Base Conversion
Two reliable methods cover every conversion. To go from base b to decimal, use Horner nesting instead of summing powers: start with the leftmost digit, then repeatedly multiply the running total by b and add the next digit. For (2 1 3) in base 5: ((2·5 + 1)·5 + 3) = 58. This avoids computing high powers and is far faster under time pressure. To go from decimal to base b, divide the number by b repeatedly, jot each remainder, and read the remainders from bottom to top — the last remainder is the leftmost digit. For bases above 10, remember the letter digits: 10 = A, 11 = B, up to 15 = F in hexadecimal. To convert between two non-decimal bases, the safe route is base-b → decimal → target base, though binary/octal/hex can shortcut through 3-bit or 4-bit grouping. Always check your answer by converting back.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Place value & conversion
| Value of a base-b number | (d_n...d_1d_0)_b = d_n·b^n + ... + d_1·b + d_0 |
|---|---|
| Horner (nested) evaluation | ((...(d_n·b + d_{n-1})·b + ...)·b + d_0) |
| Decimal → base b | Divide by b repeatedly; remainders bottom-to-top |
| Digits allowed in base b | 0, 1, 2, ..., (b−1) |
| Max value of an n-digit base-b number | b^n − 1 |
Bases & arithmetic
| Binary / Octal / Hex bases | b = 2, 8, 16 (hex digits A=10 ... F=15) |
|---|---|
| Octal ↔ binary grouping | 1 octal digit = 3 binary digits |
| Hex ↔ binary grouping | 1 hex digit = 4 binary digits |
| Carry in base b | Carry 1 when a column sum reaches b |
| Borrow in base b | A borrow adds b (not 10) to the column |