The World of Numbers

1. Chapter at a glance

• Natural numbers (ℕ) arose from one-to-one correspondence for counting; earliest records include the Ishango bone (prime groupings) and Indian place-value system up to 10¹².
• Zero (śhūnya) was formalised by Brahmagupta (628 CE) as a – a = 0, transforming philosophical śhūnyatā into an operational number with explicit arithmetic rules.
• Integers (ℤ) extend left of zero: positive numbers as dhana (fortunes), negative as ṛiṇa (debts); Brahmagupta gave complete rules for their addition and multiplication.
• Rational numbers (ℚ) are all numbers expressible as p/q (p, q integers, q ≠ 0); they are dense on the number line and include all integers and fractions (positive and negative).
• Irrational numbers (e.g., √2) cannot be written as p/q; √2 was proved irrational by Hippasus via proof by contradiction (c. 400 BCE).
• Real numbers (ℝ) are the union of all rationals and irrationals, forming a continuous unbroken line; their decimal expansions distinguish them (terminating/repeating vs. non-repeating non-terminating).
• Rational decimals are terminating (denominator prime factors only 2 and/or 5) or repeating (including pure and general repeating forms); cyclic numbers appear in blocks such as 1/7.
• Absolute value |x| gives distance from 0; distance between a and b is |a – b|.

2. Definitions, theorems and results

• Natural numbers: ℕ = {1, 2, 3, 4, …}.
• Zero (Brahmagupta): a – a = 0; rules: a + 0 = a, a – 0 = a, a × 0 = 0.
• Integers (ℤ): all positive integers, their negatives, and zero.
• Rational number: any number expressible as p/q where p, q ∈ ℤ and q ≠ 0.
• Irrational number: any real number that cannot be expressed as p/q (q ≠ 0).
• Real numbers (ℝ): union of rational and irrational numbers.
• Absolute value: |x| is the distance of x from 0 on the number line; |x| ≥ 0 for all x.
• Equality of rationals: a/b = c/d ⇔ ad = bc.
• Brahmagupta’s integer rules (still used today): fortune + fortune = fortune; debt + debt = debt; (–a) × b = –(a × b); (–a) × (–b) = ab.
• Closure: rationals closed under +, –, × and ÷ (except division by zero).
• Density: between any two rationals lies another rational (e.g., their average).
• Decimal signature: terminating or repeating decimals ⇔ rational; non-repeating, non-terminating ⇔ irrational.
• √2 is irrational — requires proof in the exam (proof by contradiction given in text).

3. Formula sheet

4. Solved-example patterns

• Representing rationals/irrationals on the number line: divide unit interval into q equal parts and locate p steps (positive right, negative left); for √2 use geometric construction (right triangle or compass arc).
• Operations with integers/rationals (Brahmagupta rules): identify signs (fortune/debt), apply addition/multiplication rules, simplify.
• Proving two rationals equal or finding sum/difference/product/quotient: reduce to common denominator or cross-multiply; follow sign rules.
• Converting repeating/terminating decimals to p/q: set x = decimal, multiply by 10ᵐ or 10ⁿ (m non-repeating, n repeating digits), subtract and solve.
• Checking terminating vs repeating without division: factorise denominator (lowest terms) — only 2 and/or 5 → terminating.
• Proving √2 irrational: assume p/q in lowest terms, square, show both p and q even → contradiction (exam proof required).
• Locating numbers between two given rationals: take average or use density property; list distinct equivalents with common denominator.
• Absolute value/distance problems: compute |a – b| directly.

5. Common mistakes and exam pitfalls

• Forgetting q ≠ 0 when writing rational numbers.
• Sign errors when adding/subtracting negatives or multiplying two negatives (– × – = +).
• Assuming every decimal is terminating or repeating (irrational decimals never repeat).
• Writing non-reduced fractions on the number line instead of the simplest p/q.
• Missing that |x| is always non-negative and |–x| = |x|.
• In proof of √2 irrational: forgetting to state “p and q co-prime” at the start or failing to reach the contradiction clearly.
• When converting decimals, miscounting non-repeating vs repeating digits (wrong powers of 10).
• Omitting units or context in word problems involving debt/fortune or temperature.