give mathematical/logical reasons as to why 0.9999~ ≠ 1?

Question

There are plenty of proofs and reasons as to how and why they are equal.

But, inductively, it seems fairly obvious that 0.999 isnt the same as 1... its just slightly smaller... infinitesimally smaller.

Can anyone give sound logic as to why they are not the same?

Answer 1

Proof by contradiction:
Assume 0.999...9 = 1
Then, 1 - 0.999...9 = 0, because if you take equal away from equal you get 0, Common Notion (known in mathematics)
But, 1 - 0.999...9 = 0.000...1
So, our assumption must be incorrect. 1 does not equal 0.999...9

hope it helps

Answer 2

Hey there!

0.9999... can equal to 1. The decimal is repeating infinitely. The decimal can be expressed as a sum of 9/10+9/100+9/1000...

Now convert the denominator into base 10.

In other words.

9/10^1+9/10^2+9/10^3...

Factor out the 9.

9(1/10^1+1/10^2+1/10^3)... -->
9((1/10)+(1/10)^2+(1/10)^3)... -->
((1/10)+(1/10)^2+(1/10)^3)... -->
(r+r^2+r^3...) -->
a=1/10, r=1/10 -->
a/(1-r) -->
(1/10)/(1-1/10) -->
(1/10)/(9/10) -->
1/9 -->
9(1/9) -->
1

This is why 0.999... is 1. If 0.999... is expressed definitely, then 1294's answer is right.

Hope it helps!

Answer 3

On the contrary, I can give a logical proof that it equals 1

Every next 9 added after the decimal reduces the difference from 1....... when the process is repeated till infinity, the difference must come to zero..... if the difference has still not come to zero, all we need is to add more and more 9's after the decimal. Hence the logical conclusion to this is that the difference must eventually come down to zero.

Hope you like this logical reply.

Answer 4

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