COMPARING IRRATIONAL NUMBERS

About "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. 

Answer : 

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. 

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, √2 + 4 > 2 + √4. 

Example 3 : 

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

Answer : 

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. 

Answer : 

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. 

Answer : 

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. 

Answer : 

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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