Every number system describes the same value differently. 42 in decimal, 101010 in binary, 52 in octal, 2A in hex — all the same quantity, four different languages. Conversion is just translation.

Decimal → Any Base

Divide repeatedly by the target base. Read the remainders bottom to top.

42 to binary (base-2):

42 ÷ 2 = 21  r 0
21 ÷ 2 = 10  r 1
10 ÷ 2 = 5   r 0
 5 ÷ 2 = 2   r 1
 2 ÷ 2 = 1   r 0
 1 ÷ 2 = 0   r 1

Read up: 101010 ✓

Same method works for octal (divide by 8) and hex (divide by 16).

Any Base → Decimal

Multiply each digit by its positional power. Sum the results.

101010 from binary:

• 0 × 1 + 1 × 2 + 0 × 4 + 1 × 8 + 0 × 16 + 1 × 32
• = 0 + 2 + 0 + 8 + 0 + 32 = 42 ✓

Binary ↔ Octal

Group binary digits in threes from the right. Each group is one octal digit.

101010 → 101 010 → 5 2 → 52 ✓

Reverse: expand each octal digit into three binary digits.

Binary ↔ Hex

Group binary digits in fours from the right. Each group is one hex digit.

101010 → 0010 1010 → 2 A → 2A ✓

Reverse: expand each hex digit into four binary digits.

The Pattern

To decimal — positional multiplication, then sum. From decimal — repeated division, remainders read upward. Binary ↔ Octal — groups of three. Binary ↔ Hex — groups of four.

Octal and hex exist because binary is exhausting to read. Conversion is the bridge — and binary is always the middle ground.