One-Line Definition
A Number System is a structured way of representing and classifying numbers based on their properties and relationships.
Core Intuition
Numbers are not all the same. For example:
- 5 is a natural number.
- 0 is a whole number.
- -8 is an integer.
- 3/4 is a rational number.
- $\sqrt{2}$ is an irrational number.
The number system helps us classify numbers based on their characteristics.
The Number System is one of the most fundamental topics in aptitude mathematics. Many CAT and other MBA entrance questions directly or indirectly rely on concepts from the number system. A strong understanding of this topic simplifies many seemingly difficult questions.
Where Is It Used?
Prerequisites Before Learning:
1. Natural Numbers ($\mathbb{N}$)
Counting numbers:
$1, 2, 3, 4, \dots$
They do not include 0 or negative numbers.
2. Whole Numbers ($\mathbb{W}$)
Whole numbers include zero:
$0, 1, 2, 3, 4, \dots$
Starts from 0 and extends positively.
3. Integers ($\mathbb{Z}$)
Integers include negative numbers, zero, and positive numbers:
$\dots, -3, -2, -1, 0, 1, 2, 3, \dots$
4. Rational Numbers ($\mathbb{Q}$)
A rational number can be written as $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.
Examples: $3/5$, $-7/3$, $8$
5. Irrational Numbers
Numbers that cannot be written as a ratio of two integers. Their decimal expansions are non-terminating and non-repeating.
Examples: $\sqrt{2}$, $\sqrt{3}$, $\pi$
6. Real Numbers ($\mathbb{R}$)
Real numbers include both Rational numbers and Irrational numbers.
Every point on the number line represents a real number.
Number Line Concept
Numbers increase as we move to the right. Numbers decrease as we move to the left. Zero separates positive and negative numbers.
SOLVED EXAMPLES (LEVEL 0 TO HARD)
Q: Classify the number 15.
Step 1: Identify natural & whole number properties ($15$ is a positive counting number starting above $0$).
Step 2: Check rational form ($\frac{15}{1}$ where denominator $q=1 \neq 0$).
Step 3: Every rational number is also a point on the number line (Real Number).
Q: Classify: $-\frac{7}{3}$.
Step 1: It is in $\frac{p}{q}$ form where $p=-7, q=3 \neq 0$, so it is a Rational Number.
Step 2: It is not an integer or whole number because it contains a fractional part ($-2.333\dots$).
Q: Is $\sqrt{49}$ irrational?
Step 1: Simplify the square root: $\sqrt{49} = 7$.
Step 2: Since $7 = \frac{7}{1}$, it can be expressed as a ratio of two integers.
Mistake 1
Assuming every square root is irrational.
For example: $\sqrt{16} = 4$ (Rational), $\sqrt{25} = 5$ (Rational). Only square roots of non-perfect squares are irrational.
Mistake 2
Confusing whole numbers with integers.
Whole numbers begin at 0, whereas integers also include negative numbers.
Mistake 3
Thinking that every decimal is irrational.
Decimals that terminate or repeat are rational.
Shortcut 1: Number Hierarchy
$\text{Natural} \subset \text{Whole} \subset \text{Integers} \subset \text{Rational} \subset \text{Real}$
Shortcut 2: Square Root Primality Check
Whenever you see a square root, first check whether the number is a perfect square.
Shortcut 3: Decimal Rationality Test
If a decimal terminates or repeats, it is rational.
1. Classify the number 0.
Answer: 0 is a Whole Number, Integer, Rational Number, and Real Number.
2. Is $\sqrt{81}$ rational or irrational?
Answer: $\sqrt{81} = 9$, which is an integer. Therefore, it is Rational.
3. Classify $-12$ into all applicable number sets.
Answer: $-12$ belongs to Integers ($\mathbb{Z}$), Rational Numbers ($\mathbb{Q}$), and Real Numbers ($\mathbb{R}$).
4. Determine whether $0.272727\dots$ is rational or irrational.
Answer: Since $0.272727\dots$ is a repeating decimal ($27/99 = 3/11$), it can be expressed in $p/q$ form. Therefore, it is Rational.
Q: Is zero a natural number?
A: In most CAT and MBA entrance exams, 0 is not considered a natural number. It is classified as a whole number.
Q: Can an integer be irrational?
A: No. Every integer can be written as a fraction with denominator 1, so every integer is rational.
Q: Are all rational numbers real numbers?
A: Yes. Every rational number is a real number, but not every real number is rational.