Math Is Fun Forum

WildSpirit

Member

Registered: 2016-02-29

Posts: 1

Does 0.9... really = 1? Not in my view!

If 0.9... is an infinite number then why do mathematicians insist it is equal to 1 which is finite.

The classic proof that it is 1 is 3 x 1/3 = 1 and since 1/3 = 0.3... then 3 x 0.3... = 1. However, you can not complete the calculation of 1/3. It is infinite. So you can not say it is 1/3 completely. In fact 1/3 is just a symbolic way of representing an unfinished division. It is easy to make the mistake that if 3 x 1/3 = 1 then 3 x 0.3... = 1.

Another flawed proof is
Let X = 0.999...
   
Then 10X = 9.999...

Subtract X from each side to give us: 9X = 9.999... - X

but we know that X is 0.999..., so: 9X = 9.999... - 0.999...
or:   
9X = 9

Divide both sides by 9 gives X = 1

The flaw in this proof is that to complete the calculations they MUST be finite values so 9 x X really in this case is 9.99 (not 9.999...).

So the complete calculation of 9X - X is 9.99 - 0.999 which is 8.991 which is clearly NOT 9.

Also, 1 is always < 0.n

Also, 1 is a different number to 0.9...

The other classic proof that 0.9...=1 is using limits.

Mathematicians declare that if something is approaching a limit then 'at  ∞' it IS the value of the limit. This step jumps over the fact that the number approaching the limit is infinite and makes it a finite number (the limit). Its one small step for mathkind, but one giant leap for valid logic! If you want an example of why this approach doesn't work in every case then try n/ ∞. By the rule that at  ∞ it is the limit we would expect therefore that n/ ∞ = 0 since 0 is the limit.

Please discuss.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Does 0.9... really = 1? Not in my view!

Hi;

If it will make it easier for you to understand just think that it is defined that .999999... = 1. Now you can move on to more interesting mathematics.

We have many threads with this topic, for instance http://www.mathisfunforum.com/viewtopic.php?id=5418 please use that from now on for this subject.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.