Math CentralQuandaries & Queries


Question from Clara, a student:

Find an irrational number between 0.53 (with 53 as repeating) and 0.54 (with 54 repeating)

I changed each to 53/99 and 54/99 with 1/99 being the difference.

Please help me.

We have three responses for you

Hi Clara,

What you have done so far is excellent. The number you want then is 53/99 + a where a is a positive irrational number less than 1/99 = 0.010101... Take some irrational number that you know like √2 - 1.4142... How can you use this to find an irratioinal number between 0 and 1/99?

I hope this helps,


Well, do you know a favourite rational number, say sqrt2? If your favourite is sqrt2, which is about 1.4, you can use this to find an irrational between any two rationals as follows: if your two given rationals are a and b, where a >b, then sqrt2/2(a-b) is irrational; what can you say about b + sqrt2/2(a-b)?



Hi Clara.

Any rational number plus an irrational number makes an irrational number. So if you can find an irrational number less than 1/99, then just add that to 53/99 and you have solved your problem.

An easy way to find such a number is to just divide a well known irrational like pi by a large enough value.

Stephen La Rocque.


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