Is a repeating decimal a Rational Number

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Is a repeating decimal a Rational Number
Is a repeating decimal a Rational Number
In today's session we are going to learn that is a repeating decimal a rational
We will discuss here that how a repeating decimal is a rational number. In the
mathematics, there are a number of rational numbers that can be converted in
the form of repeating decimal numbers.
Now, come to the point, starting the session with the help of an example. Let
say, one repeating decimal number is 0.333333333... This number is also
repeating decimal number or non terminating decimal number and it never goes
to end.

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As we know that this number can also be written as 1/3 in the form of rational
But now we'll write this number in the form of rational number (a / b) by some of
the proving with the help of a rule. So, let we take a number 0.186186186186...,
and applying some of the rule to write it in the form of rational: a =
0.186186186186 (equation 1) 10 a = 1.86186186186 100 a = 18.18618618618
1000 a = 186.18618618618 (equation 2).
Similarly, we can write it for much number of times, now we have to subtract the
equation 1 with equation2: 1000 a = 186.18618618618 - a = 0.186186186186
we will get here: 999 a = 186 Now, evaluating the value of a, we have: a= 186 /
999 so this number is a rational number.
Sometimes we face some repeating decimal numbers whose first two or three
digits are not following the pattern but other are the part of repeating decimal.
For example, non-terminating number 4.50398989898 is such type of decimal.
Now, again we perform such a process that the repeating pattern gets cancelled
and we can write it in the form of rational number.

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Taking it as example: a = 4.50398989898 (Equation 1) 10 a = 45.0398989898
100 a = 450.398989898 1000 a = 4503.98989898 (Equation 2). Now, again
same subtraction as we have done in above problem: 1000 a =
4503.98989898 - a = 4.50398989898 evaluating result, we get: 999 a =
4499.486 so, a = 4499.486 / 999 or a = 4499486 / 999000.


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