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!
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