Question

my teacher told me that the inverse of addition was subtraction and that the inverse of subtraction was addition...

could you prove it to me

-x = (-1)x

((-1)x)^-1

I don't see how I'm suppose to get + x by taking the inverse of -x i've always been told in math to just to the opposite of subbtraction which is addition but my teacher is telling me that is a lie and that it's really the inverse of subtraction is addition but I don't see the reasoning behind it

basically can you prove to me that the opposite of subractiion is addition and vise versa??? by taking the inverses????

I don't get it...

like I can prove that the opposite of multiplication is division by taking the inverse and can prove it just by defintion

(5x = 2)5^-1 = x = 5^-1 (2)

that's how you prove that relationship is really just inverses but what about addition and subtraction how are the inverse relationships...???

Thansk

Answer 1

Use numbers.

6 + 2 = 8

Therefore, 8 - 6 = 2 or 8 - 2 = 6

How can you state those relationships in abstract terms?

Answer 2

If x is some number and:

x + y = 0

then y is called an inverse (w.r.t. addition) of x

Then it follows from the same definition that x is an inverse of y. Now, what you need to prove is that inverses are unique. I.e. if for some given x

x + y = 0

and also

x + z = 0

you necessarily have y = z.

So, it then follows that the inverse of the inverse of x is x and it can't be anything else than x.

Then, if we denote the inverse of x by
-x, we can prove that:

-x = (-1)*x

THis is because:

x + (-1)*x =

1*x + (-1)*x =

(1 + (-1))*x =

0*x = 0

Here we have used that -1 is the inverse of 1.

So, (-1)*x satisfies the criterium the inverse of x which we always denote as
-x must satisfy and therefore
-x = (-1)*x

Then the fact that taking twice the inverse yields the same number implies that:

(-1)*(-1) = 1

Answer 3

i agree with all of it but still don't see how

X + B = C
we can simply solve for B by simply multiplying the whole equation by B^-1 which we note as -B because????

(X + B = C)B^-1

B cancels out

X = B^-1 C

what allows us to say that B^-1 is equal to -B

Answer 4

You are confusing terms:

Inverse is not the reciprocal. You are using reciprocal (B^-1) is reciprocal.

Now it is confusing, because the inverse operation to multiplication is division, and the inverse to division is multiplication
http://www.mathsisfun.com/definitions/inverse-operation.html

Watch the usage to "inverse", a lot of folks really mean reciprocal when they use it.