In grade \(9\), we have learnt that rational numbers have either terminating decimal expansion or non-terminating repeating decimal expansion. In this topic, we are going to learn when the rational number (\(\frac{p}{q}\)) is terminating and when it is non - terminating.
Consider the following rational numbers.
S. NoRational numberDenominator in powers of \(10\)Denominators in product of \(2\) and \(5\)
\(1\)\(0.125\)\(\frac{125}{1000} = \frac{125}{10^3}\)\(\frac{5^3}{2^3 \times 5^3}\)
\(2\)\(0.0002275\)\(\frac{2275}{10000000} = \frac{2275}{10^7}\)\(\frac{7 \times 13 \times 5^2}{2^7 \times 5^7} = \frac{7 \times 13}{2^7 \times 5^5}\)
\(3\)\(56.25\)\(\frac{5625}{100} = \frac{5625}{10^2}\)\(\frac{3^2 \times 5^4}{2^2 \times 5^2} = \frac{3^2 \times 5^2}{2^2}\)
From the above result, observe that the given number is converted into a rational number in the form of \(\frac{p}{q}\), where \(p\) and \(q\) are coprime. Since the denominators are in the powers of \(10\) which have \(2\) and \(5\) as factors and the prime factorisation of \(q\) is in the form of \(2^m 5^n\) where \(m\) and \(n\) are non-negative integers.
This leads to an important theorem which we shall learn in the upcoming topics.
Let us consider an example on how to find the termination of the decimal places of the given rational number.
After how many digits the rational number \(\frac{1}{2^6 \times 5^2}\) terminate?
Let us make the denominator as a whole number.
\(\frac{1}{2^6 \times 5^2} = \frac{1 \times 5^4}{2^6 \times 5^2 \times 5^4}\)
\(= \frac{625}{2^6 \times 5^6}\)
\(= \frac{625}{1000000}\)
\(= 0.00000625\)
Therefore, the given rational number terminates after \(6\) decimal places.
To find the termination of the rational number after how many decimal places, let us consider the following steps.
Step 1: Write the numerator and denominator of the rational number in the prime factors and cancel the common terms.
Step 2: If the denominator is not in the form of \(2^m \times 5^n\), then the rational number will have a non-terminating repeating decimal expansion.
Step 3: Determine the maximum of \(m\) and \(n\).
Consider the above example \(\frac{1}{2^6 \times 5^2}\).
Here, the denominator is in the form of \(2^m \times 5^n\).
The maximum of \(m\) and \(n\) is \(6\).
Therefore, the given rational number terminates after \(6\) decimal places.