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# If one third = 0.3(recurring) and two thirds equal 0.6(recurring)...

why does 3 thirds equal one instead of 0.9(recurring)?

25 May 2023

This is a very interesting question and the short answer is that 0.9 (recurring) is equal to 1. Recurring decimals are fascinating because they go on forever and things get weird when we deal with infinity.

If we say that a = 0.9 (recurring) then 10a = 9.9 (recurring). Therefore 10a - a = 9 as the decimal part of both 10a and a are exactly the same. But that gives us 9a = 9 so a = 1.

We can also think of a recurring decimal as an infinite sum of fractions:

0.9 (recurring) = 0.9 + 0.09 + 0.009 + 0.0009 + …

So 0.9 (recurring) = 9/10 + 9/100 + 9/1000 + 9/10000 + …

Sums containing an infinite number of terms, known as infinite series, can challenge our understanding of very basic mathematical concepts. Even though we have an infinite number of terms in the sum, the result may be finite. This particular series is known as a geometric series. Starting with 9/10, we can get the next term in the series by multiplying 9/10 by 1/10 (9/100), and the next term is again 9/100 times 1/10. So, to get to the next term, all we need to do is to multiply the current term by 1/10.

We can evaluate this infinite series using a similar method as before. Say that:

S = 9/10 + 9/100 + 9/1000 + 9/10000 + …

So 10S = 9 + 9/10 + 9/100 + 9/1000 + 9/10000 + …

We then have:

10S - S = 9 because all the fractional terms cancel out. Therefore 9S = 9 hence S = 1.

That’s why 0.9 (recurring) = 1.