Concern from lil, a student:

Why are repeating decimals thought about rational numbers?

We have two responses for you

Hi Lil,

The answer is yes, yet prior to I show why I am going to quibble via the method you asked the question. A repeating decimal is not considered to be a rational number it is a rational number. We have different means of representing numbers, for example the variety of fingers on my left hand have the right to be represented by the English word 5, or the French word cinq or the symbol 5 or the Romale character V or the fractivity 10/2 or many kind of other means. Similarly the fraction 1/3 deserve to be represented by the decimal number 0.3333... These are 2 different ways of representing the very same number.

A rational number is a number that have the right to be stood for a/b wright here a and b are integers and b is not equal to 0.

You are watching: Prove that every infinite decimal representing a rational number is recurring

A rational number deserve to likewise be stood for in decimal develop and also the resulting decimal is a repeating decimal. (I check out the decimal 0.25 as repeating since it deserve to be written 0.25000...) Also any kind of decimal number that is repeating can be written in the form a/b via b not equal to zero so it is a rational number. Let me illustrate through an instance.

Consider the repeating decimal n = 2.135135135... The repeating part (135) is 3 digits lengthy so I am going to multiply n by 103 to gain 103 n = 2135.135.135... Now I subtract


Thus n = 2133/999 and considering that 9 separated both the numerator and also denominator this have the right to be created n = 237/11.


Hi there,

Repeating decimals are considered rational numbers because they have the right to be represented as a proportion of 2 integers.

To represent any kind of pattern of repeating decimals, divide the section of the pattern to be recurring by 9"s, in the adhering to way:

0.2222222222... = 2/9

0.252525252525... = 25/99

0.1234567123456712345671234567... = 1234567/9999999

The number of 9"s in the denominator must be the same as the number of digits in the repetitive block. These rational numbers might of course be reducible, if the height is divisible by 9, or both the optimal and bottom are divisible by one more number. But this is a starting suggest which will certainly always gain you what you desire.

Why does this work? Well, we have the right to go into a bit even more information and also create out our repeating decimal, say 0.252525252525..., as an limitless series of decreasing fractions, prefer so

0.252525252525... = 2/10 + 5/100 + 2/1000 + 5/10000 + 2/100000 + 5/1000000 + ...

Now let this series be equal to x, that is

x = 2/10 + 5/100 + 2/1000 + 5/10000 + 2/100000 + 5/1000000 + ...

now multiply both sides by 100

100x = 20 + 5 + 2/10 + 5/100 + 2/1000 + 5/10000 + ...

Now subtract the first equation from the second choose so:

100x = 20 + 5 + 2/10 + 5/100 + 2/1000 + 5/10000 + 2/100000 + 5/1000000 + ... -x = - 2/10 - 5/100 - 2/1000 - 5/10000 - 2/100000 - 5/1000000 - ... -------------------------------------------------------------------------------------------------------------------- 99x = 25

currently rearrange for x and gain

x = 25/99

which is what we were looking for! So 25/99 really does equal 0.252525252525...

See more: Is The Sentence “ Could You Please Send Me A Copy Of An Official Document

I hope this helps!


Math Central is sustained by the College of Regina and also The Pacific Institute for the Mathematical Sciences.