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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L.C.M method to solve time and work problems
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