CLASS 9 • MATHS • CHAPTER 3
The World of Numbers
Short, colourful, handwritten-style notes — by EduGrown
1The Need to Count & Natural Numbers
Maths did not begin in a classroom. It began in the dirt, on tree bark, and on bones — because people needed to keep count.
📝 One-to-one correspondence
A herder drops one pebble into a pot for every cow that leaves, and removes one pebble for every cow that returns.
- Pot empty at night → the herd is safe.
- Pebbles left over → some cows are missing.
✔ A history written in bone
📜 Two famous artefacts
- Lebombo Bone (South Africa / Swaziland, about 35,000 years old) — 29 carved notches, probably a lunar phase counter.
- Ishango Bone (River Nile, Congo, about 20,000 BCE) — one column groups notches into 11, 13, 17, 19, the prime numbers between 10 and 20. Another column shows doubling.
🇮🇳 The Indian context
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- Trade at Lothal and Harappa needed standardised weights and measures — and good accounting.
- The Vedas named all powers of 10 up to 1012 (called parārdha).
- The Lalitavistara (4th century BCE) records names up to 1053 (called tallakṣhaṇa).
- Using powers of 10 set the stage for the place-value decimal system we use worldwide today.
2Śhūnya: The Birth of Zero
For thousands of years the number line started at 1. If you gave away all five apples, there was no number for what you had left — only a void.
- Babylonians and Mayans used zero only as a placeholder (a mark for an empty column), not as a number you could add or multiply.
- In Indian philosophy, Śhūnyatā (emptiness) was a revered state in the Upanishads, Buddhist texts and Patanjali’s Yoga Sutras. So Indian thinkers already had a home for the idea of ‘nothingness’.
- The Bakhśhālī Manuscript shows the physical symbol — a bold dot (bindu) for zero.
- Brahmagupta, in the Brāhmasphuṭasiddhānta (628 CE), turned zero into a real, working number by defining a – a = 0 and giving it rules.
⭐ Brahmagupta’s Rules for Zero
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- a + 0 = a (adding zero changes nothing)
- a – 0 = a (subtracting zero changes nothing)
- a × 0 = 0 (anything times zero is zero)
3Integers: Fortunes & Debts
Brahmagupta asked: if 5 – 5 = 0, then what is 3 – 5? To answer, he used the language of business.
📝 Two states
- Fortunes (Dhana) → positive numbers — wealth, assets.
- Debts (Ṙiṇa) → negative numbers — what you owe.
⭐ Brahmagupta’s laws of signs (still used today!)
- Fortune + fortune = fortune → 5 + 4 = 9
- Debt + debt = debt → (–5) + (–4) = –9
- Fortune – zero = fortune; debt – zero = debt → 7 – 0 = 7, –6 – 0 = –6
- Debt × fortune = debt → (–3) × 4 = –12
- Debt × debt = fortune → (–3) × (–4) = +12
💡 Why does negative × negative = positive?
Multiplying by a negative means removing a debt. If someone takes away 4 of your debts of ₹3 each, you are ₹12 richer. So (–3) × (–4) = +12.
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4Rational Numbers
Counting was not enough — people also had to measure. Half a cup of ghee, a third of a field… these are fractions.
📝 Definition
Combine all integers with all fractions (positive and negative) and you get the Rational Numbers, written ℚ (for quotient).
A rational number = pq where p, q are integers and q ≠ 0
🔍 Three things to remember
- ℚ contains everything before it. 5 = 51 and –10 = –101, so naturals, whole numbers and integers are all rational.
- The form is not unique. –13 = –26 = –39 = –1030 … These are equivalent fractions.
- We choose the simplest form. Out of ½, 24, 612… we pick ½, where p and q are co-prime (no common factor except 1).
Also note: in a negative fraction the minus sign can sit anywhere — –15 = –15 = 1–5.
⭐ Brahmagupta’s laws for rational numbers
- Equality: ab = cd if and only if ad = bc
- Add / subtract: make denominators the same, then ab ± cb = a ± cb
- Multiply: ab × cd = acbd
- Divide: ab ÷ cd = ab × dc = adbc (c ≠ 0)
- Addition and multiplication are commutative, and multiplication is distributive: p(q + r) = pq + pr
📝 Closure
Rational numbers are closed under addition, subtraction and multiplication — combining two rationals always gives a rational. They are closed under division too, provided you never divide by zero.
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5Rational Numbers on the Number Line
⭐ The method
To mark pq on the number line:
- Divide the unit interval (gap between two consecutive integers) into q equal parts.
- Move p parts from 0 — to the right if positive, to the left if negative.
✔ Absolute value
📝 | x |
The absolute value of x, written | x |, is its distance from 0 on the number line.
