Why is the denominator of a recurring fraction always 9?

Answer 1

No it isn't, 1/6 has a denominator of 6 and a recurring decimal of 0.1666...

However, I think you mean why is a denominator of 9 used to convert a recurring decimal into a fraction.

This requires the use of weird properties of the infinite and an infinitely long number.

For example take the recurring decimal 0.333... where the 3 recurs.

Let the fraction for this be represented by x. Then:

x = 0.333...

Now multiply both sides by 10:

10x = 3.333...

We can now subtract the first from the second to get:

10x - x = 3.333... - 0.333...

→ 9x = 3

This is the weird propert of infinity: both of the numbers had exactly the same number of 3s after the decimal point and they cancel out to leave the whole number before the decimal point. This can now be solved by dividing both sides by 9, but leaving the right hand side as a fraction which is simplified:

9x = 3

→ x = 3/9 = 1/3.

So effectively, if there was one recurring digit that recurring digit was placed over one 9 and the fraction simplified.

Similarly if there are 2 recurring decimals, eg 0.18181818...

This time multiply both sides by 100 = 10² before doing the subtraction:

x = 0.181818...

10x = 18.181818...

100x - x = 18.181818... - 0.181818...

→ 99x = 18

x = 18/99 = 2/11

In general if there are n repeating digits, then multiply the decimal by 10^n and when the subtraction is done this leaves 10^n - 1 = 9 n times, which then becomes the denominator (before simplification) when both sides are divided by it.

Or to put it another way: if there are n repeating digits, make them the numerator over n 9s as the denominator.

If there are some non repeating digits before repeating digits, this is solved as the addition of two fractions.

For example, consider 0.1666... which has the digit 1 followed by 6 repeating forever.

0.1666... = 0.1 + 0.0666... = 0.1 + 1/10 × 0.666...

Now 0.1 is a terminating decimal so is converted to a fraction by putting the one digit (1) over 10 (a 1 followed by one 0), and 0.666... is converted as above by putting the 6 over one 9. This then gives:

-0.1666... = 1/10 + 1/10 × 6/9 = 1/10 + 2/30 = 3/30 + 2/30 = 5/30 = 1/6.

This can be extended for more than one non-repeating digit:

If a decimal has m non-repeating digits and n repeating digits, then its fraction is found by:

put the non-repeating digits over 10^m (which is 1 followed by m 0s) and add 1/10^m × the repeating digits over n 9s.