color(blue)("Things to think about for the method")
These are fraction -> (count)/(size)
color(brown)("To be able to directly add or subtract counts") what you are counting color(green)("must be of the same type or size.")
(count)/(size) -> ("numerator")/("denominator")
Neither of (a+1) nor (a-1) will easily divide into each other. So you have to devise something they will. The easiest way to do this is multiply them together.
Write (a+1) times (a-1) as (a+1)(a-1)
Consider a/(a+1). How do we change (a+1) into (a+1)(a-1)?
color(green)("If we multiply it by 1 but in the form of "(a-1)/(a-1)") we change the way it looks but not its actual value. Think of 2 times 3/3 = 6/3.
~~~~~~~~~~~~~~~~~~~~~~ end of method introduction ~~~~~~~
[a/(a+1) times (a-1)/(a-1)] -[1/(a-1) times (a+1)/(a+1)]
[(a(a-1))/((a+1)(a-1))] - [(a+1)/((a+1)(a-1))]
((a^2-a) -(a+1))/((a+1)(a-1))
(a^2-2a-1)/((a+1)(a-1))
but a^2-1^2 = a^2-1 =(a+1)(a-1)
So by substitution we have:
(a^2-2a-1)/(a^2-1)
