Hexadecimal

Motivation: Easy Convertion

Computers work with binary numbers, but humans don't. We humans are used to decimal numbers, but the "binary-decimal" is not a natural idea. A better option is using hexadecimal numbers, or simply hex. Why hex? Take a look at the following convertions:

Decimal	Binary	Hex
0	0000	0
1	0001	1
2	0010	2
3	0011	3
4	0100	4
5	0101	5
6	0110	6
7	0111	7
8	1000	8
9	1001	9
10	1010	A
11	1011	B
12	1100	C
13	1101	D
14	1110	E
15	1111	F
16	0001 0000	10
17	0001 0001	11
55	0011 0111	37
195	1100 0011	C3

Key Ideas:

• 1 hex digit <=> 4 bits.

• For a long binary string, we can pad 0's on the left to some multiples of 4 and divide the binary string into chunks of length 4. For example, 10001 can be padded as 00010001 and divided into two chunks 0001 0001.

• 1 byte <=> 2 hex digits. The convertion is easy enough, even for humans.

Binary => Decimal

For example, convert 0b10110001 to decimal. From right to left:

• 1 => 1 * 2^0 = 1

• 0 => 0 * 2^1 = 0 (skipped)

• 0 => 0 * 2^2 = 0 (skipped)

• 0 => 0 * 2^3 = 0 (skipped)

• 1 => 1 * 2^4 = 16

• 1 => 1 * 2^5 = 32

• 0 => 0 * 2^6 = 0 (skipped)

• 1 => 1 * 2^7 = 128

Therefore 0b10110001 = 1 + 16 + 32 + 128 = 177.

Decimal => Binary

For example, convert 26 to binary:

• 26 / 2 = 13 ...... 0

• 13 / 2 = 6 ...... 1

• 6 / 2 = 3 ...... 0

• 3 / 2 = 1 ...... 1

• 1 / 2 = 0 ...... 1 (algorithm terminates)

Therefore 26 = 0b11010.

Hex => Decimal

For example, convert 0x125 to decimal. From right to left:

• 5 => 5 * 16^0 = 5

• 2 => 2 * 16^1 = 32

• 1 => 1 * 16^2 = 256

Therefore 0x125 = 5 + 32 + 256 = 293.

Decimal => Hex

For example, convert 293 to hex:

• 293 / 16 = 18 ...... 5

• 18 / 16 = 1 ...... 2

• 1 / 16 = 0 ...... 1

Therefore 293 = 0x125.