Operating on Negative NumbersDate: 10/21/96 at 19:35:47 From: Jake Subject: Integers Hi Dr. Math, I have lots of problems with multiplying and dividing integers. Do you have any tricks for me? I can add and subtract integers, but multiplication and division are really hard for me. The negative and positive numbers give me a really hard time. Thank you. Date: 10/29/96 at 11:26:10 From: Doctor Lim Subject: Re: Integers Dear Jake, One important thing to remember is the rules of operation. Multiplication 1) plus x plus = plus (2) x (2) = 4 2) minus x minus = plus (-2) x (-2) = 4 3) minus x plus = minus (-2) x (2) = -4 4) plus x minus = minus (2) x (-2) = -4 Division 1) plus / plus = plus (2) / (2) = 1 2) minus / minus = plus (-2) / (-2) = 1 3) minus / plus = minus (-2) / (2) = - 1 4) plus / minus = minus (2) / (-2) = -1 __________ Let me explain further. 3) minus x plus = minus (-2) x (2) = -4 4) plus x minus = minus (2) x (-2) = -4 These two are exactly the same, just as 2 x 3 = 3 x 2 This means that for example in -6 x 6, you have added negative six to itself six times. You then get the answer of - 36. It also means that you borrowed $6 from your friend six times - so how much do you have to pay your friend? 6 x 6 which is 36. This $36 will have to come out of your next allowance, so you will have a minus $36. __________ 2) minus x minus = plus (-2) x (-2) = 4 To find -6 x -6, we'd have to add negative six to itself negative six times. If we add two negative sixes together (-6 x 2), we get -6 + -6 = -12. If we add one negative six together (-6 x 1), we get -6. If we add zero negative sixes together (-6 x 0), we get 0. Notice that we are increasing by sixes each time. If we add -1 negative sixes together (-6 x -1), we get -(-6) = 6. If we add -2 negative sixes together (-6 x -2), we get -(-6) + -(-6) = 12. You can see the pattern then, and how -6 x -6 = 36. __________ So in the operation of 8 - (-5) = 8 -1(-5) -1 x -5 To find -1 x -5, we'd have to add negative one to itself negative five times. If we add two negative fives together (-5 x 2), we get -5 + -5 = -10. If we add one negative five together (-5 x 1), we get -5. If we add zero negative five together (-5 x 0), we get 0. Notice that we are increasing by five each time. If we add -1 negative five together (-5 x -1), we get -(-5) = 5. 8 - (-5) = 8 + 5 = 13 __________ minus 1 x minus 5 = 5 8 - (-5) = 8 + 5 = 13 __________ In division of negative integers, it is much easier. In a thermometer, for example, the temperature can be read from -200 degrees to +200 degrees. If the temperature drops -2 degrees each day a) what will be the temperature drop in 4 days ? one day -------> - 2 degrees four days ------> -2 x 4 degrees = -8 degrees The temperature will drop -8 degrees. __________ If the temperature drops - 2 degrees each day b) how many days will it take to drop to -12 degrees ? -2 degrees -----> one day -12 degrees -----> -12/-2 days = 6 days. It will take 6 days for the temperature to drop -12 degrees. __________ If the temperature drops - 12 degrees in four days c) what is the temperature drop in one day? four days ------> - 12 degrees one day ------> -12/4 = -3 degrees In one day, the temperature drop will be -3 degrees. __________ By putting the negative integers in things that you are familiar with, you will be able to understand better. When you get a problem like divide - 14 by 2, think of it as if you owe $14 each to 2 persons. How much do you have to pay each person? -14 / 2 = -7 You owe each person $7, which is a negative 7 from your next allowance. Got the idea? Don't worry about the method. Once you understand the basics and practise them often enough, it will be easy. The hardest step is always the first one. -Doctor Lim, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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