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Comparing Irrational Numbers

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Comparing Irrational Numbers Worksheet PDF

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Compare the values of π and √10


Solution: π (pi) is approximately 3.14159, and √10 is approximately 3.16228. Therefore, π is slightly smaller than √10.

Compare the values of π and √10.

Solution: π (pi) is approximately 3.14159, and √10 is approximately 3.16228. Therefore, π is slightly smaller than √10.


Case 1. If two irrational numbers, Radicals take the form of


(a +b ) and (c+d),


To compare irrational numbers, estimate each one’s value and compare them.


 Case 2 . If two irrational integers have the form


√a and √b,


To eliminate the square root of any irrational number, square it. Compare them next.


Solved Examples 


Example 1  : Compare (√3 + 7) and (3 + √7) and write <, >, or = in between them.

Answer :


Step 1 : Approximate √3.

√3 is between 1 and 2

Step 2 : Approximate √7.

√7 is between 2 and 3


Step 3 : Use your approximations in the above steps to estimate the values of the given irrational numbers.

√3 + 7 is between 7 and 8

3 + √7 is between 5 and 6



√3 + 7 > 3 + √7


Example 2 :

Compare 5√2 and 4√3 and write <, >, or = in between them.

Answer :


Step 1 : Square 5√2.

(5√2)2 = (5)2(√2)2

(5√2)2 = (25)(2)

(5√2)2 = 50 —-> (1)


Step 2 : Square 4√3.

(4√3)2 = (4)2(√3)2

(4√3)2 = (16)(3)

(4√3)2 = 48 —-> (2)


Step 3 : Comparing (1) and (2),

50 > 48 —-> 5√2 > 4√3

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Practice Questions :


 Q 1. Compare the values of π and √10.

Solution: π (pi) is approximately 3.14159, and √10 is approximately 3.16228. Therefore, π is slightly smaller than √10.


Q 2. Which is greater, √45 or √50?

Solution: √45 ≈ 6.708 and √50 ≈ 7.071. Since 7.071 is greater than 6.708, √50 is larger than √45.


Q 3.Which is closer to 2, √7 or √8?

Solution: √7 ≈ 2.646 and √8 ≈ 2.828. Since 2.646 is closer to 2 than 2.828 is, √7 is closer to 2.


Q 4.Compare the values of √2 and ∛3.

Solution: √2 ≈ 1.414 and ∛3 ≈ 1.442. Since 1.442 is greater than 1.414, ∛3 is larger than √2.


Q5.Which is larger, √6 or ∛8?

Solution: √6 ≈ 2.449 and ∛8 = 2. Since 2.449 is greater than 2, √6 is larger than ∛8.

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What is an irrational number?

An irrational number is a real number that cannot be expressed as a simple fraction or ratio of two integers. They are non-terminating and non-repeating decimals and cannot be represented as a fraction.

How do I compare two irrational numbers?

Comparing irrational numbers can be challenging since they do not have simple fraction representations. You can compare them by approximating their decimal representations or by using mathematical inequalities.

Can irrational numbers be equal?

Yes, two irrational numbers can be equal. For example, the square root of 2 (√2) and √2 can be considered equal since they both represent the same value.

What is the relationship between rational and irrational numbers?

Rational numbers are a subset of real numbers, including rational and irrational ones. Irrational numbers are the real numbers that are not rational. The two sets are complementary.

Can irrational numbers be added or multiplied like rational numbers?

Yes, irrational numbers can be added, subtracted, multiplied, and divided like rational numbers. However, the results may be irrational, and the operations might lead to more complicated or non-terminating decimals.

Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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