Build Your Intuition First
Think of the number line as an infinitely long ruler. The tick marks at 1, 2, 3… are the natural numbers. Zero is the starting point. Negatives go left. Between every two integers live all the fractions. And even after you squeeze in every single fraction, there are still tiny gaps — that is where irrational numbers like √2 and π live. Together they make the real number line perfectly continuous, with absolutely no gaps.
The Number Family Tree — each set is contained in all sets below it
ℕ* Natural
1, 2, 3, 4, 5 …
ℕ Whole
0, 1, 2, 3 … (adds zero)
ℤ Integer
… −3, −2, −1, 0, 1, 2, 3 … (adds negatives)
ℚ Rational
3/4, −7/2, 0.5, 0.333… (adds fractions & recurring decimals)
ℝ Real
+ irrationals: √2, √3, π, e (adds the “gaps”)
The Concepts, Explained
Rational Numbers
🍜 Real-world: money, measurements, fractions of anything
A number is rational if it can be written as p/q where p and q are integers and q ≠ 0. As a decimal, it either terminates (3/4 = 0.75) or recurs forever (1/3 = 0.333…).
Why does a rational decimal always terminate or recur? When dividing p by q, each remainder must be one of 0, 1, 2, …, (q−1). After at most q steps a remainder must repeat — so the decimal repeats. If the remainder hits 0, it terminates.
Irrational Numbers
📏 Real-world: diagonal of a square is √2; circumference uses π
A number is irrational if it cannot be written as p/q. Its decimal expansion is non-terminating and non-recurring. Famous irrationals: √2 ≈ 1.41421356…, π ≈ 3.14159265…, e ≈ 2.71828182…
Historic insight: Ancient Greek mathematicians proved √2 is irrational. Assume √2 = p/q in lowest terms. Then p² = 2q², so p is even (p = 2k). Substituting: 4k² = 2q² → q² = 2k² → q is also even. Both p and q are even — contradicting “lowest terms.” ✔
Surds
📋 Real-world: exact diagonal lengths without rounding errors
A surd is an irrational root kept in exact form: √2, √3, ∛5. Surds let you work with exact values instead of messy approximations.
Key surd rules: √a × √b = √(ab) | (√a)² = a | √(a/b) = √a / √b
Rationalising the denominator means removing surds from the bottom of a fraction. Multiply top and bottom by the conjugate: 1/(√3+1) × (√3−1)/(√3−1) = (√3−1)/2.
Why rationalise? Convention and convenience. √3/3 is easier to add to other fractions than 1/√3. Both are mathematically equal.
Properties of Real Numbers
⚙ Real-world: why you can rearrange terms in any calculation without changing the answer
- Closure: a + b and a × b are still real numbers.
- Commutativity: a + b = b + a; a × b = b × a. (Order doesn’t matter.)
- Associativity: (a + b) + c = a + (b + c). (Grouping doesn’t matter.)
- Distributivity: a(b + c) = ab + ac. (This is the engine of all algebra.)
Key Formulas
Surd multiplication
√a × √b = √(ab)
e.g. √3 × √12 = √36 = 6
Rationalise (simple)
1/√a = √a / a
Multiply top & bottom by √a
Rationalise (binomial)
1/(a+√b) = (a−√b)/(a²−b)
Multiply by conjugate; uses (a+√b)(a−√b) = a²−b
Recurring decimal → fraction
0.̅a̅b = ab / 99
e.g. 0.45̅ = 45/99 = 5/11
Common Mistakes
✘ Wrong
√(a+b) = √a + √b
✔ Correct
√(a+b) ≠ √a + √b
Try: √(9+16)=5, but √9+√16=7. Never split a root over addition!
✘ Wrong
π = 22/7 exactly
✔ Correct
22/7 ≈ π (approximation only)
π is irrational. 22/7 ≈ 3.142857 while π ≈ 3.141593 — not the same.
✘ Wrong
All decimals are irrational
✔ Correct
Only non-term, non-recur ones are
0.5 = 1/2 and 0.333… = 1/3 are both perfectly rational.
Try It Yourself
Enter a number to see exactly which number sets it belongs to.
Number Set Classifier
Enter a number above to see its classification…
Tip: type sqrt(2) for √2, pi for π, or 1/3 for fractions.
Board Questions
Q1Dhaka 2024Classify numbers4 Marks
Classify each number as natural, integer, rational, or irrational: (a) −3 (b) 7/2 (c) √5 (d) 0.232323…
a
−3 is an integer (and also rational, since −3 = −3/1). It is not natural or whole.
b
7/2 = 3.5 — terminates → rational. Not an integer.
c
√5 ≈ 2.2360679… non-terminating, non-recurring → irrational.
d
0.232323… = 23/99 — a recurring decimal → rational.
Ans
−3: Integer & Rational | 7/2: Rational | √5: Irrational | 0.23̅: Rational ✓
Q2Rajshahi 2023Rationalise denominator4 Marks
Simplify: 1 / (√3 + √2)
1
Multiply top and bottom by the conjugate (√3 − √2).
2
= (√3 − √2) / ((√3)² − (√2)²) = (√3 − √2) / (3 − 2)
3
= (√3 − √2) / 1 = √3 − √2
Ans
1/(√3+√2) = √3 − √2 ✓
Q3Chittagong 2022Recurring decimal to fraction4 Marks
Express 0.454545… as a fraction in lowest terms.
1
Let x = 0.454545…
2
Multiply by 100 (2 recurring digits): 100x = 45.454545…
3
Subtract: 99x = 45 → x = 45/99
4
Simplify: GCD(45, 99) = 9, so 45/99 = 5/11
Ans
0.45̅ = 5/11 ✓
Q4Comilla 2024Simplify surds5 Marks
Simplify: (√75 − √48) ÷ √3
1
√75 = √(25×3) = 5√3
2
√48 = √(16×3) = 4√3
3
(5√3 − 4√3) ÷ √3 = √3 ÷ √3 = 1
Ans
Answer = 1 ✓
Q5Jessore 2023Prove irrationality6 Marks
Prove that √3 is irrational.
1
Assume for contradiction that √3 = p/q where p, q are integers in lowest terms (no common factor).
2
Squaring: p² = 3q². So 3 divides p². Since 3 is prime, 3 divides p. Write p = 3k.
3
Substituting: (3k)² = 3q² → 9k² = 3q² → q² = 3k². So 3 divides q.
4
Both p and q are divisible by 3 — contradicting our assumption that p/q is in lowest terms.
Ans
Contradiction! Therefore √3 is irrational. ✓
Q6Barisal 2022Distributive property4 Marks
Evaluate 53 × 97 + 53 × 3 without a calculator, using a number property.
1
Factor out 53 using the distributive law: 53(97 + 3) = 53 × 100
2
= 5300
Ans
5300 ✓ (That’s what the distributive law is for!)
Quick Reference Card
Rational decimal
terminates OR recurs
Irrational decimal
non-term, non-recur
Surd product
√a · √b = √(ab)
Rationalise simple
1/√a = √a / a
Recurring → fraction
multiply by 10², subtract
Distributive
a(b+c) = ab + ac