Why do negative numbers behave differently when multiplied?

[dfollett76]

I'm looking for a good/simple explanation for why the rules for multiplying signed numbers are the way they are.

i.e. why does (-)*(-)= (+); (-)*(+)=(-); etc.

Also, I'm looking for some good real world examples to where these situations apply.

Thanks for you help.

 

[berkeman]

One good real world example is modulation. Modulation is multiplying two signals together for some purpose, like shifting a data waveform up to RF waveform frequencies for transmission.

So the modulated waveform looks like M(t) = A sin(w1 * t) * B sin(w2 * t)

Draw out the waveforms and see how the sign convention represents reality in the modulation process. Does that help?

 

[dfollett76]

Sorry, but not really.

I should clarify. I'm looking for an example that I can share with my 9th grade students.

 

[berkeman]

Sorry, but not really.

I should clarify. I'm looking for an example that I can share with my 9th grade students.

Doh! Well, bright 9th graders would probably understand modulation -- just make some pretty slides up with colored sine waves...

Multiplication is a non-linear process, so your examples are going to have to involve non-linear phenomena (like modulation). Let's see, what else is an example of non-linear stuff...

Boy, that's a toughie. I'd stick with modulation and try to introduce it gently to them.

 

[Hurkyl]

0 * (-1) = 0
(1 + (-1)) * (-1) = 0
1 * (-1) + (-1) * (-1) = 0
(-1) + (-1) * (-1) = 0
1 + ((-1) + (-1) * (-1)) = 0 + 1
(1 + (-1)) + (-1) * (-1) = 1
0 + (-1) * (-1) = 1
(-1) * (-1) = 1

There's a proof in excruciating detail. Well, it takes a little bit more to prove (-1) * x = (-x), but this might be enough for them.

 

[Hurkyl]

You might be able to concoct a "real world" example from the notion of a "signed distance". I.e. that (-3) meters to the right is the same thing as (+3) meters to the left.