Module 3 · Number System
Chapter 38: Base Systems
Number System
3 topics
What this chapter covers
The 3 topics below make up Base Systems on the CAT. Click any topic to open its notes, examples and practice test.
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Formula Reference Sheet
This chapter
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 |
CAT reference
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