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How can I find out if the following decimal is rational or irrational:

0.140376266280752532561505065123............

Thanks for any help.

Guest Mar 30, 2018

Best Answer 

 #2
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+1

Try the "continued fraction" method as follows:

0.140376266280752532561505065123............

Take its reciprocal:

7.123711340206185567010309278351............

Subtract 7 and take the reciprocal of the fraction:

8.08333333333333333333333333....................

Subtract 8 and take the reciprocal of the fraction:

And it converges to 12.  Take this number, 12, and multiply it by the number above it:

12 x 8.0833333333333333333...... = 97. This will be the numerator:

Multiply 97 x 7.123711340206185567010309278351

= 691 - This is the denominator. So, in this particular case, the decimal appears to be rational:

97 / 691 - Try dividing it and you will get this:

0.1403762662 8075253256 1505065123 0101302460 2026049204 0520984081 0419681620 8393632416 7872648335 7452966714 9059334298 1186685962 3733719247 4674384949 3487698986 9753979739 5079594790 1591895803 1837916063 6758321273 5166425470 3328509406 6570188133 1403762662.........etc. Its "period", or when the digits start repeating is 230 digits!.

Guest Mar 30, 2018
edited by Guest  Mar 30, 2018
 #1
avatar+628 
+1

The definition of a rational number is any number that can be expressed as the quotient or fraction p/q of two integers.

 

For example, 1/7 = 0.142857142857...

 

This keeps going, it never ends but it is still rational. 

 

π, on the other hand is irrational because there is no pattern: 3.14159265358979323846...

 

Your number, as it appears has no pattern so it must be irrational.

supermanaccz  Mar 30, 2018
 #2
avatar
+1
Best Answer

Try the "continued fraction" method as follows:

0.140376266280752532561505065123............

Take its reciprocal:

7.123711340206185567010309278351............

Subtract 7 and take the reciprocal of the fraction:

8.08333333333333333333333333....................

Subtract 8 and take the reciprocal of the fraction:

And it converges to 12.  Take this number, 12, and multiply it by the number above it:

12 x 8.0833333333333333333...... = 97. This will be the numerator:

Multiply 97 x 7.123711340206185567010309278351

= 691 - This is the denominator. So, in this particular case, the decimal appears to be rational:

97 / 691 - Try dividing it and you will get this:

0.1403762662 8075253256 1505065123 0101302460 2026049204 0520984081 0419681620 8393632416 7872648335 7452966714 9059334298 1186685962 3733719247 4674384949 3487698986 9753979739 5079594790 1591895803 1837916063 6758321273 5166425470 3328509406 6570188133 1403762662.........etc. Its "period", or when the digits start repeating is 230 digits!.

Guest Mar 30, 2018
edited by Guest  Mar 30, 2018

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