Chapter 1 — Number Systems

What is the Number System?

The number system is a way of organising all types of numbers into groups so we can understand their properties and relationships. In Class 9 we introduce the main sets of numbers and how they relate: natural numbers, whole numbers, integers, rational numbers, irrational numbers and real numbers.

Main sets (brief):

• Natural numbers (N): {1, 2, 3, ...} — counting numbers.
• Whole numbers (W): {0, 1, 2, 3, ...} — natural numbers plus 0.
• Integers (Z): {..., −3, −2, −1, 0, 1, 2, 3, ...} — whole numbers and their negatives.
• Rational numbers (Q): numbers expressible as p/q, where p and q are integers and q ≠ 0. Their decimal expansions are terminating or repeating (e.g., 3/4 = 0.75, 1/3 = 0.333...).
• Irrational numbers: numbers that cannot be written as p/q. Their decimal expansions are non-terminating and non-repeating (e.g., √2, π).
• Real numbers (R): all rational and irrational numbers together. Every point on the number line corresponds to a real number.

Decimal expansion and classification

Rational numbers have decimal expansions that either terminate or become periodic (repeat). Irrational numbers have non-terminating, non-repeating decimals. Important practical test for termination: if a rational number p/q is in lowest terms, its decimal expansion terminates iff the prime factors of q are only 2 and/or 5.

Density property

Between any two real numbers there are infinitely many rational numbers and infinitely many irrational numbers. This means rationals and irrationals are both dense on the number line.

Basic properties (quick)

• Sum or product of two rationals is rational.
• Sum or product of two irrationals can be rational or irrational (depends on the numbers).
• Every integer is rational (e.g., 5 = 5/1).

Why this matters (real-life view)

Number classification helps in measurement, computing, finance and geometry: money uses terminating decimals, measurements may need irrational numbers (diagonals, circle measurements), and repeating decimals are encountered in periodic quantities.

Note: Visual tools like a nested-Venn diagram of sets and number-line illustrations help in understanding how these sets fit together.

Statement: For any two integers a and b with b > 0, there exist unique integers q (quotient) and r (remainder) such that
a = bq + r, where 0 ≤ r < b.

Meaning: When you divide a by b, you get a quotient q and a remainder r smaller than b. The lemma guarantees existence and uniqueness of q and r.

Short proof idea (for nonnegative a): Consider the set S = {a - kb | k is an integer and a - kb >= 0}. S is nonempty (take k = 0). Let r be the least element of S. Then r = a - qb for some q; r >= 0 by definition. If r >= b, then r - b = a - (q+1)b would be a smaller nonnegative element of S, contradicting minimality. Hence r < b. This gives existence. For uniqueness, suppose a = bq1 + r1 = bq2 + r2 with 0 ≤ r1, r2 < b. Subtracting gives b(q1 - q2) = r2 - r1. The left side is a multiple of b while the right side has absolute value < b, so r1 = r2 and q1 = q2.

Key consequence (used in Euclid's algorithm): gcd(a, b) = gcd(b, r). Repeating this division with remainders gives an efficient method to compute the greatest common divisor.

Prime number: A natural number greater than 1 that has exactly two positive divisors: 1 and itself (examples: 2, 3, 5, 7, 11...).

Composite number: A natural number greater than 1 that has more than two positive divisors (examples: 4, 6, 8, 9, 12...).

Special cases: 1 is neither prime nor composite. 0 is not considered prime or composite in the usual number theory context.

How to test primality (practical rule): To check if n (>1) is prime, test divisibility by primes ≤ √n. If none divide n, then n is prime.

Prime factorization: Every integer >1 can be written uniquely (up to order) as a product of primes: n = p1^a1 · p2^a2 · ... · pk^ak. This statement is the Fundamental Theorem of Arithmetic and is essential for computing HCF and LCM.

Infinitely many primes — Euclid’s proof (concise): Suppose there are only finitely many primes p1, p2, …, pn. Consider N = p1·p2·...·pn + 1. N is not divisible by any pi (remainder 1). So either N is prime itself or it has a prime factor not in the list. In both cases the original list was incomplete. Therefore there must be infinitely many primes.

Consequences and uses: Unique prime factorization helps compute HCF and LCM using prime exponents, and primes are building blocks for many areas: cryptography, hashing, coding theory and more.

What it says: Every integer greater than 1 is either a prime number or can be expressed as a product of prime numbers. This product is unique except for the order of the primes. This statement is called the Fundamental Theorem of Arithmetic (or Unique Prime Factorization).

