Manipulating Positive and Negative NumbersDate: 8/9/96 at 20:37:45 From: terry holman Subject: Manipulating Positive and Negative Numbers I am having a little trouble understanding the rules for adding, subtracting, multiplying, and dividing positive and negative numbers. Could you please help me? Date: 8/30/96 at 17:44:17 From: Doctor James Subject: Re: Manipulating Positive and Negative Numbers Negative numbers are very interesting. There are a couple of tricks you can do to remember the rules a little better. Addition and subtraction are easy when you remember this: 3 - 1 = 3 + -1 So adding a positive and negative number is just like subtracting two positive numbers. When the negative number is first it gets a little tricky, but you can figure it out: -1 + 3 = 3 + -1 = 3 - 1 = 2 If the negative number is bigger, the final answer will be negative. For example, 2 + -5 = 2 - 5 = -3 If both numbers are negative, it's just like adding positive numbers, except that the answer is negative. -3 + -5 = -8 (compare to 3 + 5 = 8) Subtracting a negative number is just like adding a positive number. 1 - -3 = 1 + 3 So what's -3 - -8? well, -3 - -8 = -3 + 8 because subtracting a negative number is just like adding a positive number -3 + 8 = 8 + -3 because you can change the order of addition 8 + -3 = 8 - 3 because adding a negative number is just subtracting a positive number 8 - 3 = 5 Just regular subtraction! Now, the first thing to remember for multiplication is that if you multiply a positive number by a negative number, you get a negative number. Knowing that, we can say that a number like -5 is really just -1 * 5, where * means times. Remember that multiplication is 'commutative', which means that it doesn't matter what order you multiply things (as long as you are just multiplying!) So if we have 3 * -4, we can say: 3 * -4 = 3 * -1 * 4 because -4 = -1 * 4 3 * -1 * 4 = -1 * 3 * 4 because multiplication is commutative -1 * 3 * 4 = -1 * 12 because 3 * 4 = 12 -1 * 12 = -12 because -12 = -1 * 12 so we get: 3 * -4 = -12, the answer! similarly, see if you can get the following: 3 * 4 = 12. -3 * 4 = -12 -3 * -4 = 12 remember that -1 * -1 = 1 Now, if you just replace 3 and 4 with whatever two numbers with which you are working, you can do any multiplication problem! Division is similar. There are a couple things to remember, though: 1/(-1) = -1 You can see this if you multiply the top and bottom of the fraction by -1 (since you are multiplying both the top and the bottom by the same thing, its ok). This gives you -1/(-1 * -1) = -1/(1) = -1. -1/(-1) = 1 Can you figure this one out? do the same thing as the last one. Division, like multiplication, is commutative. This means that you can switch the order in division too. You have to be careful, tho! You must switch the division sign with the number following it. For example, 3 * 4/5 = 3/5 * 4, but 3 * 4/5 doesn't equal 3/4 * 5 ! Also, although division and multiplication 'commute' with each other, they don't with addition or subtraction, so be careful there too. So, if you have 6/(-2), you can do the following: 6/(-2) = 6/(-1 * 2) because -2 = -1 * 2 6/(-1 * 2) = (-1 * 6)/(-1 * -1 * 2) because you can multiply the top and bottom by the same thing. (-1 * 6)/(-1 * -1 * 2) = (-1 * 6)/(1 * 2) because -1 * -1 = 1 (-1 * 6)/(1 * 2) = (-1 * 6)/2 because 1 * 2 = 2 (-1 * 6)/2 = -1 * (6/2) because division and multiplication are commutative -1 * (6/2) = -1 * 3 because 6/2 = 3 -1 * 3 = -3 because -3 = -1 * 3 Hope this helps. If it's still confusing, please write back. -Dr. James, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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