1. Chapter at a Glance
We began with counting numbers 1, 2, 3, … then added zero, then negative numbers, then fractions. Is that every number we need? The ancient mathematicians discovered — to their great surprise — that the diagonal of a unit square has a length that is not any fraction. Such numbers are called irrational. Together, rational and irrational numbers make up the real numbers, and every point on the number line is one of them. This chapter explores this world of numbers.
- Natural, whole, integer and rational numbers — and how they nest.
- Density: between any two rationals there are infinitely many more.
- Decimal expansions: terminating, recurring and non-recurring.
- Converting recurring decimals to fractions.
- Irrational numbers and the proof that √2 is irrational.
- Locating √2, √3, … on the number line; the square root spiral.
- Operations with irrationals and rationalising denominators.
2. Families of Numbers
| Set | Symbol | Members |
|---|---|---|
| Natural numbers | N | 1, 2, 3, … |
| Whole numbers | W | 0, 1, 2, 3, … |
| Integers | Z | …, −2, −1, 0, 1, 2, … |
| Rational numbers | Q | p/q, p and q integers, q ≠ 0 |
| Irrational numbers | — | Not expressible as p/q: √2, √3, π, … |
| Real numbers | R | All rationals and irrationals |
3. Density of Rational Numbers
Between two integers there may be no integer (e.g. 3 and 4). But between any two rational numbers there are infinitely many rational numbers. This is called the density property.
Find five rational numbers between 3/5 and 4/5.
Solution: Write with denominator 60: 36/60 and 48/60. Five between them: 37/60, 38/60, 39/60, 40/60, 41/60. (The average (a + b)/2 also always lies between a and b.)
4. Decimal Expansions
| Type | Example | Number is… |
|---|---|---|
| Terminating | 7/8 = 0.875 | Rational |
| Non-terminating recurring | 1/7 = 0.142857142857… = 0.142857 | Rational |
| Non-terminating non-recurring | √2 = 1.41421356… | Irrational |
- A fraction p/q in lowest terms terminates exactly when q has no prime factors other than 2 and 5.
- When dividing by q, there are at most q − 1 possible non-zero remainders, so the digits must start repeating.
Express 0.4̅7̅ (= 0.474747…) as p/q.
Solution: x = 0.4747… → 100x = 47.4747… Subtract: 99x = 47 → x = 47/99.
Express 1.32̅ (= 1.3222…) as p/q.
Solution: x = 1.3222… → 10x = 13.222…, 100x = 132.222… Subtract: 90x = 119 → x = 119/90.
Surprising fact: 0.999… = 1. (x = 0.999…, 10x = 9.999…, 9x = 9, x = 1.)
5. Irrational Numbers
A number that cannot be written as p/q (q ≠ 0) is irrational. Its decimal expansion never ends and never repeats. Examples: √2, √3, √5, √7, π, 0.101001000100001… (the pattern of zeros grows, so no block repeats).
- Suppose √2 = p/q in lowest terms (p, q have no common factor).
- Squaring: 2q² = p², so p² is even, which means p is even. Write p = 2m.
- Then 2q² = 4m², so q² = 2m², so q² is even and q is even.
- Now p and q are both even — they share the factor 2, contradicting "lowest terms".
- So the assumption is false: √2 is irrational. ∎
The same method shows √3, √5 and √p for any prime p are irrational. In general, √n is irrational whenever n is not a perfect square.
π is irrational (proved by J. H. Lambert in 1761). 22/7 and 3.14 are only approximations of π. Āryabhaṭa (499 CE) gave π ≈ 62832/20000 = 3.1416 and described it as "āsanna" (approximate).
6. Irrationals on the Number Line
To mark √2: on the number line draw a unit square on the segment from 0 to 1. Its diagonal is √(1² + 1²) = √2. With centre 0 and radius equal to the diagonal, draw an arc to cut the number line — that point is √2.
Square root spiral: start with a right triangle of legs 1 and 1 (hypotenuse √2). On the hypotenuse, draw a perpendicular of length 1 to form a new right triangle with hypotenuse √3. Repeat to get √4 = 2, √5, √6, … The triangles curl into a beautiful spiral.
Every real number corresponds to exactly one point on the number line, and every point is a real number. That is why the real line has no "gaps".
7. Operations with Irrational Numbers
- Rational + irrational = irrational (e.g. 2 + √3).
- Non-zero rational × irrational = irrational (e.g. 5√2).
- Irrational ± or × irrational can be either: √2 × √2 = 2 (rational), √2 + (−√2) = 0, but √2 × √3 = √6 (irrational).
| Identity (a, b > 0) | Example |
|---|---|
| √a × √b = √(ab) | √3 × √12 = √36 = 6 |
| √a / √b = √(a/b) | √50 / √2 = 5 |
| (√a + √b)(√a − √b) = a − b | (√7 + √3)(√7 − √3) = 4 |
| (a + √b)(a − √b) = a² − b | (5 + √2)(5 − √2) = 23 |
| (√a + √b)² = a + b + 2√(ab) | (√2 + √3)² = 5 + 2√6 |
Rationalising the denominator
To remove a root from the denominator, multiply top and bottom by a suitable factor:
Rationalise (a) 1/√5 (b) 1/(3 + √2).
Solution: (a) √5/5. (b) Multiply by (3 − √2): (3 − √2)/(9 − 2) = (3 − √2)/7.
8. Solved Problems (Board Pattern)
Find an irrational number between 2 and 3.
