Number Systems Cheat Sheet - IIM CAT 2026

1. Set of Numbers

A deep understanding of number classifications is required to solve CAT problems involving domain constraints and diophantine equations.

Natural Numbers (\(\mathbb{N}\))

The set of counting numbers starting from 1. Defined as: \[\mathbb{N} = \{1, 2, 3, 4, \dots\}\]

Whole Numbers (\(\mathbb{W}\))

The set of natural numbers including zero: \[\mathbb{W} = \{0, 1, 2, 3, \dots\}\]

Integers (\(\mathbb{Z}\))

Whole numbers along with their negative counter-parts: \[\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}\]

Rational Numbers (\(\mathbb{Q}\))

Numbers that can be expressed as a fraction \(p/q\), where \(p, q \in \mathbb{Z}\) and \(q eq 0\). Decimal expansions are either terminating or non-terminating repeating (recurring).

Examples: \(1/2\), \(-5\), \(0.\overline{3}\)

Irrational Numbers (\(\mathbb{I}\) or \(\mathbb{Q}'\))

Numbers that cannot be written in the form \(p/q\). Their decimal representations are non-terminating and non-repeating.

Examples: \(\sqrt{2}\), \(\pi\), \(e\)

Real Numbers (\(\mathbb{R}\))

The continuous union of rational and irrational numbers: \(\mathbb{R} = \mathbb{Q} \cup \mathbb{I}\).

Complex Numbers (\(\mathbb{C}\))

Numbers formatted as \(z = a + bi\), where \(a, b \in \mathbb{R}\) and imaginary unit \(i = \sqrt{-1}\).

• Real part: \(\text{Re}(z) = a\)
• Imaginary part: \(\text{Im}(z) = b\)

2. Number Set Hierarchy & Subset Chains

Understanding the containment hierarchy of numbers helps eliminate options in CAT questions involving variables with specific set conditions.

The mathematical nested subset relationship is given by:

3. Positional Base Systems

Base systems test positional place value concepts frequently tested in the Quantitative Aptitude section.

4. Fundamental Algebraic Properties

For any real numbers \(a, b, c \in \mathbb{R}\), the fundamental field axioms hold:

• Commutative Law: \[a + b = b + a\] \[a \cdot b = b \cdot a\]
• Associative Law: \[(a + b) + c = a + (b + c)\] \[(a \cdot b) \cdot c = a \cdot (b \cdot c)\]

5. CAT PYQ Analysis & Chapter Weightage

In recent IIM CAT exam patterns (NTA/IIM TCS pattern), Number Systems contribute directly or indirectly to 10%–15% of the Quantitative Aptitude section (around 2 to 4 questions per slot).

Key Focus Areas for CAT 2026:

• Base System Conversion & Operations: Finding units digits or number of zeros in custom base representations.
• Properties of Real & Rational Numbers: Determining rational/irrational nature of expressions and identifying non-terminating decimals.
• Integral Solutions: Finding non-negative or strictly positive integral solutions based on set constraints (\(\mathbb{N}, \mathbb{W}, \mathbb{Z}\)).

6. People Also Ask (Frequently Asked Questions)