# 12.31.08

## A Minus Times a Minus

Back in grade school, I never got a satisfactory answer to the question “why is a minus times a minus equal to a plus?”

I didn’t think about it again for years, until my mom asked for an explanation to give when she taught her students. This time around, I immediately realized how you’d go about proving it. Here’s how it goes:

1 + (-1) | = | 0 | (definition of “-1″) |

(-1) * 1 + (-1) * (-1) | = | 0 | (multiply both sides by -1 and distribute) |

-1 + (-1)(-1) | = | 0 | (anything times 1 is itself) |

(-1)(-1) | = | 1 | (add 1 to both sides) |

Easy as could be, right? Unfortunately, mom preferred my brother’s explanation:

A minus times a minus is equal to a plus.

The reasons for this, we need not discuss.

Your brother said,

January 2, 2009 at 7:04 pm

Minus times minus results in a plus,

The reason for this, we needn’t discuss.

– Ogden Nash

Actually, I’m not sure that’s the exact quote either, nor even the right source. Like Yogi Berra, Nash didn’t actually say half the things he said. But at least the rhythm is a little closer to matching and I’ve found it cited that way at least once on-line. I just wouldn’t want your readers to think that your brother recites couplets where one line has three more syllables than the other. :-)

mom said,

January 3, 2009 at 7:49 am

Minus times minus equals a plus,

The reasons for this we need not discuss.

OK, that’s the version I tell my students. And I tell them about both of you as well–my fine sons! One of these days I’ll try to go over Dan’s explanation with them. But he is right. I like yours better, Jeff!

Ogden Nash? How did you come up with that?

Jagjit (jack) said,

May 10, 2010 at 5:07 am

1 * a = a, ”here ‘a’ is any number”

hence, 1 * (-1) = (-1)

also, -1 * a = -a, ”here ‘a’ is any number”

hence, -1 * (-1) = – (-1)

Negative of +1 = -1

negative of (-1) = 1, because a number plus the negative of the

number adds up to zero, for instance when -1 + x = 0, that means x=1

From above, (-1) * (-1) = – (-1) = negative of (-1) = 1

That shows (minus * minus ) yields plus

Jagjit (jack) said,

May 23, 2010 at 2:59 am

Another Soln

This is not a proof, however a demonstration for a curious kid why negative times negative is positive. It may be helpful to think that when dealing with the numbers ‘-’ or ‘+’ signs are identical to ‘-1 times’ or ‘+1 times’. Here is an example:

-5 = ‘-1 times’ 5

A number plus a negative number can result zero. Lets take the number ‘1’ that is ‘+1’, then take ‘-1’. It could have been any other number. The MAIN assumption is:

1 -1 = 0

Without losing the effect we can times the equation by ‘-’ or ‘-1’ then:

-1 -(-1) = 0

Since both equations are the same, that means:

-1 -(-1) = 1 -1

that means:

-(-1) = 1

it shows:

‘-’ times ‘-’ is plus

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This approach is also helpful to show why ‘+’ times ‘-’ is ‘-’

Let us safely assume ‘1’, which is ‘+1’, times a number results in the number, i.e. ‘1 times’ 10 is 10. Then ‘1 times’ ‘-1’ is ‘-1’ . That is:

‘+1 times’ -1 = -1

That shows: ‘-’ times ‘+’ results ‘-’

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