COMPARING IRRATIONAL NUMBERS

Key Concept

1. If two irrational numbers radicals are in the form

(a + √b) and (c + √d),

estimate the value of each irrational number and compare them.

2. If two irrational numbers are in the form

√a and √b,

square each irrational number to get rid of the square root. Then, compare them.

Example 1 :

Compare (√3 + 5) and (3 + √5) and write <, >, or = in between them.

Answer :

Step 1 :

Approximate √3.

√3 is between 1 and 2

Step 2 :

Approximate √5.

√5 is between 2 and 3

Step 3 :

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

√3 + 5 is between 6 and 7

3 + √5 is between 5 and 6

Therefore,

√3 + 5 > 3 + √5

Example 2 :

Compare (√2 + 4) and (√2 + √25) and write <, >, or = in between them.

Answer :

Step 1 :

Approximate √2.

√2 is between 1 and 2

Step 2 :

Approximate √4.

√4 is equal 2

Step 3 :

Approximate √25.

√25 is equal 5

Step 4 :

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

√2 + 4 is between 5 and 6

√2 + √25 = √2 + 5

√2 + 5 is between 6 and 7

√2 + 4 < √2 + 5

Therefore,

√2 + 4 < 2 + √25

Example 3 :

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

Answer :

Step 1 :

Square 4√2.

(4√2)2 = (4)2(√2)2

(4√2)2 = (16)(2)

(4√2)2 = 32 ----> (1)

Step 2 :

Square 3√3.

(3√3)2 = (3)2(√3)2

(3√3)2 = (9)(3)

(3√3)2 = 27 ----> (2)

Step 3 :

Comparing (1) and (2),

32 > 27 ----> 4√2 > 3√3

Example 4 :

Compare 3√8 and 4√5 and write < or > between them. 

Answer :

Step 1 :

Take square to the number 3√8.

(3√8)2 = 32(√8)2

(3√8)2 = 9(8)

(3√8)2 = 72 ----> (2)

Step 2 :

Take square to the number 4√5.

(4√5)2 = 42(√5)2

(4√5)2 = 16(5)

(4√5)2 = 80 ----> (1)

Step 3 :

From (1) and (2), we get 

72 < 80 ----> 3√8 < 4√5

Example 5 :

Compare (√12 + 6) and (12 + √6) and write <, >, or = in between them.

Answer :

Step 1 :

Approximate √12. 

√12 is between 3 and 4

Step 2 :

Approximate √6.

√6 is between 2 and 3

Step 3 :

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

√12 + 6 is between 9 and 10

12 + √6 is between 12 and 14

Therefore,

√12 + 6 < 12 + √6

Example 6 :

Compare (√5 + 6) and (5 + √6) and write <, >, or = in between them.

Answer :

Step 1 :

Approximate √5.

√5 is between 2 and 3

Step 2 :

Approximate √6.

√6 is between 2 and 3

Step 3 :

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

√5 + 6 is between 8 and 9

5 + √6 is between 7 and 8

Therefore,

√5 + 6 > 5 + √6

Example 7 :

Compare (√3 + 3) and (√3 + √9) and write <, >, or = in between them.

Answer :

√3 + 3 ----(1)

√3 + √9 = √3 + 3 ----(2)

Comparing (1) and (2),

√3 + 3 = √3 + √9

Example 8 :

Compare (√3 + 3) and (√9 + √3) and write <, >, or = in between them.

Answer :

Step 1 :

First approximate √3.

√3 is between 1 and 2

Step 2 :

Then use your approximations to simplify the expressions.

√3 + 3 is between 4 and 5 ----(1) 

Step 3 :

Find the value of √9.

√9 is equal to 3

Step 4 :

√9 + √3 = 3 + √3

3 + √3 is between 4 and 5 ----(2) 

Step 5 :

From (1) and (2), we get,

√3 + 3 = 3 + √3

Therefore,

√3 + 3 = √9 + √3

Example 9 :

Compare √3 and 𝝅 and write <, >, or = in between them.

Answer :

Step 1 :

First approximate √3.

√3 is between 1 and 2

Step 2 :

Using calculator, approximate 𝝅.

𝝅 = 3.1415......

The value of 𝝅 is between 3 and 4.

Therefore,

√3 < 𝝅

Example 10 :

Compare (√2 + 5) and (𝝅 + 2) and write <, >, or = in between them.

Answer :

Step 1 :

First approximate √2.

√2 is between 1 and 2

Step 2 :

Then use your approximation to simplify the expression.

√2 + 5 is between 6 and 7 ----(1)

Step 3 :

The value of 𝝅 is between 3 and 4.

Step 4 :

Then use your approximation to simplify the expression.

𝝅 + 2 is between 5 and 6 ----(2)

Therefore,

√2 + 5 < 𝝅 + 2

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