Rational and Non-rational numbers

RATIONAL AND NON-RATIONAL NUMBERS

Meaning of Numbers

In mathematics, we use numbers to count, measure, and compare things. But not all numbers are the same — some can be written neatly as fractions or decimals, and some cannot. That is what we mean by Rational and Non-Rational (or Irrational) numbers.

What Are Rational Numbers?

A Rational Number is any number that can be written as a fraction — that is, a number that can be written in the form:

a/b, where a and b are whole numbers, and b ≠ 0.

That means a rational number is any number that can be expressed as a part of something — like half, quarter, or even a whole number.

Examples of Rational Numbers:

1/2 (one-half)

3/4 (three-quarters)

5 (because 5 can be written as 5/1)

-2 (because -2 can be written as -2/1)

0.25 (because 0.25 = 25/100)

So, all fractions, all whole numbers, and all decimal numbers that stop or repeat are rational.

What Are Non-Rational (Irrational) Numbers?

A Non-Rational Number (or Irrational Number) is a number that cannot be written as a fraction a/b.

These numbers go on and on forever when written as decimals, and they never stop or form a repeating pattern.

Examples of Non-Rational Numbers:

√2 = 1.41421356… (goes on forever, no pattern)

π (pi) = 3.14159265… (goes on forever, no pattern)

√3 = 1.7320508…

√5 = 2.2360679…

These numbers are not neat — you cannot write them exactly as fractions. They go on forever without repeating.

Comparing Rational and Non-Rational Numbers

Rational Numbers	Non-Rational Numbers
Can be written as a fraction a/b.	Cannot be written as a fraction.
Decimal form stops or repeats.	Decimal form goes on forever without repeating.
Examples: 3/4, -2, 0.5, 7	Examples: √2, π, √3
They are easy to locate exactly on the number line.	They cannot be located exactly (we can only estimate).

Other Groups of Rational Numbers

Rational numbers include many smaller groups:

Whole numbers: 0, 1, 2, 3, 4, …

Integers: …, -3, -2, -1, 0, 1, 2, 3, …

Fractions: 1/2, 3/4, -5/6

Terminating decimals: 0.25, 0.75

Recurring decimals: 0.333…, 0.666…

All these are rational because they can be written as a/b.

How to Know if a Number is Rational or Non-Rational

Try to write the number as a fraction (a/b). If you can, it’s rational.

If the decimal stops (like 0.75) or repeats (like 0.333…), it’s rational.

If the decimal never stops and has no pattern (like √2 = 1.4142135...), it’s non-rational.

If the number comes from a square root that is not perfect (like √3, √5, √7), it’s non-rational.

Worked Examples

Example 1:
Is 3/4 a rational or non-rational number?

Step 1: It is already written as a fraction (3 ÷ 4).
Step 2: So it can be written as a/b.
Answer: Rational Number.

Example 2:
Is √2 a rational or non-rational number?

Step 1: Try to write it as a fraction — you cannot.
Step 2: As a decimal, it is 1.41421356… which never ends or repeats.
Answer: Non-Rational Number.

Example 3:
Is 0.5 a rational number?

Step 1: 0.5 can be written as 5/10 = 1/2.
Step 2: It is a fraction.
Answer: Rational Number.

Example 4:
Is π (pi) a rational or non-rational number?

Step 1: π = 3.14159265... (it goes on forever, never repeats).
Step 2: It cannot be written as a/b.
Answer: Non-Rational Number.

Example 5:
Is -2.75 a rational number?

Step 1: -2.75 = -275/100.
Step 2: It can be written as a fraction.
Answer: Rational Number.