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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.

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