**Comparing rational and irrational numbers :**

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**

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

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