# Number System and Base Conversions

Electronic and Digital systems may use a variety of different number systems, (e.g. Decimal, Hexadecimal, Octal, Binary).

A number N in base or radix b can be written as:

(N)_{b}= d_{n-1}d_{n-2}-- -- -- -- d_{1}d_{0}. d_{-1}d_{-2}-- -- -- -- d_{-m}

In the above, d_{n-1} to d_{0} is the integer part, then follows a radix point, and then d_{-1} to d_{-m} is the fractional part.

d_{n-1} = Most significant bit (MSB)

d_{-m} = Least significant bit (LSB)

**How to convert a number from one base to another?**

Follow the example illustrations:

## 1. Decimal to Binary

(10.25)_{10}

**Note: **Keep multiplying the fractional part with 2 until decimal part 0.00 is obtained.

(0.25)_{10} = (0.01)_{2}

**Answer:** (10.25)_{10} = (1010.01)_{2}

## 2. Binary to Decimal

(1010.01)_{2}1x2^{3}+ 0x2^{2}+ 1x2^{1}+ 0x2^{0}+ 0x2^{-1}+ 1x2^{-2}= 8+0+2+0+0+0.25 = 10.25 (1010.01)_{2}= (10.25)_{10}

## 3. Decimal to Octal

(10.25)_{10}(10)_{10}= (12)_{8}Fractional part: 0.25 x 8 = 2.00

**Note:** Keep multiplying the fractional part with 8 until decimal part .00 is obtained.

(.25)_{10} = (.2)_{8}

**Answer:** (10.25)_{10} = (12.2)_{8}

## 4. Octal to Decimal

(12.2)_{8}1 x 8^{1}+ 2 x 8^{0}+2 x 8^{-1}= 8+2+0.25 = 10.25 (12.2)_{8}= (10.25)_{10}

## 5. Hexadecimal to Binary

To convert from Hexadecimal to Binary, write the 4-bit binary equivalent of hexadecimal.

(3A)_{16} = (00111010)_{2}

## 6. Binary to Hexadecimal

To convert from Binary to Hexadecimal, start grouping the bits in groups of 4 from the right-end and write the equivalent hexadecimal for the 4-bit binary. Add extra 0’s on the left to adjust the groups.

1111011011001111011011(001111011011 )_{2}= (3DB)_{16}

This article is contributed by **Kriti Kushwaha**.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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