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

So,

√3 + 5  >  3 + √5

Example 2 : 

Compare  (√2 + 4) and (2 + √4) 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 :

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

√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 :

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  (√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

So,

√12 + 6  <  12 + √6

Example 5 : 

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

So,

√5 + 6  >  5 + √6

Example 6 : 

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

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