Important notes:

• 1 is neither prime nor composite, so the theorem applies to integers > 1 only.
• By "unique up to order" we mean, for example, 84 = 2 × 2 × 3 × 7 is the same factorization as 2 × 3 × 2 × 7 (order changed), but there is no different set of primes that multiply to 84.

How to get the prime factorization (methods):

• Factor tree: keep splitting a composite number into two factors until all leaves are primes.
• Repeated division: divide repeatedly by the smallest possible prime (2, then 3, then 5, etc.) until quotient becomes 1.
• Trial division: test divisibility by primes up to sqrt(n).

Sketch of a proof of uniqueness (idea): Suppose a number n had two different prime factorizations. Take a prime p from the first factorization; p divides n, so p must divide the product of primes in the second factorization. Using Euclid's lemma (if a prime divides a product, it divides at least one factor), p divides one of the primes in the second factorization. But primes have no nontrivial divisors, so that prime must equal p. Cancel p from both sides and repeat. Eventually all primes cancel, so the factorizations are the same except for order.

Why it matters:

• Prime factorization is the basis for finding all divisors of a number, computing LCM and GCD, and solving many number theory problems.
• It is also a conceptual foundation for advanced applications (e.g., modern cryptography relies on properties of primes and factorization).

Overview: Real numbers are divided into rational and irrational numbers. Rational numbers can be written as a ratio of two integers. Irrational numbers cannot be expressed as a ratio of two integers and have non-terminating, non-repeating decimal expansions.

Rational Numbers (Q)

• Definition: A number r is rational if r = p/q where p and q are integers and q ≠ 0.
• Decimal behavior: A rational number has either a terminating decimal expansion (for example, 1/4 = 0.25) or a non-terminating repeating decimal expansion (for example, 2/3 = 0.666...).
• Examples: integers (5, -3), finite decimals (0.75), repeating decimals (0.333...), fractions (7/2).
• Algebraic properties: rationals are closed under addition, subtraction, multiplication and division (except division by 0). That is, if a and b are rational and b ≠ 0, then a ± b, a·b, a/b are rational.

Irrational Numbers

• Definition: A real number that cannot be written as p/q with integers p and q (q ≠ 0). Its decimal expansion is non-terminating and non-repeating.
• Classic examples: √2, √3, π, e. These cannot be expressed as exact fractions.
• Algebraic behavior: irrationals are not closed under arithmetic operations. For example, sum of two irrationals may be rational (√2 + (1-√2) = 1) or irrational; product may be rational or irrational.

Relation between Rational and Irrational

• Every real number is either rational or irrational: R = Q ∪ (R\Q).
• Rationals are countable; irrationals are uncountable (so there are "more" irrationals than rationals).

Simple proof that √2 is irrational (sketch)

1. Assume √2 = p/q in lowest terms (p,q integers, q ≠ 0 and gcd(p,q)=1).
2. Then 2 = p^2/q^2 so p^2 = 2q^2, so p^2 is even ⇒ p is even. Let p = 2k.
3. Then (2k)^2 = 4k^2 = 2q^2 ⇒ q^2 = 2k^2 ⇒ q^2 even ⇒ q even.
4. Both p and q even contradict gcd(p,q)=1. Hence √2 is not rational.

How to identify

• If a decimal terminates or repeats, it is rational.
• If a decimal neither terminates nor repeats, it is irrational.
• Any integer is a rational number because n = n/1.

Representation on the number line: Both rational and irrational numbers are points on the number line. Rational numbers can be shown as fractions or decimals; irrational numbers appear as points that cannot be exactly expressed as fractions (for instance, mark √2 between 1 and 2, π between 3 and 4).

Definition: A rational number is any number that can be written as p/q where p and q are integers and q ≠ 0. Its decimal representation is the expansion obtained by dividing p by q (long division). Every rational number has a decimal expansion that either terminates or becomes periodic (repeats).

Two kinds of decimal expansions:

• Terminating decimal: The decimal expansion ends after a finite number of digits (for example, 0.25, 1.75).
• Repeating (recurring) decimal: After some point digits repeat in a fixed block forever (for example, 0.333..., 0.142857142857...). We write the repeating part with a bar: 0.3̅ = 0.333..., 0.142857̅ = 0.142857142857...

