# Thread: How To Multiply Repeatig Decimals

1. ## How To Multiply Repeatig Decimals

I was wondering, is it possible to multiply 0.333...*0.666... and get the correct result of 0.222... without first turning the repeating decimals into fractions.

I have played with a couple of ideas

Is it possible to multiply the 2 summations together?

I have also looked at 0.333...*0.666... = 3(6/10+6/10^2+6/10^3+....)
But I am wodering where to go from here.

Any ideas would be appreciated.

My conclusion is that turning them into fractions then multipying is the simplist way. I do want to see other methods, practical or not.

2. Originally Posted by Ranger SVO
I was wondering, is it possible to multiply 0.333...*0.666... and get the correct result of 0.222... without first turning the repeating decimals into fractions.

I have played with a couple of ideas

Is it possible to multiply the 2 summations together?

I have also looked at 0.333...*0.666... = 3(6/10+6/10^2+6/10^3+....)
But I am wodering where to go from here.

Any ideas would be appreciated.

My conclusion is that turning them into fractions then multipying is the simplist way. I do want to see other methods, practical or not.

I do not see any nice method. The proper method is to find their limits i.e. there fractions and multiply them.

Multiplying, infinite series is rather complicated.

3. If you have,
$S=a_0+a_1+a_2+...$
$S'=b_0+b_1+b_2+...
$

Then,
$SS'=a_0S'+a_1S'+a_2S'+...=b_0S+b_1S+b_2S+...$

4. I have tryed that and it works, but I think it could get a little complicated
Assume 0.3 and 0.6 is a repeating decimal. Also I should note that 0.19 is 0.1999...

The answer 0.2 is a repeating decimal.

Excuse the mess I did this in a hurry. Also 1.999... = 2 in the work above.

Any critisism is welcome

5. Originally Posted by Ranger SVO
.
Also 1.999... = 2 in the work above.
If you are asking whether it is a mistake it is not. I noticed many people thinking that such an equality is a mistake. It is a very common fallacy thinking that a decimal can be expressed in two different ways. Because you define 1.9999.... to be the value of the convergent of the real number which is 2.

Also, in set theory if you ever studied it. The Cantor's Diagnol Argument starts out as,
Originally Posted by Cantor
...if a number is able to be expressed in two different ways as a decimal then....
It is clearly true.

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### Multiply 0.666 into 10

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