**Number System Conversions-**

Before you go through this article, make sure that you have gone through the previous article on **Basics of Number System**.

In number system,

- It is very important to have a good knowledge of how to convert numbers from one base to another base.
- Here, we will learn how to convert any given number from any base to any other base.

**Conversion of Bases-**

A given number in base x can be converted to any other base y using the following steps-

**Step-01:**

Convert the number from base x to base 10 using expansion method.

**Read More-** **Conversion to Base 10**

**Step-02:**

Convert the number from base 10 to base y using division & multiplication method.

**Read More-**

**PRACTICE PROBLEMS BASED ON CONVERSION OF BASES-**

**Problem-01:**

Convert (1056)_{16} to ( ? )_{8}

**Solution-**

**Step-01: Conversion To Base 10-**

**(1056)**_{16} → ( ? )_{10}

_{16}→ ( ? )

_{10}

Using Expansion method, we have-

(1056)_{16}

= 1 x 16^{3} + 0 x 16^{2} + 5 x 16^{1} + 6 x 16^{0}

= 4096 + 0 + 80 + 6

= (4182)_{10}

From here, (1056)_{16} = (4182)_{10}

**Step-02: Conversion To Base 8-**

**(4182)**_{10} → ( ? )_{8}

_{10}→ ( ? )

_{8}

Using Division method, we have-

From here, (4182)_{10} = (10126)_{8}

Thus, (1056)_{16} = (10126)_{8}

**Problem-02:**

Convert (11672)_{8} to ( ? )_{16}

**Solution-**

**Step-01: Conversion To Base 10-**

**(11672)**_{8} → ( ? )_{10}

_{8}→ ( ? )

_{10}

Using Expansion method, we have-

(11672)_{8}

= 1 x 8^{4} + 1 x 8^{3} + 6 x 8^{2} + 7 x 8^{1} + 2 x 8^{0}

= 4096 + 512 + 384 + 56 + 2

= (5050)_{10}

From here, (11672)_{8} = (5050)_{10}

**Step-02: Conversion To Base 16-**

**(5050)**_{10} → ( ? )_{16}

_{10}→ ( ? )

_{16}

Using Division method, we have-

From here, (5050)_{10} = (13BA)_{16}

Thus, (11672)_{8} = (13BA)_{16}

**Problem-03:**

Convert (2724)_{8} to ( ? )_{5}

**Solution-**

**Step-01: Conversion To Base 10-**

**(2724)**_{8} → ( ? )_{10}

_{8}→ ( ? )

_{10}

Using Expansion method, we have-

(2724)_{8}

= 2 x 8^{3} + 7 x 8^{2} + 2 x 8^{1} + 4 x 8^{0}

= 1024 + 448 + 16 + 4

= (1492)_{10}

From here, (2724)_{8} = (1492)_{10}

**Step-02: Conversion To Base 5-**

**(1492)**_{10} → ( ? )_{5}

_{10}→ ( ? )

_{5}

Using Division method, we have-

From here, (1492)_{10} = (21432)_{5}

Thus, (2724)_{8} = (21432)_{5}

**Problem-04:**

Convert (3211)_{4} to ( ? )_{5}

**Solution-**

**Step-01: Conversion To Base 10-**

**(3211)**_{4} → ( ? )_{10}

_{4}→ ( ? )

_{10}

Using Expansion method, we have-

(3211)_{4}

= 3 x 4^{3} + 2 x 4^{2} + 1 x 4^{1} + 1 x 4^{0}

= 192 + 32 + 4 + 1

= (229)_{10}

From here, (3211)_{4} = (229)_{10}

**Step-02: Conversion To Base 5-**

**(229)**_{10} → ( ? )_{5}

_{10}→ ( ? )

_{5}

Using Division method, we have-

From here, (229)_{10} = (1404)_{5}

Thus, (3211)_{4} = (1404)_{5}

**Problem-05:**

Convert (1001001100)_{2} to ( ? )_{6}

**Solution-**

**Step-01: Conversion To Base 10-**

**(1001001100)**_{2} → ( ? )_{10}

_{2}→ ( ? )

_{10}

Using Expansion method, we have-

(1001001100)_{2}

= 1 x 2^{9} + 0 x 2^{8} + 0 x 2^{7} + 1 x 2^{6} + 0 x 2^{5} + 0 x 2^{4} + 1 x 2^{3} + 1 x 2^{2} + 0 x 2^{1} + 0 x 2^{0}

= 512 + 64 + 8 + 4

= (588)_{10}

From here, (1001001100)_{2 }= (588)_{10}

**Step-02: Conversion To Base 6-**

**(588)**_{10} → ( ? )_{6}

_{10}→ ( ? )

_{6}

Using Division method, we have-

From here, (588)_{10 }= (2420)_{6}

Thus, (1001001100)_{2} = (2420)_{6}

To gain better understanding about Conversion of Bases,

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