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MAT 030/060 – Lesson 15 Integers (negative and positive whole numbers) Numbers greater than zero are called positive numbers. Numbers less than zero are called negative numbers. Zero is neither a negative nor a positive number. To represent a negative number, you must write a negative sign in front of the number: -12 -3 -217 If a number does not have a sign in front of it, it is assumed to be a positive number. Two numbers that are the same distance from zero on the number line but on opposite sides of zero are called opposites or additive inverses. -4 and 4 are opposites 17 is the additive inverse of -17 The absolute value of a number is the distance between zero and that number on the number line; therefore, the absolute value of a number is a positive number. The symbol for absolute value is | | |5| = 5 |−18| = 18 |33| = 33 |−112| = 112 If there is a negative sign before the absolute value sign, read “the opposite of” -|12| reads “the opposite of the absolute value of 12” which = -12 -|−27| reads “the opposite of the absolute value of -27 which = -27 Adding and subtracting integers Rule for Adding Two Integers To add integers with the same sign, add the numbers and keep the sign. To add integers with different signs, find the difference between the numbers, and attach the sign of number with the larger absolute value. Examples: 127 + 28 = 155 -6 + 13 = 7 -15 + (-67) = -82 The difference between 6 and 13 is 7 The larger absolute value of these two numbers is 13 Attach a positive sign to the answer (we don’t normally write a positive sign) __________________________________________________________________________________________ 162 + (-247) = -85 The difference between 162 and 247 is 85 The larger absolute value of these two numbers is 247 Attach a negative sign to the answer Practice: -162 + 98 = -2 + (-7) + 4 + (-6) = -154 + (-37) = -5 + (-2) + 9 + (-3) = Rule for Subtracting Two Integers To subtract two integers, change the subtraction sign to addition, Change the sign of the second number to its opposite and follow the rules for addition of integers Examples: 8 – 15 8 + (-15) Change the subtraction sign to addition and change the sign of the 15. 7 Follow rules for addition: the difference between 8 and 15 -7 The larger absolute value of the two numbers is 15, so attach – sign. __________________________________________________________________________________________ 13 – (-24) 13 + 24 Change the subtraction sign to addition and change the sign of the 24 37 Follow rules for addition __________________________________________________________________________________________ (-15) – 75 (-15) + (-75) Change the subtraction sign to addition and change the sign of the 75 - 90 Follow rules for addition Practice: (-12) – 8 = 3 – (-15) = -8 – 30 – (-12) -7 = 4 – (-3) – 12 – (-7) – 20 = Rule for Multiplying and Dividing Two Integers To multiply and divide numbers with the same sign, the answer is positive To multiply and divide numbers with different signs, the answer is negative Examples: 4 · 8 = 32 (-4)(-8) = 32 27 ÷ 9 = 3 -27 ÷ (-9) = 3 -4 · 8 = -32 (4)(-8) = -32 -27 ÷ 9 = -3 27 ÷ (-9) = -3 __________________________________________________________________________________________ Practice: (-3)(-5) = -42 · 62 = -7(-8)(9)(-2) = (-120) ÷ (-8) = -81 ÷ 3 = 112 ÷ 2 = -5(-4)(6)(-3) = 95 ÷ (-5) =