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Every non-terminating repeating decimal expansion can be represented in (p/q) form where p and q are integers and q≠0

Let’s first start with the definition of rational numbers. I know that definitions in Mathematics appear to be boring but remember, they are the foundations of mathematics. The strength of your foundation will decide the heights of your mathematical concepts.

So let’s start with the definition.

A number ‘r’ is called a rational number, if it can be written in  form (p/q) where p and q are integers and q≠0

And there is one more interesting fact related to rational numbers. It is that non-terminating repeating decimal is a rational number. So obviously, every non-terminating repeating decimal expansion can be represented in  form (p/q) where p and q are integers and q≠0

Let’s observe this with the help of an example

Let a non-terminating repeating decimal expansion be 0.635635……

Let x=0.635635……

Multiply both sides by 1000

1000x=635.635………

1000x=635+x       where  x=0.635635……

999x=635

x = (635/999)

Where p=635 and q=999

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