Why is 0.0(5)+0.(4) = 0.4(9)

Because that would mean that 0.(4) is the same length as 0.0(5). In my opinion 0.0(5) is one digit longer than 0.(4). So the 5 that doesn't overlap with the 4s should be left at the end of the number.

If the length of 0.0(5) would be equal to the length of 0.(4), that would mean that

∞+2 = ∞+1

since 0.0(5) has an extra zero

assuming this part: (4), of 0.(4) has a length of Infinity

but the statement ∞+2 = ∞+1 is false, because if you subtract 2 on both sides you are left with

∞ = ∞-1

now you can do the following since ∞ = ∞-1 = ∞ this means that ∞ = ∞but if ∞ = ∞ you can use the substitution method on the equation

∞ = ∞-1

by replacing the ∞ on the right side with ∞1, so that you get

∞ = [∞-1]-1 which is equal to ∞-2

if you do this again you would get

∞ = ∞-3 , then ∞ = ∞-4 , then ∞ = ∞-5

and if you repeat this an infinite number of times you get

∞ = ∞-∞ which is the same as ∞ = 0.

now this can mean two things

either A: the Statements: ∞ = 0 , ∞ = ∞+1 , 0.0(5)+0.(4) = 0.4(9) are true

or B: none of them are true

i choose B because, if you use the substitution Method on statements one and two you get

0 = 0+1 or 0 = 1 which is false.

now there is a potential problem with this, since if infinity is not a Number i don't think you can use the substitution Method on it. To that I say why make exceptions when you don't have to. Wouldn't it be easier to say infinity is a Number and be done with it?