Monday, March 16, 2009

Are repeating numbers irrational ?

The question of the nature of repeating numbers comes up when we convert fractions into binary, as even apparently simple fractions in decimal becomes an infinitely long string in binary, for example:

0.110 = 0.0001100110011 ....2 = 0.000112.

On first appearances, we seem to have "changed" the type of number we are representing, just through the change of base. Have we in effect converted a rational number into an irrational number ?

No, we haven't ! This new representation of the number is still rational. The proof is as follows:

A rational number is defined as a number that can expressed as the quotient of two integers (e.g. 0.1 = 1/10).

We can express an repeating number as a geometric series:
e.g. 0.997997997997 ..... = 0.997 + 0.997 (1/1000)1 + 0.110(1/1000)2 + ...... etc.

where a = 0.997 and r = (1/1000)

It is well know that the sum of geometric series of this type = a/(1 -r), which will result in a ratio of integers (in this example 997/999).

This, because an infinitely repeating numbers sequence can be represented as a geometric series and the sum of a geometric series can expressed as a ratio of integers, such numbers must be rational.

QED (Quite easily done for non Latin speakers)

Note: Thank you to associate Sergey Suslov for his thoughts on this topic.

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