# Prove that -4 x -4 = 16

1. Feb 6, 2005

### Raza

Can someone prove that -4 x -4 =16?

2. Feb 6, 2005

### jcsd

Lets say that 4 and 16 are members of a ring such that 4*4 = 16

In any associative ring:

1) a*0 = a*0 + 0 = a*(b + -b) + ab + -(ab) = a(b + -b + b) + -(ab) = ab + -(ab) = 0

2) 0*a = 0*a + 0 = (b + -b)*a + ba + -(ba) = (b+ -b + b)a + -(ba) = ba + -(ba) = 0

3) a*-a = a*-a + 0 = a*-a + (a*a + -(aa)) = a(-a + a) + -(aa) = a*0 + -(aa) = -(aa)

4) -a*-a = -a*-a + 0 = -a*-a + (a*-a + aa) = (-a + a)*-a + aa = 0*-a + aa = aa

edited to add: here are the axioms of a ring http://mathworld.wolfram.com/Ring.html

I haven't shown which axioms is use din each step but i hope you can see thta each step does follow on directly from the ring axioms and what was shown before it.

Last edited: Feb 6, 2005
3. Feb 6, 2005

### Zurtex

It's a simple extension of the properties of real number to prove that -1*a = -a (where -a is the additive inverse), also that -1*-1 = 1 and also that (-a)*b = -(a*b). I think the proof is fairly elementary once you have properties like these.

Edit: Well it appears the person above me had done a lot more general proof with full details so nevermind.

4. Feb 6, 2005

### cronxeh

what a de ja vu... i swear ive seen this before somewhere

5. Feb 6, 2005

### dextercioby

And that's actually déjà vu as in nonadapted orginal French... :tongue2:

Daniel.

6. Feb 7, 2005

### WORLD-HEN

If I understand correctly, you want me to prove that negative times negative is positive, right?

$$-a(a-a) = 0$$

This statement is true since $$(a-a) = 0$$

Using the distributive law :

$$-a(a-a)=0$$

$$-a^2 + (-a X -a) = 0$$

$$-a X -a = a^2$$

So, substituting 4 for a gives
$$-4 X -4 = 4^2$$
$$-4 X -4 = 16$$

Last edited: Feb 7, 2005
7. Feb 7, 2005

### Zurtex

WORLD-HEN we then to use a * when absolutely needed to denote multiplication. But (-4)(-4) = 16 might be easier.

8. Feb 7, 2005

### dextercioby

So that big "ex" $X$ was actually $\times$ or $\cdot$ all along... Ingenious... :rofl:

Daniel.

9. Feb 7, 2005

### Raza

Thank you all, I finally get this.
A substitute teacher was asking me this question and I couldn't answer her at all

10. Feb 7, 2005

### jcsd

The last time a simalir question was asked someone quoted W. H. Auden:

"Minus times minus is plus, the reasons for this we need not discuss"

If you don't want to bother with all the ring theoretic stuff just tell your teacher that

11. Feb 8, 2005

### WORLD-HEN

Sorry, havent used latex much before

12. Feb 8, 2005

### Raza

your probability won't work if they are two different numbers, let say -5 and -3. How will you work that out?

13. Feb 8, 2005

### mathwonk

what does -1 mean? it means the number which added to 1 gives zero. and hence -(-1) is the number which added to -1 gives zero, so -(-1) = 1.

morover (0+0) = 0, so 0a = (0+0)a = 0a + 0a, so subtracting gives 0a = 0.

hence 0 = 0a = [1+(-1)]a = a + (-1)a.

hence (-1)a =-a.

thus (-a)(-a) = (-1)(-1)a^2 = -(-1)a^2 = a^2.

gee, this is as long and tedious as alkl the oithers.

obviously this kind of nonsense was never meant to be fun.

14. Feb 8, 2005

### WORLD-HEN

The exact same way.

-5(3-3)=0

(-5)(3) + (-3)(-5) = 0

-15 + (-3)(-5) = 0

-3 x -5 = 15

More generally

-a(b-b) = 0
-ab + (-b)(-a) = 0

(-b)(-a) = ba

15. Feb 8, 2005

### mathwonk

world hen, you are assuming that (-a)(b) = -ab, which i proved above.

that is why your post looks shorter.

16. Feb 8, 2005

### eNathan

Is that simple enought. No, it probally not. But its fairly simple. The concept of a negative number is totally different from that of real numbers. A negative number is not only the asbsense of a real number, but its used to make up for the non-existance of the real numbers. This is why a negative times a negative is a positive.

17. Feb 9, 2005

### jcsd

Negative numbers ARE real numbers.

There's a simple amendmnt you make to the proof I made earlier so that it proves (-a)(-b) = ab rather than (-a)(-a) = aa.

18. Feb 9, 2005

### eNathan

Yes, your right on that one. I tend to get lost with all the terms :yuck:
but my concept of what negative numbers are correct, its like saying "I am not not going to the store", which really means you ARE going to the store. Its sort of hard to explain, and for even me to understand (...) but a negative a negative amount of times is really a positive, because like I said, a negative number is the absense of positive numbers. If our mathamatical system was not like that :surprised We could travel back in time! But of course, there is the hole thing about the fact that negative numbers dont have true roots, but imaginary roots. I guess, mathematicly, to get the square root of p, when p is -negative, you would do 0-(Sqr(0-p)) That should work

19. Feb 9, 2005

### Zurtex

eNathan I can't help thinking when reading your posts that you don't really understand the number system that well.

A negative number: -a is defined such that: a + (-a) = 0. That is it, there is no more to it, however from this and the other rules of real numbers we can prove that -1*(a) = -a, -1*-1=1, (-a)*b = -(a*b) and many other useful identities.

Just because the square root of a negative number is imaginary it does not mean it is not the 'true' square root of the number.

I think you would find it rather helpful to look up the mathematical definitions of the words 'real', 'imaginary' and 'complex'. As well as looking up what Natural Number, Integers and Quotients are.

20. Feb 9, 2005

### mathwonk

this has nothing to do with negative numbers, -a is not a negative number, it is the additive inverse of a.

a need not be a real number at all, a could belong to any additive group.

In any case even if a were a real number, -a is mereoly the opposite of a, if a is zero, -a is zero not negative, and if a is negative then -a is positive.

-a is not read "negative a", but "minus a" or "additive inverse of a", or "the number you add to a to get zero".

the word negative means "less than zero". this am,kes no sense in amny situations where -a makes perfect sense, for example complex numbers where there are no negative complex numbers, since "greater than" and "less than" are not defined for complex numbers, for the same reason you cannot say which horse on a merry go round is in "front".