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How To Convert A Recurring Number Into A Fraction?

Although it may seem counter-intuitive, every single recurring decimal in the world can be represented as a fraction.

The process is surprisingly straightforward and utilises the repetitive nature of the number and our base-10 number system.

Let’s try to represent the decimal above as a fraction:

1. Observe how many numbers are being repeated

If we wrote the recurring decimal out in full, we would get 0.582582… meaning there are three numbers being repeated.

2. Subtract two different powers of 10 multiplied by this recurring decimal from each other such that the recurring part after the decimal point is the same

We know that three numbers are being repeated meaning if we multiplied the recurring decimal by 10³, we would 582.582582… where the bit after the decimal point is the same as before. Now we can subtract the original value (i.e ×10¹) to get a whole number which is essentially 999 times the original value.

3. Rearrange to get fractional representation

Knowing that this new whole number is 999 times the original value, we can rearrange this equation to get a fraction which is the original recurring decimal as a fraction. Hope that makes sense?



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Akhil Sonthi

Tech Enthusiast | Entrepreneur | Music Artist | MEng @ Cambridge