Binary & Octal
Binary (base 2) uses only the digits 0 and 1; its place values are 1, 2, 4, 8, 16, 32, 64, 128 — the powers of 2 you should know on sight. Octal (base 8) uses digits 0 to 7, with place values 1, 8, 64, 512. The key idea is place value: in any base b the rightmost digit is worth 1, the next worth b, the next b², and so on. A digit must always be less than the base, so 8 can never appear in octal and 2 can never appear in binary — spotting an illegal digit is a quick CAT sanity check. A neat link the exam loves: one octal digit equals exactly three binary digits, because 2³ = 8. So 101 110 (binary) groups straight into 56 (octal) with no arithmetic. Learn the first eight powers of 2 cold and most binary work becomes instant addition.
✅ 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 |