# COMPARING IRRATIONAL NUMBERS

Comparing irrational numbers :

To compare and order irrational numbers, we can approximate irrational numbers as decimals.

## Comparing irrational numbers - Examples

Example 1 :

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

Step 1 :

First approximate √3.

√3 is between 1 and 2

Next approximate √5.

√5 is between 2 and 3.

Step 2 :

Then use your approximations to simplify the expressions.

√3 + 5 is between 6 and 7.

3 + √5 is between 5 and 6.

So, √3 + 5 > 3 + √5

Example 2 :

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

Step 1 :

First approximate √2.

√2 is between 1 and 2

Next approximate √4.

√4 is equal 2.

Step 2 :

Then use your approximations to simplify the expressions.

√2 + 4 is between 5 and 6.

2 + √4 is equal to 4.

So, √2 + 4 > 2 + √4

Example 3 :

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

Key concept :

Square both irrational numbers and compare them.

Step 1 :

Take square to the number 4√2.

(4√2)²  =  (4)²(√2)²

(4√2)²  =  (16)(2)

(4√2)²  =  32 --------> (1)

Step 2 :

Take square to the number 3√3.

(3√3)²  =  (3)²(√3)²

(3√3)²  =  (9)(3)

(3√3)²  =  27 --------> (2)

Step 3 :

From (1) and (2), we get

32 > 27  -----> 4√2 > 3√3

Example 4 :

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

Step 1 :

First approximate √12.

√12 is between 3 and 4

Next approximate √6.

√6 is between 2 and 3.

Step 2 :

Then use your approximations to simplify the expressions.

√12 + 6 is between 9 and 10.

12 + √6 is between 12 and 14.

So, √12 + 6 < 12 + √6

Example 5 :

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

Step 1 :

First approximate √5.

√5 is between 2 and 3

Next approximate √6.

√6 is between 2 and 3.

Step 2 :

Then use your approximations to simplify the expressions.

√5 + 6 is between 8 and 9.

5 + √6 is between 7 and 8.

So, √5 + 6 > 5 + √6

Example 6 :

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

Step 1 :

√3 + 3 --------> (1)

Step 2 :

Find the value of √9.

√9 is equal to 3.

Step 3 :

√3 + √9  =  √3 + 3 --------> (2)

Step 4 :

From (1) and (2), we get

√3 + 3  =  √3 + √9

After having gone through the stuff given above, we hope that the students would have understood "How to compare irrational numbers".

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