Base Systems
A "base" is just the size of the bucket you count in. In our everyday decimal system the base is 10, so each place is worth a power of 10 — units, tens, hundreds, and so on. Change the base to b and the place values become powers of b instead. Base systems matter because they expose what a number really is: a sum of digit times place value. Computers run on base 2 (binary) and group bits into base 8 (octal) and base 16 (hexadecimal), which is why this topic surfaces in CAT-style logical reasoning, in coding-aptitude rounds, and directly in XAT, SNAP, NMAT and IIFT number-system sets. This chapter builds three skills in order. First, reading any base-b number through place value, with binary and octal as the worked cases. Second, converting cleanly both ways — base-b to decimal using Horner-style nesting, and decimal to base-b using repeated division and reading remainders bottom-up. Third, doing arithmetic — addition, subtraction and multiplication — directly inside a base without detouring through decimal. Master the carry-and-borrow rule (you carry when a column reaches the base, not when it reaches ten) and most base questions collapse into ordinary, fast mental work.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Know the powers of 2 (1,2,4,8,16,32,64,128) and 8 (1,8,64,512) on sight — most binary/octal reads become pure addition.
- Base b → decimal: use Horner nesting (multiply running total by b, add next digit) instead of summing high powers.
- Decimal → base b: divide by b repeatedly and read the remainders bottom-to-top; the last remainder is the leftmost digit.
- Octal ↔ binary in one step: 1 octal digit = 3 binary bits; hex ↔ binary: 1 hex digit = 4 bits. Group, do not compute.
- Carry/borrow at the base, not at 10: in base b you carry 1 when a column reaches b, and a borrow adds b to the column.
- The largest n-digit number in base b is b^n − 1, so adding 1 rolls it to a clean power (666 + 1 = 1000 in base 7).
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Writing a digit equal to or larger than the base (an 8 in octal, a 2 in binary) — every digit must be 0 to b−1.
- Reading the division remainders top-to-bottom instead of bottom-to-top, which reverses the answer.
- Carrying or borrowing at 10 out of habit instead of at the actual base b.
- Forgetting the letter digits in hex (A=10 ... F=15) and writing 10 as two characters.
- Converting between two non-decimal bases directly by mismatched grouping — go through decimal unless the bases are 2/8/16.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Base Systems when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (3 topics) | 3/3 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
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 |