Rational and Irrational Numbers

You probably deal with numbers every day; splitting bills, converting units, or calculating discounts. But there are two big types of numbers that behave very differently. Rational numbers include all the fractions and decimals you are familiar with. Irrational ones are a bit more complicated types, like square roots and pi. They never end or repeat. Learning the rational number definition, differences between the two makes everyday math quicker, smoother, and way less stressful. So, let’s dive in!

1.0What Is a Rational Number?

A rational number is simply a number which can be represented in the form of a fraction, meaning it’s the ratio of two integers. In math terms, that’s:

Rational Number = a/b

In this, both a and b are integers. Also, it is important that b is not equal to 0. 

So, whether it’s a positive number, a negative number, or even zero, if it is possible to write it as a fraction, it’s rational.

Fun Fact:

The word ‘rational’ is derived from “ratio,” which makes perfect sense because rational numbers are all about comparing two integers!

Example:

3/2 = 1.5

Here, 3 and 2 are both integers, and the number is formulated in the form of a fraction, so it’s a rational number.

2.0What Is an Irrational Number?

An Irrational number is any real number that cannot be written as a fraction of two integers. Their decimal forms go on forever without repeating any pattern.

You can still write them in decimal form, but they never end, and they never fall into a neat, predictable pattern.

Example:

$\sqrt{8}$ = 2.8284271

The digits keep going, never repeat, and it can’t be written as a simple fraction—so it’s irrational!

3.0Rational vs Irrational Number

Not sure whether a number is rational or irrational? Here’s how you can tell the difference between rational and irrational numbers:

4.0Examples 

Examples of Rational Numbers:

• 9 - Can be written as 9/1
• 0.5 - Equals 1/2​, which is a fraction
• $\sqrt{81}$ = 9- A perfect square root that simplifies to an integer
• 0.7777... - Repeating decimal, so it’s rational

Examples of Irrational Numbers:

• Π ≈ 3.14159 - Goes on forever without repeating
• 2 ≈ 1.414.. - Not a perfect square
• 0.212112111... - Non-terminating, non-repeating; cannot be written as a fraction

The irrational numbers list goes on and on.

5.0Features of Irrational and Rational Numbers

Now, let’s see how rational and irrational numbers behave when we add or multiply them.

Rational Numbers

1. When two rational numbers are added, the outcome is rational.

½ + ⅓ = ⅚ 

2. The product of two rational numbers is rational.

½ × ⅓ = ⅙ 

Irrational Numbers

3. The sum of two irrational numbers is not always irrational.

$\sqrt{2}+\sqrt{2}=2\sqrt{2}$ (Still irrational)

But:

$2+2\sqrt{5}+(-2\sqrt{5})=2$ (Rational)

4. The product of two irrational numbers is not always irrational.

$\sqrt{2}\times \sqrt{3}=\sqrt{2}$ (Irrational)

$\sqrt{2}\times \sqrt{2}=2$ (Rational)

So, irrational numbers like to keep things interesting. Sometimes their behaviour surprises us!