# COMPARING RATIONAL AND IRRATIONAL NUMBERS

## About "Comparing rational and irrational numbers"

Comparing rational and irrational numbers :

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

## Comparing rational and irrational numbers - Examples

Example 1 :

Compare  (√3 + 5) and 22/3 and write < or > between them.

Step 1 :

First approximate √3.

√3 is between 1 and 2

Step 2 :

Then use your approximations to simplify the expressions.

√3 + 5 is between 6 and 7.

Step 2 :

Now, find the value of 22/3.

The approximate value of 22/3 is 7.33.

So, we have

(√3 + 5) < 22/3

Example 2 :

Compare  (√2 + 4) and (2 + √4) and write < or > 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, we have

√2 + 4 > 2 + √4

Example 3 :

Compare  4√5 and 3√9 and write < or > between them.

Key concept :

Square both the numbers and compare them.

Step 1 :

Take square to the number 4√5.

(4√5)²  =  (4)²(√5)²

(4√5)²  =  (16)(5)

(4√5)²  =  80 --------> (1)

Step 2 :

Take square to the number 3√3.

(3√3)²  =  (3)²(√9)²

(3√3)²  =  (9)(9)

(3√3)²  =  81 --------> (2)

Step 3 :

From (1) and (2), we get

80 < 81  -----> 4√5 < 3√9

Example 4 :

Compare  (√3 + 3) and (√9 + √16) and write <, >, or = in between them.

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 6 and 7--------(1)

Step 2 :

Find the value of √9.

√9 is equal to 3.

Step 3 :

Find the value of √16.

√16 is equal to 4.

Step 4 :

√9 + √16  =  3 + 4

√9 + √16  =  7 --------(2)

Step 4 :

From (1) and (2), we get

√3 + 3  <  √9 + √16 After having gone through the stuff given above, we hope that the students would have understood "How to compare rational and irrational numbers".