Solution: √5 (since 4 < 5 < 9), or 2.101001000100001…
Without dividing, decide whether 13/40 and 7/24 terminate.
Solution: 40 = 2³ × 5 → terminates (13/40 = 0.325). 24 = 2³ × 3 → has factor 3 → recurring (0.291666…).
Simplify (3 + √3)(2 − √3).
Solution: 6 − 3√3 + 2√3 − 3 = 3 − √3.
9. Master Formula Table
| Rule | Statement |
|---|---|
| Rational number | p/q, q ≠ 0 |
| Terminating test | q = 2ᵐ × 5ⁿ (lowest terms) |
| Pure recurring (k digits) | 0.a̅b̅… = (repeating block) / (99…9, k nines) |
| Number between a and b | (a + b)/2 |
| Product of roots | √a √b = √(ab) |
| Conjugate product | (√a + √b)(√a − √b) = a − b |
| Rationalise 1/(a + √b) | (a − √b)/(a² − b) |
10. Master Summary Table
| Number type | Decimal form | Examples |
|---|---|---|
| Rational (terminating) | Ends | 0.25, 3/8, −1.6 |
| Rational (recurring) | Repeats forever | 1/3, 2/7, 0.4̅7̅ |
| Irrational | Never ends, never repeats | √2, √3, π |
| Real | Any of the above | All points on the number line |
Chart — length of the repeating block of 1/q:
11. Mind Map
- p/q
- Dense; terminate or recur
- √2 proof
- Non-recurring
- √2 by construction
- Root spiral
- √a√b = √ab
- Rationalise
12. Tips and Tricks
- Recurring decimal → fraction: multiply by 10 for each digit in the repeating block, then subtract.
- For "is √n rational?", just check whether n is a perfect square.
- To rationalise, use the conjugate: change the sign between the two terms.
- Thinking π = 22/7 exactly.
- Writing √(a + b) = √a + √b (false: √(9 + 16) = 5, but √9 + √16 = 7).
- Assuming the sum of two irrationals is always irrational.
13. Important One-liners
- Every integer is rational, but not every rational is an integer.
- Between any two rationals there are infinitely many rationals.
- A rational number's decimal either terminates or recurs.
- √2 is irrational — proved by contradiction.
- Every point on the number line is a real number.
- 0.999… = 1.
14. Exam-pattern Questions with Detailed Answers
Which of these is irrational?
- √16
- 0.3̅
- √7
- 22/7
Show answer
√16 = 4, 0.3̅ = 1/3, 22/7 is a fraction. √7: 7 is not a perfect square.
Answer: (C)
Write 0.6̅ as a fraction.
Show answer
10x − x = 6 → x = 6/9 = 2/3.
Answer: 2/3
Will 17/125 have a terminating decimal expansion?
Show answer
125 = 5³ — only the prime 5. Yes: 0.136.
Answer: Yes (0.136)
Express 2.1̅5̅ (= 2.151515…) as p/q.
Show answer
x = 2.1515…, 100x = 215.1515… → 99x = 213 → x = 213/99 = 71/33.
Answer: 71/33
Rationalise 6/(√5 − √3).
Show answer
Multiply by (√5 + √3): 6(√5 + √3)/(5 − 3) = 3(√5 + √3).
Answer: 3√5 + 3√3
Give an example of two irrational numbers whose product is rational.
Show answer
√3 × √12 = √36 = 6, or (2 + √5)(2 − √5) = −1.
Answer: e.g. √3 and √12
Simplify (√11 − √7)(√11 + √7).
Show answer
11 − 7 = 4.
Answer: 4
Find two irrational numbers between 0.5 and 0.55.
Show answer
Non-recurring, non-terminating decimals in the range: 0.5101001000100001… and 0.5303003000300003…
Answer: e.g. 0.5101001000…, 0.5303003000…
Prove that √3 is irrational.
Show answer
Suppose √3 = p/q in lowest terms. Then p² = 3q², so 3 divides p², hence 3 divides p (3 is prime). Let p = 3m: 9m² = 3q² → q² = 3m², so 3 divides q. Both p and q divisible by 3 — contradiction. Hence √3 is irrational. ∎
Answer: Proof by contradiction as shown
If x = 2 + √3, find the value of x + 1/x.
Show answer
1/x = 1/(2 + √3) = (2 − √3)/(4 − 3) = 2 − √3. Sum = 4.
Answer: 4
Show that 3 + 2√5 is irrational (given that √5 is irrational).
Show answer
Suppose 3 + 2√5 = r, rational. Then √5 = (r − 3)/2, which is rational — contradiction. So 3 + 2√5 is irrational.
Answer: Proof by contradiction
Find a and b if (5 + 2√3)/(7 + 4√3) = a + b√3.
Show answer
Multiply by (7 − 4√3): denominator 49 − 48 = 1. Numerator: 35 − 20√3 + 14√3 − 24 = 11 − 6√3. So a = 11, b = −6.
Answer: a = 11, b = −6
15. Quick Revision Summary
- N ⊂ W ⊂ Z ⊂ Q ⊂ R; irrationals fill the gaps between rationals.
- Rationals: terminating (q = 2ᵐ5ⁿ) or recurring; dense on the number line.
- Irrationals: non-terminating, non-recurring; √2 proof by contradiction.
- Locate √n with right triangles; the square root spiral.
- √a√b = √ab; rationalise using conjugates.