**Conversion of Bases-**

In number system, it is very important to have a good knowledge of how to convert numbers from one base to another base.

In the previous articles, we learnt-

**How to convert a number from any given base to base 10?****How to convert a number from base 10 to any other base?**

Before you proceed further, make sure that you have gone through these articles. We will use the concepts learnt in these articles here.

In this article, we will learn how to convert a number from any given base to any other base.

Suppose the given number has base x1 and we want to convert the number in base x2.

(Given Number)→_{base x1 } (?)_{base x2} |

To perform such kind of conversions, we will follow the following steps-

**Step-01:**

Convert the number from base x1 to base 10 using expansion method.

**Step-02:**

Convert the number from base 10 to base x2 using division / multiplication method.

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

**Problem-01:**

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

**Solution-**

**(1056)**_{16} → ( ? )_{8}

_{16}→ ( ? )

_{8}

**Step-01: Converting 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}

**(1056) _{16} = (4182)_{10}**

**Step-02: Converting to base 8-**

**(4182)**_{10} → ( ? )_{8}

_{10}→ ( ? )

_{8}

Using Division method, we have-

**(4182) _{10} = (10126)_{8}**

Thus,

(1056)_{16} = (10126)_{8} |

**Problem-02:**

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

**Solution-**

**(11672)**_{8} → ( ? )_{16}

_{8}→ ( ? )

_{16}

**Step-01: Converting 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}

**(11672) _{8} = (5050)_{10}**

**Step-02: Converting to base 16-**

**(5050)**_{10} → ( ? )_{16}

_{10}→ ( ? )

_{16}

Using Division method, we have-

**(5050) _{10} = (13BA)_{16}**

Thus,

(11672)_{8} = (13BA)_{16} |

**Problem-03:**

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

**Solution-**

**(2724)**_{8} → ( ? )_{5}

_{8}→ ( ? )

_{5}

**Step-01: Converting 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}

**(2724) _{8} = (1492)_{10}**

**Step-02: Converting to base 5-**

**(1492)**_{10} → ( ? )_{5}

_{10}→ ( ? )

_{5}

Using Division method, we have-

**(1492) _{10} = (21432)_{5}**

Thus,

(2724)_{8} = (21432)_{5} |

**Problem-04:**

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

**Solution-**

**(3211)**_{4} → ( ? )_{5}

_{4}→ ( ? )

_{5}

**Step-01: Converting 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}

**(3211) _{4} = (229)_{10}**

**Step-02: Converting to base 5-**

**(229)**_{10} → ( ? )_{5}

_{10}→ ( ? )

_{5}

Using Division method, we have-

**(229) _{10} = (1404)_{5}**

Thus,

(3211)_{4} = (1404)_{5} |

**Problem-05:**

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

**Solution-**

**(1001001100)**_{2} → ( ? )_{6}

_{2}→ ( ? )

_{6}

**Step-01: Converting 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}

**(1001001100) _{2} = (588)_{10}**

**Step-02: Converting to base 6-**

**(588)**_{10} → ( ? )_{6}

_{10}→ ( ? )

_{6}

Using Division method, we have-

**(588) _{10} = (2420)_{6}**

Thus,

(1001001100)_{2} = (2420)_{6} |

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