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- | 53 | = 53 and | –53 | = 53 and | 0 | = 0
- Absolute value is never negative: | x | ≥ 0
- Distance between two numbers a and b = | a – b |. e.g. between –4 and 3 it is | 3 – (–4) | = 7
6Density of Rational Numbers
📝 Rational numbers are dense
No matter how close two rational numbers are, you can always squeeze another one between them — just take their average.
Number between a and b = a + b2
✏️ Example
Between 1 and 2 lies 32. Between 1 and 32 lies their average:
1 + 3/22 = 54
You can keep repeating this forever → there are infinitely many rational numbers between any two points.
🤔 But…
It feels as though rationals must completely fill the number line with no gaps left. Do they? No! Read on…
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7Irrational Numbers
Around 800 BCE, while writing his Śhulbasūtra (a manual for building geometric fire altars), Baudhāyana met lengths that simply refused to be fractions. The Greeks hit the same crisis a few centuries later.
✏️ The square that broke the rules
Take a square of side 1 unit. By the Baudhāyana–Pythagoras Theorem the diagonal d satisfies
1² + 1² = d² → d² = 2 → d = √2
And √2 cannot be written as a fraction!
📝 Definition
Numbers on the number line that cannot be expressed as a ratio of integers are called Irrational Numbers (𝕀).
Examples: √2, √3, √10, π, e
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Examples: √2, √3, √10, π, e
8Proof: √2 is Irrational
The first proof came from Hippasus of the Pythagorean school (c. 400 BCE). He used a technique called Proof by Contradiction — assume the opposite of what you want to prove, then show it leads to a logical disaster.
⭐ The proof in 5 quick steps
- Assume √2 is rational, so √2 = pq in simplest form (p and q co-prime).
- Square & rearrange: 2 = p²q² → 2q² = p²
- So p² is even → p is even. Write p = 2k.
- Substitute: 2q² = (2k)² = 4k² → q² = 2k² → q² is even → q is even.
- Contradiction! p and q are both even, so they share the factor 2 — but we said they were co-prime.
💡 Useful fact
If the square of a number is even, the number itself must be even. This little fact is the engine of the whole proof.
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9Constructing √n on the Number Line
⭐ Marking √2 with ruler & compass
- On the number line take OA = 1 unit, and draw a perpendicular at A.
- On this perpendicular mark B with AB = 1 unit, and join OB. By Pythagoras, OB = √2.
- With O as centre and OB as radius, draw an arc cutting the number line at P. Then OP = √2, so P represents √2.
💡 Keep going!
Repeat the same idea from √2 to get √3, then √4, √5… This lets you construct a segment of length √n for any positive integer n.
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10The Story of Pi (π) & Mādhava’s Series
- π is the ratio of a circle’s circumference to its diameter — and it is irrational.
- Āryabhaṭa (499 CE) gave the very accurate 39271250 = 3.1416, but honestly called it only an āsanna (approximation).
- Johann Lambert proved π is irrational in 1761.
- Mādhava of Sangamagrama (14th century), founder of the Kerala School of Mathematics, realised an irrational number needs an infinite sum, not a single fraction.
⭐ Mādhava’s infinite series
π = 4 × ( 1 – 13 + 15 – 17 + … )
Adding infinitely many terms means finding the value we get closer and closer to as we add more and more terms.
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11Real Numbers
📝 Definition
Unite the dense web of rational numbers with the gap-filling irrational numbers and you get the unbroken, continuous line of the Real Numbers ℝ.
Every length, every temperature, every real physical measurement has a home on this line.
Every length, every temperature, every real physical measurement has a home on this line.
12Decimal Expansions: Terminating or Repeating
The easiest way to tell a rational from an irrational is to look at its decimal expansion.
| Type of number | Decimal expansion | Example |
|---|---|---|
| Rational — terminates | The division reaches remainder 0 and stops | 38 = 0.375 |
| Rational — repeats | Never reaches remainder 0; digits loop forever | 511 = 0.454545… = 0.45 |
| Irrational | Never ends and never repeats — no pattern at all | √2 = 1.41421356… π = 3.14159265… |
🔍 Why do decimals repeat?
Divide by 7 and the only possible remainders are 1, 2, 3, 4, 5, 6 (never 0, or it would terminate). Since there are only finitely many choices, a remainder must repeat sooner or later — and once a remainder repeats, the whole division loops!
✔ Predicting the type without doing long division
⭐ The prime-factor test
Take pq in its lowest terms. Then:
The decimal terminates ⇔ the prime factors of q are only 2, only 5, or both
Because then we can turn the denominator into a power of 10.
✏️ Example
320 → 20 = 2² × 5 (only 2s and 5s ✔)
320 = 3 × 520 × 5 = 15100 = 0.15 (terminating)
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13Converting Decimals into p/q Form
✔ Case 1: Terminating decimals
Just write the digits over the matching power of 10 and simplify.
e.g. 0.35 = 35100 = 720
✔ Case 2: Pure repeating decimals
Repetition starts immediately after the decimal point.
