Question

What do you do to check whether a number is rational or irrational?

In your explanation, use an example of an irrational and rational number.

Answer 1

Step-1: Identifying Rational and Irrational Numbers:

Step $\left(a\right)$: Check if the number is an integer or a fraction with an integer numerator and denominator. If it is, it is rational. If not, move to step $2$

Step $\left(b\right)$: Write any other numbers in decimal form. If the decimal stops at some point, it is rational. If not, move to step $3$.

Step $\left(c\right)$: If the decimal that continues forever has a repeating pattern, it is rational. If not, it is irrational.

Step-2: Rational number

A rational number is a number which can be written in the form of $\frac{\mathrm{p}}{\mathrm{q}}$ , where $\mathrm{p}$and $\mathrm{q}$ are integers and $q\ne 0$.

Step-3: Explain with an example.

Example-$1$.$\frac{4}{5}$

In the above example each numerator and denominator is integer and denominator $\ne 0$.Hence these are rational numbers.

Example-$2$ .$-9.75$

It is a decimal that stops, so it is a rational number.

Example-$3$. $-0.333...$

It is a decimal number that has repeating pattern, so it is a rational number.

Step-4: Irrational number

An irrational number cannot be expressed in $\frac{\mathrm{p}}{\mathrm{q}}$ form, and the decimal expansion of an irrational number is non-repeating and non-terminating.

Step-5: Explain with an example.

Example-$1$..$\sqrt{2}=\mathrm{1.41421356....}$

Hence, It is a decimal number which is non-terminating and does not have repeating pattern, so it is an irrational number.