 Irrational numbers are one type of number in mathematics that cannot be expressed as a fraction. A number line is a straight line with numbers placed at equal intervals along its length. On this page, you can learn the simple way to represent a square root number on the number line using the Pythagoras theorem. Check more details like how to represent irrational numbers on the number line and problems on it.

## Representing Irrational Numbers on The Number Line

We use a simple concept called the Pythagoras theorem for the representation of irrational numbers on the number line. We already know that, irrational number can never be written in the form of $$\frac { a }{ b }$$. Some of the examples are √2, √5, √7, and so on.

Let us take a right-angled triangle ABC with AB, BC, CA as perpendicular, base, hypotenuse sides and AB = x units, BC = y units, the hypotenuse AC = √(x² + y²). Following are the steps that are helpful to point the irrational number on the number line.

• Draw a number line and mark the centre point as zero.
• Mark the right side of zero as positive numbers and the left side as negative numbers.
• Decide to take the left side or right side based on the given irrational number.
• Draw a perpendicular line to the real number such that the new line has a length of 1 unit.
• Join the point (0) and the end of a new line of unity length to form a right-angled triangle.
• Name perpendicular side as AB, base as BC, and hypotenuse as AC of the right-angled triangle ABC.
• The length of hypotenuse AC can be calculated by applying Pythagoras theorem.
• Now take AC as the radius, C as the centre and cut an arc on the same number line and name that point as D.
• Since AC is the radius of the arc and CD is the length of the irrational number.
• Hence, D is the representation of an irrational number on the number line.

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### Questions on Representation of Irrational Numbers on The Number Line

Question 1:
Represent √2 on the number line.

Solution:
To represent √2 on the number line, draw a number line.
Draw a line perpendicular to point 1 and its length should be 1 unit.
Join points 0, end of the new line and that right-angle triangle.
Using the Pythagorean theorem,
AC² = AB² + BC²
= 1² + 1² = 2
AC = √2 Take 0 as centre C, with AC as radius, cut the number line at point D which is √2 Question 2:
Represent √5 on the number line.

Solution:
To represent √5 on the number line, draw a number line.
Draw a line perpendicular to point 1 and its length should be 2 units.
Join points 0, end of the new line and that right-angle triangle.
Using the Pythagorean theorem,
AC² = AB² + BC²
= 2² + 1² = 5
AC = √5
Take 0 as centre C, with AC as radius, cut the number line at point D which is √5. Question 3:
How to represent √3 on the number line?

Solution:
To represent √3 on a number line, we have to represent √2 on it.
At √2 on the number line, draw a line that is perpendicular to AB from point A such that the new line has unit length and a new point is E.
Join C and E. Using Pythagoras theorem,
AE² + Ac² = EC²
EC² = 1² + (√2)² = 1 + 2 = 3
EC = √3
With C as centre and EC as radius, cut the number line at F which is √3. 