Key test (in lowest terms): For a fraction p/q reduced to lowest terms, its decimal expansion terminates iff the denominator q has only 2 and/or 5 as prime factors (q = 2^a · 5^b). If q has any other prime factor, the decimal expansion is repeating.

How to get the decimal: Perform long division of p by q. If remainder becomes 0, the decimal terminates. If a remainder repeats, the decimal digits between first appearance and the repeat form the repeating block.

Converting repeating decimals to fractions (algebra method):

• Pure repeating: if x = 0.a̅ (a has n digits), then 10^n x − x = integer formed by a, so x = a / (10^n − 1).
• Mixed repeating: if x = nonrepeat.repeating where nonrepeat has m digits and repeating block has n digits, then x = (integer formed by nonrepeat+repeat − integer formed by nonrepeat) / (10^{m+n} − 10^m).

Important notes: A terminating decimal can be written with infinite trailing zeros (0.25 = 0.25000...) and some decimals have two representations (e.g., 0.5000... = 0.4999... = 0.5).

Real-life contexts: money and prices (rupees and paise use two decimal places), measurements (lengths in metres shown to limited decimals), time fractions (hours to decimal), recipes (fractions converted to decimals for scaling).

What are real numbers? Real numbers include all rational numbers (fractions and integers) and all irrational numbers (numbers that cannot be written as a fraction). Every real number corresponds to a unique point on the number line, and every point on the number line corresponds to a real number.

Rational numbers (Q): Can be written as p/q where p and q are integers and q ≠ 0. Their decimal expansions are either terminating or repeating (recurring).

Irrational numbers: Cannot be expressed as p/q. Their decimal expansions are non-terminating and non-repeating (examples: √2, π, √3).

Representation on the number line:

• Integers and rational fractions: To represent m/n, divide a unit segment into n equal parts and count m parts from 0 (use negative direction for negative signs).
• Irrationals (square roots): Use geometric construction — e.g., to locate √2, construct a right triangle with legs 1 and 1; the hypotenuse has length √2. Transfer that length to the number line using a compass.
• Every point on the line is a real number; between any two distinct real numbers there are infinitely many rational and infinitely many irrational numbers (density property).

Key properties useful on the number line:

• Order: If a is left of b, then a < b.
• Distance between points a and b is |a − b| (absolute value).
• Midpoint of a and b is (a + b)/2.
• Decimal characterization: after reducing p/q to lowest terms, its decimal terminates iff q's prime factors are only 2 and/or 5.

Completeness idea (informal): Real numbers have no 'gaps' — every Cauchy-like sequence of rationals that should converge has a limit in the real numbers. (This is an advanced idea you may see informally in Class 9 as the number line being 'continuous'.)

What are real numbers? Real numbers (R) include all rational numbers (fractions and integers whose decimal expansions terminate or repeat) and all irrational numbers (numbers whose decimal expansions are non‑terminating and non‑repeating). Real numbers can be placed on a number line and compared using their positions: larger numbers lie to the right.

Comparing two real numbers — practical steps

• If signs differ, the positive number is greater.
• If both are positive or both negative, compare magnitudes. For fractions, use common denominators or cross‑multiply: a/b ? c/d ⇔ ad ? bc (provided b,d>0).
• For decimals, align decimal points and compare digits from left to right. If necessary convert a fraction to a decimal (or vice versa) for easy comparison.
• For negative numbers remember ordering reverses: e.g. −3 < −2 because −3 is to the left.

Key properties when manipulating inequalities

• If a > b, then a + c > b + c (adding same number preserves inequality).
• If a > b and c > 0, then ac > bc (multiplying by positive preserves order).
• If a > b and c < 0, then ac < bc (multiplying by negative reverses order).

Density of rational and irrational numbers

‘Density’ means: between any two distinct real numbers there are infinitely many rational numbers and infinitely many irrational numbers.

Simple proof idea (rationals): If a < b, the midpoint (a + b)/2 is a rational if a and b are rational; more generally you can always choose a rational in any interval because you can find a decimal with finitely many digits lying strictly between the endpoints.

Simple constructive proof (irrationals): Let √2 be a fixed irrational > 0. If a < b then a/√2 < b/√2. By density of rationals choose a rational q with a/√2 < q < b/√2. Then s = √2·q is irrational (product of nonzero rational and irrational is irrational) and satisfies a < s < b. Thus an irrational lies between a and b.

Decimal representations — what they tell us

• Rational ⇔ decimal either terminates (e.g. 0.75) or eventually repeats (e.g. 0.333... = 1/3).
• Irrational ⇔ decimal is non‑terminating and non‑repeating (e.g. √2 = 1.4142135...).

