4. Converting Between Binary, Hex, and Decimal by Hand

The concept

Lessons 2 and 3 already cover every conversion this lesson needs — nothing here is a new technique. What's new is the strategy for combining them, and that strategy isn't the same in both directions.

Lesson 2 gave a direct path between binary and decimal (place value, in both directions), and Lesson 3 gave a direct path between binary and hex (the nibble table, in both directions). That leaves decimal and hex without a path of their own — but only one direction actually needs one:

decimal → hex   no shortcut: pivot through binary
                 (Lesson 2, then Lesson 3)

hex → decimal   direct: same place-value idea as Lesson 2,
                 just with powers of 16 instead of powers of 2

Building a hex digit from a decimal number means asking, at each position, "how many times does this place value fit?" — a division, not the simple yes/no test Lesson 2's binary method uses at each of its 8 positions. So for decimal → hex, pivoting through binary really is the easier hand method: it trades one division problem for eight yes/no checks. But reading a decimal value out of hex digits that already exist is just multiplication and addition — no division, no detour needed. That's the same place-value idea Lesson 2 opened with ("each position is worth the base raised to a power"), just instantiated at base 16 instead of base 2.

Example 1 — decimal to hex, through binary

Take decimal 214. First, Lesson 2's method: find which place values sum to it.

128 ≤ 214 → bit 7 = 1, remainder 214 − 128 = 86
 64 ≤  86 → bit 6 = 1, remainder 86 − 64 = 22
 32 ≤  22? no → bit 5 = 0
 16 ≤  22 → bit 4 = 1, remainder 22 − 16 = 6
  8 ≤   6? no → bit 3 = 0
  4 ≤   6 → bit 2 = 1, remainder 6 − 4 = 2
  2 ≤   2 → bit 1 = 1, remainder 2 − 2 = 0
  1 ≤   0? no → bit 0 = 0

That gives 1101 0110. Now Lesson 3's method: convert each nibble.

1101 → 13 → d
0110 → 6

214 is 1101 0110 in binary, and 0xd6 in hex.

Example 2 — hex to decimal, directly

Take 0x9e. No binary needed this time — Lesson 3's table already gives each hex digit's decimal value (its middle column), so just multiply each digit by its place value and add. A two-digit hex number only has a ones place (16⁰ = 1) and a sixteens place (16¹ = 16):

9 × 16 = 144
e (14) × 1 = 14

144 + 14 = 158

0x9e is 158 — one lookup, one multiplication, one addition.

Example 3 — three names, one value

Binary, hex, and decimal are never three different numbers — they're three different ways of writing the same one. Take 1010 0111:

binary:   1010 0111
hex:      a → 1010, 7 → 0111, so 0xa7
decimal:  bits 7, 5, 2, 1, 0 → 128 + 32 + 4 + 2 + 1 = 167

1010 0111, 0xa7, and 167 are the same value, just written three ways. Whichever one a problem hands over, the other two are only a nibble-table lookup and a place-value sum away.