0.999....=1

[jigokutamago]

Zkuq said:

jigokutamago said:
If d = infinitely small number
x = 0.999... = 1 - d
x = 1 - d
10x = 10 - d
10x = 9 + 1 - d
10x = 9 + x
9x = 9
x = 1

This makes sense right? An infinitely small number multiplied by 10 is still an infinitely small number.

Not really I think. If 10d = d then dividing by d gives you 10 = 1. My guess is that the problem takes advantage of this fact.

If 10d = d, then d must equal 0. If d = 0, you can't divide by d. Or, in other words, your d must equal 0. It's the decimal notation that has a problem, nothing else.

I see. My thought was that the problem takes advantage of the fact of assuming an infinitely small number to be equal to zero. Then obviously

d = infinitely small number = 0
0.999... = 1 - d = 1 - 0 = 1

but can we assume infinitely small number = 0 ?