Whats the proof for the multiplication sign law that a negative number times a negative number gives positive?

Question

We all know the sign laws for multiplication:
Positive times positive is positive
Negative times positive is negative
Negative times negative is positive.

It's easy to proove the first two. But how can the last one be prooved in an inductive way?

Thanks,

Answer 1

im not sure if this is an inductive proof, but here:

multiplication of negative numbers can be thought of as repeated subtraction. so 3 x -4 = -4 - 4 - 4 = -12

if youre multiplying two negative numbers, signs will "cancel" out

-3 x -4 = -(-4) - (-4) - (-4) = 12

ALSO, you could think of -3 x -4 as
-1 x (3 x -4)
= -1 x (-12)
= 12

hope this helps

Answer 2

A Mathematical rationalization If we can agree that a unfavourable volume is basically an excellent volume expanded through -a million, then we can continuously write the manufactured from 2 unfavourable numbers this form: (-a)(-b) = (-a million)(a)(-a million)(b) = (-a million)(-a million)ab as an party, -2 * -3 = (-a million)(2)(-a million)(3) = (-a million)(-a million)(2)(3) = (-a million)(-a million) * 6 So the authentic question is, (-a million)(-a million) = ? and the reply is that right here convention has been followed: (-a million)(-a million) = +a million This convention has been followed for the straightforward reason that the different convention might want to reason something to break. as an party, if we followed the convention that (-a million)(-a million) = -a million, the distributive resources of multiplication would not artwork for unfavourable numbers: (-a million)(a million + -a million) = (-a million)(a million) + (-a million)(-a million) (-a million)(0) = -a million + -a million 0 = -2 As Sherlock Holmes said, "once you've excluded the no longer plausible, regardless of continues to be, even with the undeniable fact that unbelievable, should be the reality." because that each little thing except +a million might want to be excluded as no longer plausible, it follows that, even with the undeniable fact that unbelievable it sort of feels, (-a million)(-a million) = +a million.

Answer 3

The definition of multiplying a * b is adding the number a, b times.
If b is negative, you subtract a, abs(b) times. If a is also negative, it means you are subtracting a negative number, abs(b) times, which is the same as adding abs(a), abs(b) times. So (-2) * (-2) = 0 - (-2) - (-2) = 0 + 2 +2 = 4

Answer 4

Since you accept the first two rules, then the third one follows simply by rearranging the second algebraically.

a*b=c (a is positive, b & c are negative)
a = c/b = c*(1/b) = c*d (where I defined d=1/b, obviously d must be negative)
a = c*d

Hence, positive = negative * negative.

If you really wanted to get picky, it's actually redundant to even write both rule #2 and rule #3.

Answer 5

A Proof

Let a and b be any two real numbers. Consider the number x defined by

x = ab + (-a)(b) + (-a)(-b).
We can write

x = ab + (-a)[ (b) + (-b) ] (factor out -a)
= ab + (-a)(0)
= ab + 0
= ab.

Also,

x = [ a + (-a) ]b + (-a)(-b) (factor out b)
= 0 * b + (-a)(-b)
= 0 + (-a)(-b)
= (-a)(-b).

So we have

x = ab
and
x = (-a)(-b)

Hence, by the transitivity of equality, we have

ab = (-a)(-b).

Answer 6

Let n,m be positive. Then
0 = n * 0
0 = n * (-m + m)
0 = n*(-m) + n*m
-[n*m] + n*m = n*(-m) + n*m
-[n*m] = n*(-m)
This shows that n*(-m) = -(n*m) is negative.

Now the last statement.
0 = 0 * 0
0 = (-n + n) * (-m + m)
0 = (-n)*(-m + m) + n*(-m + m)
0 = (-n)*(-m) + (-n)*m + n*(-m) + n*m

Use the previous conclusion, cancel last two terms
0 = (-n)*(-m) + -(n*m) + -(n*m) + n*m
0 = (-n)*(-m) + -(n*m)
n*m + -(n*m) = (-n)*(-m) + -(n*m)
n*m = (-n)*(-m)
This shows that (-n)*(-m) = n*m is positive.

Answer 7

It's not a proof, it's a definition. If it were defined otherwise, inconsistent result would occur: at least one of the laws of multiplication would fail.

Answer 8

I used to know the answer to this question a long time ago.

Now all I have to do to prove it is use a calculator.

Answer 9

i guess its because multiplying means making more so it has to become positive that's a good question i never though of that

Answer 10

welll..
trust is everything.
Kiddin, it's pretty hard to dem in words. Board is much easy and talkin.