Why this matters: Density means that no matter how close two numbers are, you can always find other rational and irrational numbers between them. This underlies ideas of limits, approximation, and continuity later in mathematics and is important in measurement and computation.

What is approximation? Approximating a number means replacing it by a simpler number that is close to it. We often approximate to a certain place value (nearest ten, nearest integer, or to a fixed number of decimal places) to make calculations easier or to express measured values with limited precision.

Two common methods:

• Truncation (Cutting off): Remove extra digits after the required place without changing the remaining part. Example: truncate 3.786 to 2 decimal places → 3.78. It always moves toward zero for positive numbers.
• Rounding (Round off): Replace the number by the nearest value at the required place. Look at the first discarded digit:

• If the first discarded digit < 5, keep the retained part as it is (round down).
• If the first discarded digit ≥ 5, increase the last retained digit by 1 (round up).

Steps to round to n decimal places (n ≥ 0):

• Multiply the number by 10^n.
• Round the result to the nearest integer using the rule above.
• Divide back by 10^n.

Example (stepwise): Round 3.786 to 2 decimal places.

• Multiply by 10^2 = 100: 3.786 × 100 = 378.6
• Nearest integer to 378.6 is 379 (since 0.6 ≥ 0.5)
• Divide by 100: 379 / 100 = 3.79

Error bounds (how far an approximation can be from the true value):

• If R is the value rounded to n decimal places, then the absolute rounding error satisfies |x − R| ≤ 0.5 × 10^(−n) (for the usual round-half-up rule).
• If T is the value obtained by truncation to n decimal places, then |x − T| < 1 × 10^(−n) (strictly less than 10^(−n)).

Practical notes:

• Rounding gives a closer approximation on average than truncation because it allows going up or down; truncation always cuts down (for positive numbers).
• When working with negative numbers, truncation means moving toward zero (e.g., truncate −2.349 to 2 d.p. → −2.34), while rounding follows the same digit rule but affects sign appropriately (−2.345 → −2.35 if using round-half-up).
• Always state how many decimal places or significant digits you are using when giving an approximate value.

Where it appears in real life: money (prices rounded to nearest paisa/cent), measurement readings (lengths measured to mm or cm), displays (digital meters show limited decimals), summary statistics (reporting population in thousands).

Overview: This topic applies concepts of the Number System (natural, whole, integers, rational, irrational and real numbers) to solve practical and competitive problems. Key ideas used are conversion between fractions and decimals, identification of terminating/repeating decimals, prime factorization, HCF (GCD) and LCM, and the Euclidean algorithm.

Core ideas and techniques:

• Classify numbers: Decide whether a number is rational (can be written as p/q, q ≠ 0) or irrational (non-terminating, non-repeating decimal). For example, √2 is irrational, 3/4 is rational.
• Fraction ⇄ Decimal: Convert fractions to decimals by division. A rational number's decimal expansion either terminates or repeats.
• Termination test: A fraction p/q in simplest form has a terminating decimal iff q has only 2 and/or 5 as prime factors (q = 2^a5^b).
• Repeating decimals to fraction: Use algebra: put x = repeating decimal, multiply by appropriate power of 10, subtract to get an integer, solve for x.
• Prime factorization: Express numbers as product of primes. It helps find HCF and LCM: HCF uses common lowest powers, LCM uses all prime factors with highest powers.
• Euclid's algorithm: Efficient method to find HCF: repeatedly apply remainder rule: HCF(a,b) = HCF(b, a mod b) until remainder 0.
• Relation between HCF and LCM: For two positive integers a and b, HCF(a,b) × LCM(a,b) = a × b.

Problem-solving strategy (step-by-step):

1. Read the problem; identify what numbers/quantities are rational or integers and which operations are needed (conversion, HCF/LCM, comparison).
2. Convert all numbers to a common form if needed (fractions or decimals).
3. Use prime factorization or Euclid's algorithm to get HCF/LCM when dealing with divisibility, sharing or scheduling problems.
4. Apply the termination-test and repeating-decimal conversion techniques when asked about decimal behaviour.
5. Check answer: simplify fractions, verify with multiplication or direct substitution.

Real-life applications: scheduling (LCM for cycles/timetables), packing or grouping (HCF to maximize size of equal groups), currency/measurement conversions (fractions and decimals), computer representation of numbers (terminating vs repeating), and error/approximation handling.