Number System

Build a strong, exam‑oriented understanding of numbers from the earliest grades to competitive exams. Progress from basics to proficient level through clear explanations, worked examples, and targeted practice.

Learning path: Natural → Whole → Integers → Fractions → Decimals → Operations → Properties → LCM/GCD → Divisibility → Rounding → Practice Sets.

Natural Numbers

Natural numbers are counting numbers: 1, 2, 3, … . They are positive and do not include zero.

Key Ideas

  • Even numbers are divisible by 2; odd numbers are not.
  • Prime numbers have exactly two factors: 1 and the number itself.
  • Composite numbers have more than two factors.
Example: Factors of 18 are 1, 2, 3, 6, 9, 18; so 18 is composite. 17 has only 1 and 17; so 17 is prime.

Whole Numbers

Whole numbers extend natural numbers by including 0: 0, 1, 2, 3, …

Key Ideas

  • Zero is the additive identity: n + 0 = n.
  • Whole numbers are closed under addition and multiplication.
Example: 0 + 45 = 45, 12 × 7 = 84 are whole numbers.

Integers

Integers include negative numbers, zero, and positive numbers: … −3, −2, −1, 0, 1, 2, 3, …

Key Ideas

  • Absolute value |n| is the distance from 0 on the number line.
  • Add a negative to subtract; subtract a negative to add.
Example: |−7| = 7; 5 − (−3) = 8.

Number Line & Place Value

Use the number line to visualize ordering and distance. Place value explains how digits represent units, tens, hundreds, and so on.

  • Comparing: a is greater than b if it lies to the right of b.
  • Place value: 4,503 = 4 thousands + 5 ones + 0 hundreds + 3 ones.

Fractions

A fraction a/b represents a parts out of b equal parts. Proper fractions have a < b; improper fractions have a ≥ b; mixed numbers combine a whole and a fraction.

Operations

  • Like denominators: add/subtract numerators.
  • Unlike denominators: use LCM to make common denominators.
  • Multiply: (a/b)×(c/d) = ac/bd; divide: (a/b)÷(c/d) = a/b × d/c.
Example: 2/3 + 1/6 = 4/6 + 1/6 = 5/6.

Decimals

Decimals express fractions with denominators that are powers of 10. Place value extends to tenths, hundredths, thousandths.

  • Compare decimals by aligning place values.
  • Operations follow place value alignment.
Example: 3.408 rounded to hundredths is 3.41; to tenths is 3.4.

Fraction–Decimal Conversion

  • Terminating decimals when denominator factors only into 2 and 5.
  • Repeating decimals otherwise.
Example: 3/8 = 0.375; 1/3 = 0.333…; 7/20 = 0.35.

Comparing & Ordering

  • Convert to common denominators or to decimals to compare.
  • Use number line placement for ordering.

Basic Operations

  • Add/subtract/multiply/divide with integers, fractions, and decimals.
  • Order of operations: parentheses > exponents > multiplication/division > addition/subtraction.
Example: 6 + 2 × (3 + 1) = 6 + 2 × 4 = 14.

Number Properties

  • Commutative: a + b = b + a; a × b = b × a.
  • Associative: (a + b) + c = a + (b + c).
  • Distributive: a(b + c) = ab + ac.
  • Identity: 0 for addition; 1 for multiplication.

Divisibility Rules

  • 2: last digit even; 3: sum of digits divisible by 3; 4: last two digits divisible by 4.
  • 5: last digit 0 or 5; 6: divisible by 2 and 3; 8: last three digits divisible by 8.
  • 9: sum of digits divisible by 9; 10: last digit 0; 11: alternating sum rule.

LCM & GCD

  • Prime factorization method to find LCM and GCD.
  • LCM × GCD = product of two numbers.
Example: 24 = 2³×3, 36 = 2²×3²; LCM = 2³×3² = 72, GCD = 2²×3 = 12.

Rounding & Estimation

  • Round to nearest place using the next digit.
  • Use estimation to check reasonableness of results.

Practice Set: Basic

Which is prime? 21, 29, 33, 35
Answer: 29
29 has factors 1 and 29 only. Others are composite.
Additive identity of whole numbers
Answer: 0
n + 0 = n for any whole number n.
Compute |−12|
Answer: 12
Absolute value is distance from 0.
Add 2/5 and 1/10
Answer: 1/2
LCM 10. 2/5=4/10; 4/10+1/10=5/10=1/2.

Practice Set: Intermediate

Round 8.476 to nearest tenth
Answer: 8.5
Hundredths is 7 ≥ 5, so tenths rounds up.
Two numbers have product 360 and GCD 6. Find LCM.
Answer: 60
LCM × GCD = product ⇒ LCM = 360 ÷ 6 = 60.
Check divisibility by 11: 918273
Answer: Yes
Alternate sum: (9+8+7) − (1+2+3) = 24 − 6 = 18; 18 not multiple of 11, so No. Corrected: 918273 is not divisible by 11.
Compute (3/4) × (8/9)
Answer: 2/3
Cancel 4 with 8 → 2; 3 with 9 → 3; result 2/3.

Practice Set: Proficient

Evaluate −18 ÷ 3 + 5 × (−2)
Answer: −16
−18 ÷ 3 = −6; 5 × −2 = −10; sum = −16.
Order: 0.7, 0.65, 0.705, 0.699 ascending
Answer: 0.65, 0.699, 0.7, 0.705
Compare digit by digit.
Find smallest number divisible by 12, 15, 20
Answer: 60
LCM(12,15,20)=60.

Exam Practice: Olympiad, Sainik/JNV

Which is rational? √5, 3/7, π, √2
Answer: 3/7
Rational numbers are ratios of integers with non‑zero denominator.
A tank is 3/5 full. After adding 1/4 of its capacity, how full?
Answer: 17/20
3/5 + 1/4 = 12/20 + 5/20 = 17/20.
Three lights blink every 12s, 18s, 24s together next after?
Answer: 72 seconds
LCM(12,18,24)=72.
Distance between −9 and 5 on number line
Answer: 14
|−9 − 5| = |−14| = 14.