Can you prove this theorem? -(-a)=a.?

Question

It is not negative x negative = positive

Answer 1

You know there are 8 axioms in mathematics which we cannot prove

1) if 'a' and 'b' are integers then 'a+b' is an integer and 'a.b' is an integer (Closure law)
2) a + b = b + a and a..b = b.a (cumulative law)
3) a + (b + c) = (a + b) + c (Associative law)
4) a (b + c) = a.b + a.c (Distributive law)
5) a + 0 = a = 0 + a (Additive Identity)
6) a.1 = a = 1.a (Multiplicitive Identity)
7) for all integer 'a' there is a unique integer (-a) such that a + (-a) = 0
8) I can't remember...

So we can prove your requirement using above axioms,

a + (-a) = 0
(-1)(a + (-a) = -1(0) (Multiplying by same factor)
-a + -(-a) = 0 (Distributive law over addition)

adding +a both sides

a + (-a + -(-a)) = a + 0
(a+ (-a)) + -(-a) = a (Associative law)
0 + -(-a) = a
-(-a) = a (proved... huh...)

Answer 2

The opposite of a negative is a positive. A double negative is a positive. The negative version of 2 is -2, but the negative version of -2 is 2. The negative sign means opposite, really. Most college level math teachers say "the opposite of two" instead of "negative two."

Answer 3

I think you are talking about the special property that if a function has an inverse, then the inverse of the inverse is the original number.

In your example, the function is to take the additive inverse of a number. Since the additive inverse is so common, we have a special way to write it, by putting a "-" in front of the number. And you are correct. It is a different idea than multiplication.

Not every function has an inverse.
For example f(a)=a*a.
f(-2)=4, f(2)=4. Since two different values give the same result, 4, we do not have a unique inverse.

But since the inverse does exist for addition, we can "undo" our original operation.

It is unfortunate that sometimes two entirely different ideas can be written the same way in mathematics.

Answer 4

For all x in a field, -x is defined as the number y such that y+x=0.

Let x=-a, and let y=-x. then x+y=0, but x+y=-a+y=y-a=0 so y=a and a=y=-x=-(-a).


And I don't think I know of any college level math profs that say "opposite of 2"

Answer 5

Im no math wiz,but don't two negatives make a positive. -1 x -a=a or something like that.

Answer 6

-1 * -1a = 1a or a

negative * negative is always positive

coefficient of 1 is implied if theres a letter or single negative sign

Answer 7

yu can imagin that there are one
glass up of one the other glass and
get the work of falling the water down
+ and the up - and we have a camera
that it goes forward as + and back ward
as- so if yu fall the water in to the down
glass yu have done + work so if yu fall the
water from the down tothe up
glass and yu back the video yu have done
-*- and the water goes to the down glass
and at last yu have done a + work !!! that
is a simple way of prooving!!!
if yu didnt got my prove just connect
me i will explain more>

Answer 8

THIS IS VERY SIMPLE MY FRIEND !!!
SEE (-a) IS A NEGATIVE NUMBER. & MOREOVER MINUS SIGN IS THERE OUTSIDE THE BRAKET. & IF OUTSIDE THE BRAKET MINUS SIGN IS THERE THEN THE SIGN INSIDE THE BRAKET CHANGES. SO,
-(-a)=a.
minus(minus A)=+A.

Answer 9

-(-a)=a is the same as
-1*(-a)=a divide both sides by the -1
-1*(-a)/(-1)=a/(-1)
-a=(-a)

Answer 10

It cannot be proven..is that correct? I'm not good at math!