – (-8) = 14 3. 6 – 8 = -2 4. -6 – (-8) = 2
Slide 5 - 13 Properties Multiplication Property of Zero Division For any a , b , and c where b 0, means that c b = a . × = × = 0 0 0 a a a b = c

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Slide 5 - 14 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 .
Slide 5 - 15 Examples Evaluate: a) (3)( - 4) b) ( - 7)( - 5) c) 8 • 7 d) ( - 5)(8) Solution: a) (3)( - 4) = - 12b) ( - 7)( - 5) = 35 c) 8 • 7 = 56 d) ( - 5)(8) = - 40

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Slide 5 - 16 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 .
Slide 5 - 17 Example Evaluate: a) b) c) d) Solution: a) b) c) d) 72 9 - 72 9 - 72 - 8 72 - 8 = 72 9 8 - = - 72 9 8 - - = 72 8 9 = - - 72 8 9

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Slide 5 - 18 5.3 The Rational Numbers
Slide 5 - 19 The Rational Numbers The set of rational numbers , denoted by Q , is the set of all numbers of the form p/q , where p and q are integers and q 0. The following are examples of rational numbers: 1 3 , 3 4 , - 7 8 , 1 2 3 , 2, 0, 15 7

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Slide 5 - 20 Fractions Fractions are numbers such as: The numerator is the number above the fraction line.
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ch05MGF1106 - Chapter 5 Number Theory and the Real Number...

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