Chapter 3: The World of Numbers - Ultimate 100/100 Master Study Guide

Created by Team Vardaan with Powered by

Topic 1: Dawn of Mathematics, Tally Bones & Indian History

One-to-One Correspondence & Counting History

One-to-One Correspondence: Matching each physical object with a pebble or notch. For instance, cattle herders along the Saraswati river placed pebbles in a pot as cows left to graze and removed them as they returned. This gave birth to Natural Numbers ($\mathbb{N} = \{1, 2, 3, 4, \dots\}$).
Lebombo Bone (35,000 BP): Found in Lebombo Mountains (South Africa/Swaziland). Contains 29 carved notches used as a lunar phase counter or menstrual calendar.
Ishango Bone (20,000 BCE): Found near Nile headwaters (DR Congo). Features 3 notched columns grouping prime numbers ($11, 13, 17, 19$) and demonstrating multiplication by 2 (doubling).
Ancient Indian Context:
- Harappa & Lothal standard weights/measures for international trade with Mesopotamia.
- Vedic powers of 10 up to $10^{12}$ (*parārdha*).
- *Lalitavistara* (4th century BCE): Buddha named powers of 10 up to $10^{53}$ (*tallakṣhaṇa*).

Immediate Solved Application

Example 1.1: Trade Exchange Ratio NCERT Ex 3.1

Question: A merchant in Lothal exchanges 15 copper ingots for every 2 bags of spices. If he brings 12 bags of spices, how many copper ingots does he get?

Solution:

Ratio of ingots to spice bags $= \frac{15}{2}$.

$$\text{Total Ingots} = 12 \times \frac{15}{2} = 6 \times 15 = \mathbf{90\text{ copper ingots}}$$

Topic 1: Test Your Understanding

The Ishango bone groups notches into $11, 13, 17, 19$. What do these numbers share in common? List the next 3 terms. [Ans: All are prime numbers; next three are $23, 29, 31$]
Are Natural Numbers closed under subtraction? Justify with an example. [Ans: No; $3 - 5 = -2 \notin \mathbb{N}$]

Topic 2: The Revolution of Śhūnya (Zero) & Brahmagupta's Laws

Philosophical & Mathematical Origins

Philosophical Origin (Śhūnyatā): Upanishads, Buddhist literature, and Patanjali's *Yoga Sutras* (3rd century BCE) revered *Śhūnyatā* (emptiness/zeroness) as the meditative state of mind stillness (*vṛtti-nirodha*).
Bakhshālī Manuscript: Earliest physical representation of zero as a bold dot (*bindu*).
Brahmagupta (628 CE, *Brāhmasphuṭasiddhānta*): Formally transformed zero into an operational number defined by $a - a = 0$.

$$\mathbf{\text{Brahmagupta's Laws for Zero:}}$$ $$\mathbf{1.\; a + 0 = a} \qquad \mathbf{2.\; a - 0 = a} \qquad \mathbf{3.\; a \times 0 = 0}$$

Immediate Solved Application

Example 2.1: Operations with Zero NCERT

Question: Evaluate: (i) $7 - 0$, (ii) $-6 - 0$, (iii) $15 \times 0$, (iv) $0 - (-14)$.

Solution:

(i) $7 - 0 = \mathbf{7}$. (ii) $-6 - 0 = \mathbf{-6}$.

(iii) $15 \times 0 = \mathbf{0}$. (iv) $0 - (-14) = 0 + 14 = \mathbf{14}$.

Topic 2: Test Your Understanding

Why is division by zero undefined? [Ans: If $a \div 0 = k \implies k \times 0 = a$, which is impossible for any non-zero $a$]
Calculate $(-25) \times 0 + 18$. [Ans: $0 + 18 = 18$]

Topic 3: Integers ($\mathbb{Z}$) & Brahmagupta's Laws of Arithmetic

Economic Foundation: Fortunes & Debts

Fortunes (Dhana): Positive numbers representing wealth/assets.
Debts (Ṛiṇa): Negative numbers representing debts/losses.
Set of Integers ($\mathbb{Z}$): $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$ (from German *Zahlen*).

$$\mathbf{\text{Brahmagupta's Arithmetic Rules for Integers:}}$$ $$\mathbf{1.\; \text{Fortune } + \text{ Fortune } = \text{ Fortune: } 5 + 4 = 9}$$ $$\mathbf{2.\; \text{Debt } + \text{ Debt } = \text{ Debt: } (-5) + (-4) = -9}$$ $$\mathbf{3.\; \text{Debt } \times \text{ Fortune } = \text{ Debt: } (-3) \times 4 = -12}$$ $$\mathbf{4.\; \text{Debt } \times \text{ Debt } = \text{ Fortune: } (-3) \times (-4) = +12}$$ Key Concept: Multiplying by a negative number represents the removal of a debt! Removing 4 debts of ₹3 makes you ₹12 richer!

