COMPARING RATIONAL AND IRRATIONAL NUMBERS

Key Concept :

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

Example 1 :

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

Answer :

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. 

Answer :

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.

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 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

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