6 + 2 = 8
Therefore, 8 - 6 = 2 or 8 - 2 = 6
How can you state those relationships in abstract terms?
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
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
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
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
Watch the usage to "inverse", a lot of folks really mean reciprocal when they use it.
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