Number Systems

What you'll learn

Distinguish between natural numbers, whole numbers, and integers
Read and write numbers using Indian and international place value systems
Understand fractions and their basic forms
Identify terminating and non-terminating decimals

Key concepts

Natural and Whole Numbers

Natural numbers are the counting numbers we use every day.

Type	Symbol	Definition	Examples
Natural Numbers	N	Counting numbers starting from 1	1, 2, 3, 4, 5, …
Whole Numbers	W	Natural numbers + zero	0, 1, 2, 3, 4, …

Key difference: Every natural number is a whole number, but 0 is a whole number that is NOT a natural number.

Number line — Whole Numbers:

0 --- 1 --- 2 --- 3 --- 4 --- 5 --- 6

Properties of Whole Numbers:

Property	Addition	Multiplication
Closure	a + b is a whole number	a × b is a whole number
Commutative	a + b = b + a	a × b = b × a
Associative	(a+b)+c = a+(b+c)	(a×b)×c = a×(b×c)
Identity	a + 0 = a	a × 1 = a
Distributive	a × (b + c) = a×b + a×c	—

Worked Example: Verify closure for addition: 7 + 5 = 12 (whole number) ✓

Integers

Integers extend whole numbers to include negative numbers.

Integers = { …, −3, −2, −1, 0, 1, 2, 3, … }

Number line — Integers:

… −4 --- −3 --- −2 --- −1 --- 0 --- 1 --- 2 --- 3 --- 4 …

Type	Examples
Negative integers	−1, −2, −3, −100
Zero	0 (neither positive nor negative)
Positive integers	1, 2, 3, 100

Ordering integers: On the number line, the number to the right is always greater.

−3 < −1 (−1 is to the right of −3)
−5 < 0 < 3

Worked Example: Arrange in ascending order: 4, −2, 0, −7, 1 Answer: −7, −2, 0, 1, 4

Place Value — Indian System

The Indian system groups digits as: ones, tens, hundreds, then groups of two (thousands, lakhs, crores).

Periods in Indian System:

Period	Place Values
Ones	Ones, Tens, Hundreds
Thousands	Thousands, Ten-Thousands
Lakhs	Lakhs, Ten-Lakhs
Crores	Crores, Ten-Crores

Example — 4,75,83,296:

Cr	T-L	L	T-Th	Th	H	T	O
4	7	5	8	3	2	9	6

Reading: Four crore seventy-five lakh eighty-three thousand two hundred ninety-six.

Place Value — International System

The international system groups digits in threes: thousands, millions, billions.

Period	Place Values
Ones	Ones, Tens, Hundreds
Thousands	Thousands, Ten-Thousands, Hundred-Thousands
Millions	Millions, Ten-Millions, Hundred-Millions
Billions	Billions, Ten-Billions, …

Example — 475,832,960:

H-M	T-M	M	H-Th	T-Th	Th	H	T	O
4	7	5	8	3	2	9	6	0

Reading: Four hundred seventy-five million eight hundred thirty-two thousand nine hundred sixty.

Conversion table:

Indian	International
1 Lakh	100 Thousand
10 Lakh	1 Million
1 Crore	10 Million
100 Crore	1 Billion

Fractions Basics

A fraction represents a part of a whole.

Fraction = Numerator / Denominator

Term	Meaning	Example (3/5)
Numerator	Parts taken	3
Denominator	Total equal parts	5

Types of Fractions:

Type	Condition	Examples
Proper fraction	Numerator < Denominator	2/5, 3/7, 1/4
Improper fraction	Numerator ≥ Denominator	7/3, 5/5, 9/4
Mixed fraction	Whole number + proper fraction	2¾, 1⅓
Unit fraction	Numerator = 1	1/2, 1/7, 1/100

Equivalent fractions — multiply/divide numerator and denominator by the same number:

1/2 = 2/4 = 3/6 = 4/8 (multiply by 2, 3, 4 respectively)

Worked Example: Convert 2¾ to improper fraction. 2¾ = (2 × 4 + 3) / 4 = 11/4

Types of Decimals

A decimal is a way to write fractions with denominators that are powers of 10.

Place value in decimals:

Hundreds	Tens	Ones	.	Tenths	Hundredths	Thousandths
100	10	1	.	1/10	1/100	1/1000

Example: 34.725

3 tens + 4 ones + 7 tenths + 2 hundredths + 5 thousandths

Types of Decimals:

Type	Description	Example
Terminating decimal	Decimal digits end after finite steps	0.5, 1.25, 3.875
Non-terminating repeating	Digits repeat in a pattern	0.333… (= 1/3), 0.142857142857…
Non-terminating non-repeating	No pattern, never ends	π = 3.14159…

Converting fractions to decimals:

3/4 → divide 3 by 4 → 0.75 (terminating)
1/3 → divide 1 by 3 → 0.333… (repeating; written as 0.3̄)

Worked Example: Is 7/8 terminating or non-terminating? 7 ÷ 8 = 0.875 — it ends, so terminating.

Rule: A fraction p/q (in lowest terms) gives a terminating decimal only if the denominator q has no prime factors other than 2 and 5.

Quick check

What is the difference between natural numbers and whole numbers?
Arrange in descending order: −6, 2, −1, 0, −9, 5
Write 3,75,42,806 in the international system with commas.
Convert 5⅔ to an improper fraction.
Is 11/6 a terminating decimal? Why or why not?

Open the Practice tab for graded questions on Number Systems.