Immediate Solved Application

Example 3.1: Financial Account Balance NCERT Ex 3.2

Question: A spice trader takes a loan of ₹850. Next day he earns profit of ₹1,200. Following week he suffers loss of ₹450. Find his final financial standing.

Solution:

Loan (Debt) $= -850$, Profit (Fortune) $= +1200$, Loss (Debt) $= -450$.

$$\text{Final Balance} = (-850) + (+1200) + (-450) = 350 - 450 = \mathbf{-100}$$

He has a net debt of ₹100.

Topic 3: Test Your Understanding

Ladakh temperature at noon is $4^\circ\text{C}$. By midnight it drops by $15^\circ\text{C}$. Find midnight temperature. [Ans: $4 - 15 = \mathbf{-11^\circ\text{C}}$]
Calculate: (i) $(-12) \times 5$, (ii) $(-8) \times (-7)$, (iii) $(-20) \div 4$. [Ans: (i) $-60$, (ii) $+56$, (iii) $-5$]

Topic 4: Rational Numbers ($\mathbb{Q}$), Absolute Value & Density

$$\mathbf{\text{Rational Number: } x = \frac{p}{q} \quad (p, q \in \mathbb{Z}, \; q \neq 0, \; \gcd(p, q) = 1)}$$ $$\text{Absolute Value: } |x| \ge 0. \quad \text{Distance between } a \text{ and } b \text{ is } |a - b|.$$ $$\mathbf{\text{Density Property: Rational between } a \text{ and } b = \frac{a + b}{2}}$$

Immediate Solved Application

Example 4.1: Density & Average Method NCERT Ex 3.4

Question: Find three distinct rational numbers between $-\frac{1}{2}$ and $\frac{1}{4}$.

Solution:

1. Make denominators equal: $-\frac{1}{2} = -\frac{2}{4}$ and $\frac{1}{4}$.

2. Multiply by $\frac{10}{10}$: $-\frac{20}{40}$ and $\frac{10}{40}$.

Three rational numbers: $\mathbf{-\frac{10}{40} = -\frac{1}{4}}$, $\mathbf{0}$, $\mathbf{\frac{5}{40} = \frac{1}{8}}$.

Topic 4: Test Your Understanding

Evaluate $\left|-\frac{5}{3}\right|$ and distance between $-4$ and $3$. [Ans: $\frac{5}{3}$ and $|3 - (-4)| = 7$]
Find a rational number halfway between $1$ and $\frac{3}{2}$. [Ans: $\frac{1 + 3/2}{2} = \mathbf{\frac{5}{4}}$]

Topic 5: Irrational Numbers, Proof of $\sqrt{2}$ & Mādhava's $\pi$ Series

Baudhāyana-Pythagoras & Proof by Contradiction

Baudhāyana's *Śhulbasūtra* (800 BCE): Unit square of side 1 has diagonal $d^2 = 1^2 + 1^2 = 2 \implies d = \sqrt{2}$.
Proof by Contradiction (Hippasus, 400 BCE): $\sqrt{2}$ cannot be written as $\frac{p}{q}$ ($\gcd(p, q) = 1$).
$2q^2 = p^2 \implies p$ is even ($p = 2k$). $2q^2 = 4k^2 \implies q^2 = 2k^2 \implies q$ is even. Contradiction! Thus $\sqrt{2}$ is **Irrational**.

$$\mathbf{\text{Mādhava's Infinite Series for } \pi \text{ (14th Century Kerala School):}}$$ $$\mathbf{\pi = 4 \left( 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \dots \right)}$$ $$\text{Āryabhaṭa (499 CE) Approximation: } \pi \approx \frac{3927}{1250} = 3.1416$$

Immediate Solved Application

Example 5.1: Geometrical Construction of $\sqrt{2}$ NCERT

Question: Describe how to construct $\sqrt{2}$ on the number line using a compass and straightedge.

Solution:

1. Mark origin $O(0)$ and point $A(1)$ on number line ($OA = 1\text{ unit}$).

2. Draw perpendicular $AB = 1\text{ unit}$ at $A$. By Pythagoras, hypotenuse $OB = \sqrt{1^2 + 1^2} = \sqrt{2}$.

3. With $O$ as center and radius $OB = \sqrt{2}$, draw an arc cutting the number line at point $P$. $OP = \mathbf{\sqrt{2} \approx 1.414}$.

