Multiplying Positive and Negative NumbersDate: 10/27/2003 at 18:15:17 From: kirsten Subject: math In a problem like -3 x 7 x 4 = ? what do I do with the negative numbers? Date: 10/27/2003 at 22:26:37 From: Doctor Ian Subject: Re: math Hi Kirsten, Here's the basic idea. A negative number is the same as -1 times the corresponding positive number. For example, -3 is just -1*3, -18 is just -1*18, and so on. (I'm using '*' to represent multiplication.) How does this help? Well, suppose we have a bunch of positive and negative numbers being multiplied, like 3 * -4 * 5 * -2 * -7 = ? We can start by replacing the negatives with -1 times the corresponding positives: 3 * -1 * 4 * 5 * -1 * 2 * -1 * 7 = ? How does this help us? Well, since multiplication is commutative, order doesn't matter; so we can move all the -1's out to one side: -1 * -1 * -1 * 3 * 4 * 5 * 2 * 7 = ? The next thing to know is that -1 * -1 = 1. So whenever we have a pair of -1's multiplied together, we can replace them with a 1... and since multiplying by 1 doesn't change anything, we can just eliminate the 1: (-1 * -1) * -1 * 3 * 4 * 5 * 2 * 7 = ? (1) * -1 * 3 * 4 * 5 * 2 * 7 = ? -1 * 3 * 4 * 5 * 2 * 7 = ? Now, we just have to multiply the other numbers together, and since we still have a -1 hanging around, the sign of the whole thing is negative. Now, there is a shortcut we can use, but it's only worth doing if we realize _why_ it works. Instead of dragging the -1's out front to kill each other off, we can just kill of pairs of signs in place. For example, given 3 * -4 * 5 * -2 * -7 = ? we could kill off a pair of signs: 3 * -4 * 5 * -2 * -7 = ? ^ ^ |________|_________________ these signs cancel 3 * 4 * 5 * 2 * -7 = ? Now that there's only one sign left, we can bring it out front: -(3 * 4 * 5 * 2 * 7) = ? Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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