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 — — — — d₁ d₀ . d_-1 d_-2 — — — — d_-m

In the above, d_n-1 to d₀ is integer part, then follows a radix point, and then d_-1 to d_-m is 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


(10)₁₀ = (1010)₂

Note: Keep multiplying the fractional part with 2 until decimal part .00 is obtained.
(.25)₁₀ = (.01)₂
Answer: (10.25)₁₀ = (1010.01)₂

2. Binary to Decimal

(1010.01)₂
1×2³ + 0x2² + 1×2¹+ 0x2⁰ + 0x2 ^-1 + 1×2 ^-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 and Binary

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


(3A)₁₆ = (00111010)₂
To convert from Binary to Hexadecimal, group the bits in groups of 4 and write the hex for the 4-bit binary. Add 0's to adjust the groups.
1111011011
(001111011011 )₂ = (3DB)₁₆