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 = dn-1 dn-2 -- -- -- -- d1 d0 . d-1 d-2 -- -- -- -- d-m

In the above, d_n-1 to d₀ 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 
1x23 + 0x22 + 1x21+ 0x20 + 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 81 + 2 x 80 +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. 

1111011011
0011 1101 1011
(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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