### Is 0.9999... really the same as 1?

Posted:

**Fri Mar 03, 2017 6:24 am**I've gone back and forth on this issue and am curious what the general public thinks about the answer given my reservations about it.

Firstly there's something called the Archimedean property or Archimedean axiom which states that there's no such thing as infinity or an infinitesimal real numbers. In that sense there is no difference between 1 and 0.99999... If we take 1-0.99999... = 0.0000....1 but since the 0's never end and there's no such thing as infinitesimal real numbers this is the same as 0.

Other proofs exist such as 1/3 + 2/3 = 3/3 or 0.33333... + 0.66666... = 0.99999... = 3/3 = 1

Let x = 0.9999999...

then 10x = 9.9999999...

Then subtract the x from 10x:

10x = 9.9999999...

x = 0.9999999...

=================

9x = 9

x = 1

however, I've come to notice the "=" sign is being used in a slightly different context than something like 1+1=2. It's behaving more like the equal sign in a limit

Like the example lim (x->infinity) 1 - 10^(-x) = 1

We can see that lim (x->infinity) 1 - 10^(-x) approaches 1 here:

f(1) = 0.9

f(2) = 0.99

f(3) = 0.999

etc...

f(x) may approach 1 as x increases but still will never reach 1. But the "=" sign in this context means "approaches" and not "of the same value", or is it?

Firstly there's something called the Archimedean property or Archimedean axiom which states that there's no such thing as infinity or an infinitesimal real numbers. In that sense there is no difference between 1 and 0.99999... If we take 1-0.99999... = 0.0000....1 but since the 0's never end and there's no such thing as infinitesimal real numbers this is the same as 0.

Other proofs exist such as 1/3 + 2/3 = 3/3 or 0.33333... + 0.66666... = 0.99999... = 3/3 = 1

Let x = 0.9999999...

then 10x = 9.9999999...

Then subtract the x from 10x:

10x = 9.9999999...

x = 0.9999999...

=================

9x = 9

x = 1

however, I've come to notice the "=" sign is being used in a slightly different context than something like 1+1=2. It's behaving more like the equal sign in a limit

Like the example lim (x->infinity) 1 - 10^(-x) = 1

We can see that lim (x->infinity) 1 - 10^(-x) approaches 1 here:

f(1) = 0.9

f(2) = 0.99

f(3) = 0.999

etc...

f(x) may approach 1 as x increases but still will never reach 1. But the "=" sign in this context means "approaches" and not "of the same value", or is it?