Real Numbers and Decimals

Rational numbers are in the form of fractions. We can find equivalent rational numbers as we find for fractions.

The word rational refers to something reasonable, understandable, within reason.

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Example :

Note that, every integer is a rational numbers.

Consider these rational numbers, .

The first quotient 0.375 ends up and gives the remainder 0, and this kind of decimal is called as terminating decimal.

But, the second and the third quotients, 1.11â€¦and 4.66â€¦. will not give the remainder 0 and the number 1 and 6 repeats itself so many times. Such kind of decimals is called as non-terminating decimals.

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Let us see some examples :

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Example
Show that 1.8282â€¦.= in the form of , where a and b are integers and b â‰  0.
Solution
Let us take x = 1.8282â€¦. ----------(i)

Multiply by 100 on both the sides.

100 x = 182. 82â€¦â€¦ -----------(ii)

Subtract equation (i) from equation (ii)

100 x â€“ x = 182.82â€¦.. â€“ 1.8282â€¦.

99 x = 181

x =

Hence, we can conclude that every non-terminating recurring decimal number can be expressed as a rational number.

Now, let us see one more example, in which a rational number has been expressed in the decimal form.

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Example

Find the maximum number of repeating block of digits in the decimal expansion of . ( Use division method to find the decimal expansion).

Solution

Here, the rational number can be expressed as a repeating decimal number.

= .

So, the decimal value of a rational number is either a terminating decimal or a non-terminating decimal.
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The decimal value of an irrational number is a non-terminating and non-recurring decimal.

Let us take an example for the non-terminating and non-recurring decimal.
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Example

Show that 92 Ã· 7 gives a non-terminating and non-recurring decimal.

Solution
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The quotient which we get 13.1428â€¦ is a non-terminating and non-recurring decimal.

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Examples : -

Irrational + Irrational = Rational

Irrational â€“ Irrational = Rational

Irrational

Â´ Irrational = Rational

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Irrational Â¸ Irrational = Rational

Rational Â´ Irrational = Irrational

Irrational â€“ Irrational = Irrational

Irrational + Irrational = Irrational

Irrational Â¸ Irrational = Irrational