Rational Numbers in Terminating and Non-Terminating Decimals

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 \(\frac{p}{2^{n} × 5^{m}}\), 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) \(\frac{5}{8}\) = \(\frac{5}{2^{3} × 5^{0}}\). So, \(\frac{5}{8}\) is a terminating decimal.

(ii) \(\frac{9}{1280}\) = \(\frac{9}{2^{8} × 5^{1}}\). So, \(\frac{9}{1280}\) is a terminating decimal.

(iii) \(\frac{4}{45}\) = \(\frac{4}{3^{2} × 5^{1}}\). Since it is not in the form \(\frac{p}{2^{n} × 5^{m}}\), So, \(\frac{4}{45}\) is a non-terminating, recurring decimal.

For example let us take the cases of conversion of rational numbers to terminating decimal fractions:

(i) \(\frac{1}{2}\) is a rational fraction of form \(\frac{p}{q}\). When this rational fraction is converted to decimal it becomes 0.5, which is a terminating decimal fraction.

(ii) \(\frac{1}{25}\) is a rational fraction of form \(\frac{p}{q}\). When this rational fraction is converted to decimal fraction it becomes 0.04, which is also an example of terminating decimal fraction.

(iii) \(\frac{2}{125}\) is a rational fraction form \(\frac{p}{q}\). 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) \(\frac{1}{3}\) is a rational fraction of form \(\frac{p}{q}\). When we convert this rational fraction into decimal, it becomes 0.333333… which is a non terminating decimal.

(ii) \(\frac{1}{7}\) is a rational fraction of form \(\frac{p}{q}\). When we convert this rational fraction into decimal, it becomes 0.1428571428571… which is a non terminating decimal.

(iii) \(\frac{5}{6}\) is a rational fraction of form \(\frac{p}{q}\). When this is converted to decimal number it becomes 0.8333333… which is a non terminating decimal fraction.

Irrational Numbers:

We have different types of numbers in our number system such as whole numbers, real numbers, rational numbers, etc. Apart from these number systems we have Irrational Numbers. Irrational numbers are those which do not terminate and have no repeating pattern. Mr. Pythagoras was the first person to prove a number as irrational number. We know that all square roots of integers that don’t come out evenly are irrational. Another best example of an irrational number is ‘pi’ (ratio of circle’s circumference to its diameter).

π = (3.141592653589793238462643383279502884197169399375105820974.....)

First three hundred digits of ‘pi’ are non-repeating and non-terminating. So, we can say that ‘pi’ is an irrational number.

Rational Numbers

Decimal Representation of Rational Numbers

Rational Numbers in Terminating and Non-Terminating Decimals

Recurring Decimals as Rational Numbers

Laws of Algebra for Rational Numbers

Comparison between Two Rational Numbers

Rational Numbers Between Two Unequal Rational Numbers

Representation of Rational Numbers on Number Line

Problems on Rational numbers as Decimal Numbers

Problems Based On Recurring Decimals as Rational Numbers