About one... and its infinite cousin - Jesse Hyttinen

Hi and welcome to the Matladpi blog!

The question if 0.999... = 1 is mathematically an interesting one. I have collected three viewpoints that can be regarded as answers. Each of them agree that 0.999... actually is 1.

Viewpoint 1:

1/3 = 0.333...

3(1/3) = 3(0.333...)

1 = 0.999...

0.999... = 1

Viewpoint 2:

a = 0.999...

10a = 9.999...

10a - a = 9.999... - 0.999...

9a = 9

a = 1

0.999... = 1

Viewpoint 3:

1 - (1/10) = 0.9

1 - (1/100) = 0.99

1 - (1/1000) = 0.999

1 - (1/10000) = 0.9999

1 - lim k->oo(10^(-k))= 0.999...

1 - 0 = 0.999...

1 = 0.999...

0.999... = 1

If you have ever wondered why a^0 = 1 with a not being zero, consider the following:

a^0 = a^(p - p) = (a^p)/(a^p) = (a/a)^p = 1^p = 1

Actually, from the step (a/a)^p we can see that if you plug in zero, we get (0/0)^p, which is indeterminate, in another words can not be calculated, due to the rules of math. Thus, also 0^0 can not be calculated. Njäf! said.

I am Jesse Sakari Hyttinen and I will see you in the next post!