Topic 5: Test Your Understanding

Prove by contradiction that $\sqrt{3}$ is irrational. [Hint: Assume $\sqrt{3} = p/q \implies 3q^2 = p^2 \implies p, q$ share factor 3]
State Lambert's contribution in 1761. [Ans: Formally proved that $\pi$ is irrational]

Topic 6: Real Numbers ($\mathbb{R}$), Decimals & Cyclic Numbers

Decimal Signature Rules

Terminating Decimals: Denominator $q$ prime factorization contains ONLY powers of 2 and/or 5 ($q = 2^m \times 5^n$). Example: $\frac{3}{20} = \frac{3}{2^2 \times 5} = 0.15$.
Non-Terminating Repeating Decimals: $q$ contains prime factors other than 2 and 5. Example: $\frac{1}{3} = 0.\overline{3}, \frac{5}{11} = 0.\overline{45}$.
Irrational Decimals: Non-terminating, non-repeating. Example: $\sqrt{2} = 1.4142135\dots, \pi = 3.14159265\dots$.

$$\mathbf{\text{Magic Cyclic Number } \frac{1}{7} = 0.\overline{142857}:}$$ $$142857 \times 1 = 142857, \quad 142857 \times 2 = 285714, \quad 142857 \times 3 = 428571, \dots$$

Immediate Solved Application

Example 6.1: Converting Repeating Decimals to $\frac{p}{q}$ NCERT

Question: Convert (i) $0.\overline{6}$ and (ii) $2.35\overline{7}$ into $\frac{p}{q}$ form.

Solution:

(i) Let $x = 0.666\dots \implies 10x = 6.666\dots$. Subtracting: $9x = 6 \implies x = \frac{6}{9} = \mathbf{\frac{2}{3}}$.

(ii) Let $x = 2.35777\dots$. Shift non-repeating digits (2 digits): $100x = 235.777\dots$.
Shift repeating digit (1 digit): $1000x = 2357.777\dots$.
Subtracting: $900x = 2122 \implies x = \mathbf{\frac{2122}{900} = \frac{1061}{450}}$.

Topic 6: Test Your Understanding

Prove algebraically that $0.\overline{9} = 1$. [Ans: Let $x = 0.999\dots \implies 10x = 9.999\dots \implies 9x = 9 \implies x = 1$]
Without division, state if $\frac{18}{125}$ terminates. [Ans: $125 = 5^3 \implies$ Terminating after 3 decimal places $= 0.144$]

Topic 7: Frontiers Beyond Real Numbers — Imaginary Unit $i = \sqrt{-1}$

$$\mathbf{i = \sqrt{-1} \implies i^2 = -1}$$

Topic 8: Case Studies & Exam HOTS Problems

Case Study 1: Silk Tailor Fabric Division CASE STUDY

A tailor in Varanasi has $15\frac{3}{4}\text{ metres}$ of fine silk. Making 1 kurta requires $2\frac{1}{4}\text{ metres}$ of silk.

Questions:
1. Convert both mixed fractions into improper fractions.
2. Calculate how many kurtas he can make.
3. Find the length of remaining silk, if any.

Solution:

1. Total silk $= 15\frac{3}{4} = \frac{63}{4}\text{ m}$. Per kurta $= 2\frac{1}{4} = \frac{9}{4}\text{ m}$.

2. $\text{Kurtas count} = \frac{63}{4} \div \frac{9}{4} = \frac{63}{4} \times \frac{4}{9} = \mathbf{7\text{ kurtas}}$.

3. Remaining silk $= \mathbf{0\text{ metres}}$ (exact division!).

HOTS 1: Zero Rational Proof HOTS / RD Sharma

Question: Three rational numbers $x, y, z$ satisfy $x + y + z = 0$ and $xy + yz + zx = 0$. Show that $x, y, z$ must be simultaneously zero.

Solution:

Square $x + y + z = 0 \implies (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) = 0$.

Substitute $xy + yz + zx = 0 \implies x^2 + y^2 + z^2 + 2(0) = 0 \implies x^2 + y^2 + z^2 = 0$.

Since the sum of non-negative real squares is zero, each term must individually be zero: $x^2 = 0, y^2 = 0, z^2 = 0 \implies \mathbf{x = y = z = 0}$.

9. 100/100 Quick Revision Checklist

Master Checklist

Natural Numbers ($\mathbb{N}$): $\{1, 2, 3, \dots\}$, open tally history (Lebombo, Ishango bones).
Zero ($\text{Śhūnya}$): Defined by Brahmagupta ($a - a = 0$). Rules: $a+0=a, a-0=a, a \times 0 = 0$.
Integers ($\mathbb{Z}$): Fortunes ($+$) and Debts ($-$). $(- \times - = +)$.
Rational ($\mathbb{Q}$): $p/q, q \neq 0$. Terminating if $q = 2^m \times 5^n$, repeating otherwise.
Irrational ($\mathbb{I}$): $\sqrt{2}, \pi$. Non-terminating non-repeating decimals. $\pi = 4(1 - 1/3 + 1/5 - \dots)$ (Mādhava).
Real Numbers ($\mathbb{R}$): Union of Rational and Irrational numbers.