Decimal Representation of Irrational Numbers

Decimal Representation of Irrational Numbers
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Irrational Numbers (Real) | Arithmetic | CK-12 Foundation


Hey, kids! We all know what irrational numbers are.

But ever wondered how we can represent irrational numbers in the decimal form if there is no actual limit to the digits of these numbers?

Let's learn about it in the article given here! 

Lesson Plan 

What Are Irrational Numbers?

Irrational numbers are real numbers that cannot be simplified into fractions.

Thus, the conversion of decimals to fractions for such numbers is also not possible. 

Rational and irrational numbers | mho maths


What Do You Mean by the Decimal Representation of Any Number?

Decimal representation is merely showing any given number in the form of decimal numbers. 

All rational and irrational numbers will have the following type of decimal representation:

Decimal representation of numbers

Irrational numbers are the ones that will have non-terminating decimals with non-repeating digits. 


How to Represent Irrational Numbers in Decimal Form?

The decimal representation of irrational numbers means expressing the most accurate value of the irrational number in the form of decimal numbers.

Try entering the values of any rational or irrational number in the given irrational number to decimal calculator. Check the output in the form of a decimal number yourself!

important notes to remember
Important Notes
  • Irrational numbers are real numbers that cannot be simplified into fractions.
  • Irrational numbers will have non-terminating decimals with non-repeating digits.

Solved Examples

Example 1



Jim bought \(100\) apples from a nearby fruit vendor, but later found out that \(5\) of them were rotten. Can you tell the fraction as decimals of the rotten apples to the total apples bought by Jim?


Here, we have \(5\) rotten apples out of \(100\).

So our fraction becomes,


How do we write it as decimals?

Such problems are solved by dividing the numerator by the denominator.

Here, we need to divide 5 by 100

To divide 5 by 100, we will simply shift it by 2 decimal places on the right.

The number of decimal places we can shift in the numerator depends upon the trailing zeroes the whole number in the denominator has.
Thus, after division, we get:


\(\therefore\) Rotten apples to the fresh apples in decimal form is \(0.05\). 
Example 2



What could be the decimal representation of \( \sqrt{16}\)?


We know that 16 is a square number.

Hence, the exact value of its square root will be a rational number. 

Hence, \( \sqrt{16} = \sqrt{4^2} = 4\)

\(\therefore\) It will be represented as 4.
Challenge your math skills
Challenging Questions

If the exact values of irrational numbers are known by us, then why do their decimal representations seem less exact?

Do you think there is a difference in the exact values and the decimal representations of such numbers?


Let's Summarize

The mini-lesson targeted the fascinating concept of Decimal representation of irrational numbers. The math journey around Decimal representation of irrational numbers starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp, but will also stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

Frequently Asked Questions (FAQs)

1. What is a decimal fraction?

Ans. Any simple fraction which has its denominator as a power of 10, will be known as a decimal fraction. 

2. How are irrational numbers expressed as decimals? 

Ans. Irrational numbers are expressed as non-terminating, non-recurring decimals.

3. Are irrational numbers integers?

Ans. No, an integer 'n' can be considered as \( \frac{{n}}{{1}}\) thus satisfying the definition of rational numbers.

4. Are all irrational numbers terminating?

Ans. Irrational numbers have non-terminating, non-recurring decimals. Thus no irrational number has a terminating decimal. 

5. Is the square root of 3 a non-terminating decimal? 

Ans. Yes, all square roots which aren't integers, are irrational numbers. Thus square root of 3 is a non-terminating decimal. 

6. Is 0.333 recurring an irrational number?

No, it is a rational number, as it is a non-terminating, repeating decimal number. 

7. Is 0.7 an irrational number?

No, it is a rational number since it is a terminating decimal.


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