✏️ Convert 0.6
- Let x = 0.6
- 1 digit repeats → multiply by 101: 10x = 6.6
- Subtract: 10x – x = 6.6 – 0.6 = 6 → 9x = 6
- So x = 69 = 23
✔ Case 3: General repeating decimals
Some non-repeating digits first, then a repeating block.
✏️ Convert 0.16
- Let x = 0.16
- 1 non-repeating digit → ×10: 10x = 1.6
- 1 repeating digit → ×10 again: 100x = 16.6
- Subtract: 90x = 15 → x = 1590 = 16
| Decimal type | Steps to follow |
|---|---|
| Pure repeating | Let x = the decimal → multiply by 10n (n = number of repeating digits) → subtract from the original → solve for x. |
| General repeating | Let x = the decimal → multiply by 10m (m = non-repeating digits) → then by 10n (n = repeating digits) → subtract → solve for x. |
💡 A surprising fact
Decimal forms are not unique. Every terminating decimal has a twin ending in repeating 9s:
1.000… = 0.999… and 2.47000… = 2.46999…
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1.000… = 0.999… and 2.47000… = 2.46999…
14The Magic of Cyclic Numbers
17 = 0.142857142857… = 0.142857. The repeating block 142857 is a cyclic number. Watch what happens:
| × 1 | × 2 | × 3 | × 4 | × 5 | × 6 |
|---|---|---|---|---|---|
| 142857 | 285714 | 428571 | 571428 | 714285 | 857142 |
✨ Look closely
The same six digits appear every time — they just rotate in a circle! This hidden symmetry is a beautiful hallmark of the rational number 17.
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15The Number Family Tree
| Set | Symbol | What it contains |
|---|---|---|
| Natural Numbers | ℕ | The counting numbers 1, 2, 3, … |
| Integers | ℤ | ℕ plus zero and the negatives: …, –2, –1, 0, 1, 2, … |
| Rational Numbers | ℚ | All p/q with q ≠ 0 — i.e. terminating or repeating decimals |
| Irrational Numbers | 𝕀 | Cannot be written as fractions (√2, π, √10) — non-terminating, non-repeating |
| Real Numbers | ℝ | Rational + irrational together — the whole number line |
Notice how each set sits inside the next: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. Only the irrationals stand apart from ℚ, and together they complete ℝ.
🧠 Is the journey over?
What is √–1? Since 1 × 1 = 1 and (–1) × (–1) = 1, no real number squares to a negative. So mathematicians stepped off the line altogether and invented Imaginary Numbers (the letter i) — essential today for electrical engineering, quantum mechanics and the tech inside your phone. But that is a story for a later grade!
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16Common Mistakes to Avoid
❌ Don’t do this
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- Forgetting the condition q ≠ 0 when writing p/q. Division by zero is undefined.
- Thinking integers are not rational. Every integer is rational — 7 = 71.
- Writing a negative absolute value. | x | is never negative.
- Applying the prime-factor test before reducing the fraction to lowest terms.
- Calling 0.1010010001… rational just because it has a ‘pattern’. A rational decimal must have a fixed repeating block, and here the block keeps growing → it is irrational.
- Assuming √ of every number is irrational. √81 = 9, which is perfectly rational!
- Forgetting the bar. 0.45 means 0.454545…, which is very different from 0.45.
17Chapter Summary — Quick Revision
🏆 All of Chapter 3 in 10 lines
- Natural Numbers (ℕ) {1, 2, 3, …} were born from humanity’s need to count — evidence goes back tens of thousands of years (Lebombo and Ishango bones).
- Zero (Śhūnya) grew out of the Indian philosophical idea of Śhūnyatā and was made a working number by Brahmagupta (628 CE), who also introduced negative numbers.
- Integers (ℤ) extend the line to the left of zero — Brahmagupta’s debts (ṛiṇa) against fortunes (dhana).
- Brahmagupta’s laws gave the first rigorous rules for signed arithmetic, including ‘the product of two debts is a fortune’ (– × – = +).
- Rational Numbers (ℚ) are all numbers of the form p/q with p, q integers and q ≠ 0.
- Rational numbers are dense — there is always another rational between any two rationals.
- Irrational Numbers like √2 and π cannot be written as fractions. Hippasus proved √2 irrational by contradiction; Lambert proved π irrational in 1761.
- Real Numbers (ℝ) = rational + irrational = the complete, unbroken number line.
- Decimal expansions are the signature: rational → terminating or repeating; irrational → non-terminating and non-repeating.
- Cyclic numbers such as 142857 (the block of 1/7) show the hidden symmetry inside rational numbers. Beyond ℝ lie the Imaginary Numbers.
✨ Happy Learning — @edugrown ✨