represent a rational number as terminating or non terminating repeating decimal recurring
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Answer:
Integers are positive and negative whole numbers including zero, such as {-3, -2, -1, 0, 1, 2, 3}.
When these whole numbers are written in the form of ratio of whole numbers it is known as rational numbers. So, rational numbers can be positive, negative or zero. So, a rational number can be expressed in the form of p/q where ‘p’ and ‘q’ are integers and ‘q’ is not equal to zero.
Rational Numbers in Decimal Fractions:
Rational numbers can be expressed in the form of decimal fractions. These rational numbers when converted into decimal fractions can be both terminating and non-terminating decimals.
Terminating decimals: Terminating decimals are those numbers which come to an end after few repetitions after decimal point.
Example: 0.5, 2.456, 123.456, etc. are all examples of terminating decimals.
Non terminating decimals: Non terminating decimals are those which keep on continuing after decimal point (i.e. they go on forever). They don’t come to end or if they do it is after a long interval.
For example: π =(3.141592653589793238462643383279502884197169399375105820974.....) is an example of non terminating decimal as it keeps on continuing after decimal point.
If a rational number (≠ integer) can be expressed in the form p2n×5m, where p ∈ Z, n ∈ W and m ∈ W, the rational number will be a terminating decimal. Otherwise, the rational number will be a nonterminating, recurring decimal.
For example:
(i) 58 = 523×50. So, 58 is a terminating decimal.
(ii) 91280 = 928×51. So, 91280 is a terminating decimal.
(iii) 445 = 432×51. Since it is not in the form \(\frac{p}{2^{n} × 5^{m}}\), So, 445 is a non-terminating, recurring decimal.
For example let us take the cases of conversion of rational numbers to terminating decimal fractions:
(i) 12 is a rational fraction of form pq. When this rational fraction is converted to decimal it becomes 0.5, which is a terminating decimal fraction.
(ii) 125 is a rational fraction of form pq. When this rational fraction is converted to decimal fraction it becomes 0.04, which is also an example of terminating decimal fraction.
(iii) 2125 is a rational fraction form pq. When this rational fraction is converted to decimal fraction it becomes 0.016, which is an example of terminating decimal fraction.
Now let us have a look at conversion of rational numbers to non terminating decimals:
(i) 13 is a rational fraction of form pq. When we convert this rational fraction into decimal, it becomes 0.333333… which is a non terminating decimal.
(ii) 17 is a rational fraction of form pq. When we convert this rational fraction into decimal, it becomes 0.1428571428571… which is a non terminating decimal.
(iii) 56 is a rational fraction of form pq. When this is converted to decimal number it becomes 0.8333333… which is a non terminating decimal fraction.
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Answer:
convert a terminating decimal into rational number p/q form