Proving the Positivity of Negative Times Negative: An Algebraic Approach

[Taturana]

What's the proof for a negative number times another negative number gives a positive number?

Thank you

 

[Stephen Tashi]

Are you studying the arithmetic of numbers? Or are you studying something more advanced like algebra.

If you are studying algebra, perhaps what you want to see is a proof of something like
(-x)(-y) = xy. ( Strictly speaking a number like "-x" isn't necessarily "a negative" For example if x = -3 then -x is a positive number. )

 

[Taturana]

Are you studying the arithmetic of numbers? Or are you studying something more advanced like algebra.

If you are studying algebra, perhaps what you want to see is a proof of something like
(-x)(-y) = xy. ( Strictly speaking a number like "-x" isn't necessarily "a negative" For example if x = -3 then -x is a positive number. )

I'm studying arithmetic of numbers. I want to understand why the product of, say, -3 for -4 is 12 and not -12.

Thanks

 

[Mark44]

0 = 0 * (-4) = (-3 + 3)(-4) = (-3)(-4) + (3)(-4)

Presumably you know that 3(-4) = -12. Since (-3)(-4) + (3)(-4) = 0, (-3)(-4) and (3)(-4) must be additive inverses, which implies that (-3)(-4) = 12.

 

[g_edgar]

In order to prove anything, you have to start with something. What are you starting with?

 

[micromass]

Basically, it's true because we want it to be true. If we would choose something else, then things would not behave so nicely. For example, we wouldn't have distributivity.

Alternatively, if you're into geometry, then you can easily see why we want it to be true. Take a vector. Multiplying by -1 flips the vector. And multiplying again, flips the vector again. So doing -1 and -1 again gives you nothing. So (-1)(-1)=1.
Of course, this is not a proof, but merely an indication of why we want it to be true. Like many things is math, this is just a convention. But it's a convention which works really well with physics and the real world!

 

[jhae2.718]

Isn't it just axiomatic?

 

[gb7nash]

Either it's axiomatic or you could use this:

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

0 = -1 + (-1)(-1), and finally 1 = (-1)(-1)

Like previous posters have said, it depends what you're allowed to use. a lot of these operations we take for granted.

 

[Stephen Tashi]

I'm studying arithmetic of numbers. I want to understand why the product of, say, -3 for -4 is 12 and not -12.

Thanks

Mark44 gave a good answer. I can't see your textbook, so I don' know what you have already studied from it. I have to guess what you already accept as true about numbers. The steps

0 = (0)(4) = (3 + (-3)) (-4) uses various laws of arithmetic.
The distributive law gives us:
0 = (3)(-4) + (-3)(-4)

if you accept that (3)(-4) = -12 then we have
0 = (-12) + (-3)(-4)
So (-3)(-4) is something that when added to (-12) gives you 0. So it must be the same thing as 12.

If you want physical argument instead of a mathematical one, you'd have to say something like: If I had a film of a car going 3 feet per second in the negative direction and I played it backwards for 4 seconds, then the car would appear to go 12 feet in the positive direction.

 

[sahil_time]

a+b=0
a=(-b)...(1)
(a+b)^2=a^2+b^2+2ab...Simple Algebra
therefore
0=a^2+b^2+2ab...(2)
now squaring eqn no 1
can lead to 2 results
1)a^2=(-b^2)
or
2)a^2=b^2
Now put both results in Eqn second and verify!:)

Hint:a+b=0 Take a and b to be non zero and you will arrive at the result that a^2=b^2 coz if a^2=(-b^2) then in Eqn two ull find that "2ab=0" But that violates our assumption that a and b are non zero!:)