3. Hex: Binary in Groups of Four

The concept

Binary is precise, but it's painful for a person to read, write, or copy by hand. An 8-bit number is already eight symbols to keep straight, and real machine words are often twice or four times that long — a long run of visually similar 0s and 1s is exactly the kind of thing an eye slips on while copying it down. Hex exists to fix that readability problem, not to give the computer a third way to store numbers — underneath, it's still plain binary. Hex is just a shorthand for writing binary more compactly.

The shorthand works because of a coincidence of arithmetic: a group of 4 bits has exactly 16 possible patterns (0000 through 1111), and 16 is also how many symbols a single hex digit can hold. That match means every possible 4-bit pattern maps to exactly one hex digit, with none left over and none overlapping. So the whole trick is: split a binary number into groups of 4 bits (called nibbles) starting from the right, then replace each nibble with the one hex digit that stands for it.

Since a nibble can hold 16 different values, hex needs 16 different digit symbols — more than decimal's 0-9. It reuses 0-9 for the first ten values, then continues with the letters a through f for the remaining six:

binary   decimal   hex
0000        0       0
0001        1       1
0010        2       2
0011        3       3
0100        4       4
0101        5       5
0110        6       6
0111        7       7
1000        8       8
1001        9       9
1010       10       a
1011       11       b
1100       12       c
1101       13       d
1110       14       e
1111       15       f

This lesson writes hex numbers with a 0x prefix and lowercase letters — 0x4c, not 4C or $4c — so a hex number is never mistaken for a decimal one or a variable name. And unlike binary, hex numbers don't need internal grouping to stay readable: each digit already is one nibble, so hex is already about as compact as this shorthand gets.

Example 1 — binary to hex

Take 0110 1101 — the same 8-bit number from Lesson 2's Example 2, whose decimal value was 109. It's already split into two nibbles. Convert each one using the table above:

0110 → 6
1101 → 13 → d

So 0110 1101 is 0x6d in hex. Nothing about the value changed — 109, 0110 1101, and 0x6d are three different ways of writing the exact same number.

Example 2 — hex to binary, then to decimal

Take 0xb4 and reverse the process: expand each hex digit back into its 4-bit nibble.

b → 1011
4 → 0100

So 0xb4 is 1011 0100 in binary. From there, Lesson 2's place-value method gives the decimal value: bits 7, 5, 4, and 2 hold a 1, with place values 128, 32, 16, and 4:

128 + 32 + 16 + 4 = 180

0xb4 is 180.

Example 3 — why octal doesn't fit as neatly (a brief aside)

Some historic machines grouped bits in threes instead of fours (octal, digits 0-7) rather than hex — it's the same shorthand idea, just with a 3-bit group instead of a 4-bit one. The catch is that 8-bit widths don't divide evenly by 3: splitting 1011 0100 into groups of three from the right gives 10 110 100 — two clean 3-bit groups, but a leftover 2-bit group at the front that needs an extra padding zero (010) before it means anything. Hex never has this problem for the widths this course uses, because 8, 16, and 32 are all multiples of 4 — groups of four always come out even. That clean fit, not any deeper advantage, is why hex is this course's default shorthand and octal only gets a passing mention.