Why do you convert hex to binary?

Answer 1

Hex is a convenient method for humans to notate binary numbers. Binary is a base-2 numbering system, but any base that is itself a power of 2 can be used to notate binary numbers using fewer digits. For instance, base-4 has only 4 digits: 0, 1, 2 and 3. Each of these digits can be mapped to any pair of binary digits (bits):

0 = 00

1 = 01

2 = 10

3 = 11

Thus the 24-bit binary sequence 100001000011111101110010 can be rewritten as 201003331302 in base-4 notation (replacing each pair of bits with its equivalent base-4 digit). In other words, instead of 24 binary digits we only need to write 12 base-4 digits. To convert back to binary, we simply replace each base-4 digit with its base-2 equivalent.

Base-8, or octal notation, has 8 digits: 0, 1, 2, 3, 4, 5, 6 and 7. Each of these can be mapped to any group of 3 bits:

0 = 000

1 = 001

2 = 010

3 = 011

4 = 100

5 = 101

6 = 110

7 = 111

Thus the 24-bit binary sequence 100001000011111101110010 becomes 41037561 in octal notation (replacing each group of 3 bits with its equivalent octal digit), which requires just one-third the number of digits. to convert back to binary, we simply replace each octal digit with its equivalent 3-bit grouping.

Hexadecimal (or hex) notation is base-16 and therefore uses 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F (0 to 15 decimal, respectively) and each hex digit can be mapped to a 4-bit grouping:

0 = 0000

1 = 0001

2 = 0010

3 = 0011

4 = 0100

5 = 0101

6 = 0110

7 = 0111

8 = 1000

9 = 1001

A = 1010

B = 1011

C = 1100

D = 1101

E = 1110

F = 1111

Thus the 24-bit binary sequence 100001000011111101110010 becomes 843F72 in hexadecimal notation, thus we use one-quarter the number of digits used in binary. Again, we can easily convert back to binary by replacing each hex digit with its equivalent 4-bit grouping.

A 24-bit value is typically represented as 3 separate bytes (8-bit groupings): 10000100 00111111 01110010. Splitting multi-byte values in this way is useful because each byte resides at a separate memory address and it is useful to know which specific value resides at which specific address, even when those values are combined to form a larger value. Hexadecimal typically follows suit, thus we get 84 3F 72.

Since 4-bits is half a byte, we often refer to hex digits as being nybbles.

Converting between hex and binary is an extremely trivial algorithm for humans to perform and anything that can be reduced to a trivial algorithm can therefore be easily programmed into a computer. Thus the computer can be programmed to convert our hexadecimal notation to its own native binary representation and vice versa. Converting from binary to decimal is not quite so straightforward, but the computer can still be programmed to convert from its native binary to any base we desire, whether it be decimal, octal, hexadecimal or even sexagecimal (base-60).

Believe it or not, we use sexagecimal every day without even realising it. The 24-hour clock is base-60 since we have 60 seconds to the minute and 60 minutes to the hour. 12 is also a factor of 60, as are 2, 3, 4, 5 and 6. It is this high range of divisibility that makes base-60 so useful, hence there are also 360 degrees of rotation.