Why Does a Negative Times a Negative Equal a Positive?Date: 03/01/98 at 21:12:30 From: Jessica Subject: Multiplying Negative numbers How does a negative number times another negative number equal a positive number? Date: 03/02/98 at 09:15:10 From: Doctor Bruce Subject: Re: Multiplying Negative numbers Hello Jessica, I detect in your question a measure of annoyance at having to learn the rule for multiplying negative numbers. I'll bet it seems like someone just made the rule up out of thin air, with no particular reason why the answer should be positive. I want to reassure you that this rule is not just "made up." There is a chain of reasoning -- a mathematical "argument" -- that shows why the rule *has* to be that negative times negative equals positive. Mathematical argument takes a little getting used to. This might look rather strange at first. Here's how the reasoning goes: (1) Zero times anything equals zero. (2) Every number has exactly one additive inverse. This means if N is a positive number, then -N is its additive inverse, so that N + (-N) = 0. Likewise, the additive inverse of -N is N. (3) We want negative numbers to obey the distributive law. This says that a*(b+c) = a*b + a*c. (4) Now, we are forced to accept a new law, that negative times positive equals negative. This is because we can use the distributive law on an expression like 2*(3 + (-3)). This equals 2*(0), which is zero. But by the distributive law, it also equals 2*3 + 2*(-3). So 2*(-3) does the job of the additive inverse of 2*3, and therefore 2*(-3) is the additive inverse of 2*3. But the additive inverse of 6 is just -6. So 2 times -3 equals -6. (5) Next, we are forced to accept another new law, that negative times negative equals positive. It's a lot like the example in (4). We use the distributive law on, say, -3*(5 + (-5)). This is again equal to zero. But by the distributive law, it also equals -3*5 + (-3)*(-5). We know the first thing, (-3*5) equals -15 because of the law in (4). So (-3)*(-5) is doing the job of the additive inverse of -15. We know -15 has exactly one additive inverse, namely 15. Therefore, (-3)*(-5) = 15. I hope this doesn't frighten you! The main thing is, keep right on questioning the things that don't make sense. In mathematics, you are always entitled to an explanation of WHY things are the way your teacher (or I) say they are. Good luck, -Doctor Bruce, The Math Forum Check out our web site http://mathforum.org/dr.math/ |
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