Proof of Multiplying by Fraction = Dividing by Inverse

[jimgavagan]

Supposedly this proof answers my question.

8 / 16 = .5
8 / 8 = 1
8 / 4 = 2
8 / 2 = 4
8 / 1 = 8
8 / .5 = 16
8 * 2/1 = 8 / .5

I'm just wondering how this proof answers my question?

 

[Dr. Seafood]

It doesn't ... and that's not a proof o__O

It follows kind of from "definition" of division: x \over y actually means x \cdot y^{-1} where y^{-1} is the number such that y \cdot y^{-1} = 1, called the "multiplicative inverse" or "reciprocal" of y. This number is unique. Obviously ({a \over b})^{-1} = {b \over a} (since {a \over b} \cdot {b \over a} = 1) as long as we have a, b \ne 0. (Also of note: 0^{-1} does not exist!

So it turns out that since {x \over y} = x \cdot y^{-1}, setting y = {a \over b} we get

"{{x} \over {a \over b}}" = {x} \cdot ({a \over b})^{-1} = {x} \cdot {b \over a} (again provided a, b \ne 0). I hope this explanation helps.

 

[jimgavagan]

ooooooooooooooooooooooooooh awesome! :D

Very much appreciated!