Math Is Fun Forum

wasee

Member

Registered: 2007-08-12

Posts: 2

Negative *Negative

Can somebody prove to me why negative times negave is positive? I hope to learn what I probably missed in class 4.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Negative *Negative

Facts (can be proven if you rigorously define what a number is, but that takes advanced knowledge of set theory):

0 + a = a
a - a = 0 (that is, -a means the additive inverse of a)
ab = ba
a + b = b + a
If a = b, then ac = bc
If a = b, then a + c = b + c
a * (b + c) = ab + ac
a(bc) = (ab)c
a + (b + c) = (a + b) + c

Proof:

0 + 0 = 0, and by multiplying on the left side by any number a, we get a0 + a0 = a0.  Subtracting a0 from both sides gives a0 + a0 - a0 = a0 - a0.  Thus, a0 + 0 = 0, or rather, a0 = 0.  That is, any number multiplied by 0 is 0.

Using "reverse distribution" (factoring), it must be that -a(-b) + -a(b) = -a(-b + b) = -a(0) = 0.  Eliminating the middle two steps leaves us with -a(-b) + -a(b) = 0, and so adding a(b) to both sides gives us -a(-b) = a(b). Thus, a negative times a negative is positive.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Negative *Negative

We don't want you to not understand

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Negative *Negative

hehehe, but does that state that we want Wasee to understand?

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A logarithm is just a misspelled algorithm.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Negative *Negative

Exactly! Two negatives make a positive

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Negative *Negative

Perhaps a more suitable explanation could be given if you could tell us what level your maths is at.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Negative *Negative

A comedian was on stage ranting about how two negatives make a positive.  "But you never see two positives make a negative!" he said.  Just then, a man stood up in the back of the crowd and shouted, "Yea right."

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Negative *Negative

Okay, let's see

B_____A_______C

B is 5 metres left of A.

B=A-5

And C is 7 metres right of A.
C=A+7

How long is C right of B?
The answer should be:
C-B=A+7-(A-5)
The answer, as you can varify, is 12.
Ah-ha!
How do you calculate --5?  = +5.

Why?

Because left is negative/reverse for right, and left-end is negative in the formula of d=C-B  then the two negative effect cancells each other out. (you can go to the explaination now)

Or another model.
A, B, C = 3 persons. right= higher, left= shorter.
Now do it again. You will find that the shorter the shortest person than the middle person, the taller the tallest one than the shortest.

When comparison appears, you have this interesting cancelling out effect, which may be the origins of negative negative= positive.

BTW: The Explaination
If I say A is 5 metres right of B, and C is 7 metres right of A, you probably will tell that C is 12metres right of B without any hesitation.

But now we have only B 5 metres left of A.
First, left is reverse to right. So B is -5 metres "right" of A, as we can define. "-" here only means the reverse.
Now, if one thing is right of another, the other is just left of this one thing with the same distance.
So when you swap A and B in the sentence B is x metres right/left of A, you get a reverse between left and right.
As mentioned before, left itself is the reverse/negative of right, and now you reverse left, then you definately have:
Right___Left___Right:
Reverse then reverse comes back. Or reverse the reverse gets it back.

Thus A is 5 metres right of B.
we can abstract this "Reverse then reverse comes back" effect to a simple expression
--=+ or --x=x
as in the case -(-5)=5, which also stands in a bit more complex situation 7-(-5)=12

Last edited by George,Y (2007-08-16 15:01:57)

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X'(y-Xβ)=0

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Negative *Negative

Logically negative the negative get's the positive.

Like "I am nothing but special". (looking familiar in Harry Porter?)

Here "nothing" negates the description behind it just like not, and "but" again negates the description behind it. So together "special" is a positive description in the meaning of this sentence.

Of course this sentence has another meaning "I am only special". The "only" component is a little bit complex, you don't need to know now, just get the general idea that negate then negate means affirmation.

Or simplier:

I never stop/quit typing.
Means I am typing.

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X'(y-Xβ)=0

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Negative *Negative

Exactly! Two negatives make a positive

I'm not sure that it does in that case. If we say that we don't want him to not understand, that merely means we have no desire to impede his comprehension, but that does not necessarily mean we wish to help him?

Geez I'm bored...

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A logarithm is just a misspelled algorithm.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Negative *Negative

I was thinking something along those lines, but then I realised that it doesn't work because he used the word not. "Not understand" encompasses everything that isn't "understand". There are only those two groups, and so if we don't want him to be in one, then we have to want him to be in the other.

Two sentences that can be different are things like:
"I want you to like this."
"I don't want you to dislike this."

In the first one, you need to like this, but for the second one you're allowed to love it, like it, hate it, whatever. It works because the 'not' isn't explicitly there.

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Why did the vector cross the road?
It wanted to be normal.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Negative *Negative

OK, here is another explanation: Multiplying Negatives Makes A Positive

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Negative *Negative

"I want you to like this."
"I don't want you to dislike this."

In logic, the negative of "like" is not exactly "dislike", because surely there are other options like neutral.

So how about this pair?
"I think you like this."
"I don't think you don't like this."

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X'(y-Xβ)=0

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Negative *Negative

OK, here is another explanation: Multiplying Negatives Makes A Positive

Great! Here backward stepping plays one negative factor.

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X'(y-Xβ)=0