Chapter 1 — Real Numbers

Statement: For any 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.

Explanation / Existence: Think of dividing a objects into groups of size b. The number of complete groups is q and the leftover objects are r. Formally, take q = floor(a/b) (the greatest integer ≤ a/b). Then r = a − bq satisfies 0 ≤ r < b, so such q and r exist.

Uniqueness: If a = bq1 + r1 = bq2 + r2 with 0 ≤ r1, r2 < b, then b(q1 − q2) = r2 − r1. The left side is a multiple of b, but the right side has absolute value < b unless r1 = r2. Hence r1 = r2 and therefore q1 = q2; q and r are unique.

Key consequence (Euclidean algorithm): If a = bq + r, then gcd(a,b) = gcd(b,r). Repeating this step (replacing a by b and b by r) reduces the numbers until the remainder becomes 0; the last nonzero remainder is gcd(a,b). This yields an efficient algorithm to compute GCDs.

Notes: CBSE usually treats a as a nonnegative integer and b > 0. The lemma is the foundation for division, gcd computations, and the Fundamental Theorem of Arithmetic (unique prime factorization).

What is GCD? The greatest common divisor (GCD or HCF) of two integers a and b (not both zero) is the largest positive integer that divides both a and b exactly.

Euclidean algorithm (GCD by repeated division) — idea in one line: If a = bq + r (0 ≤ r < b), then gcd(a, b) = gcd(b, r). Replacing the pair (a, b) by (b, r) repeatedly reduces the numbers until the remainder becomes 0; the last nonzero remainder is the gcd.

Why it works (brief proof):

• If d divides both a and b then d divides a − bq = r, so every common divisor of a and b is a common divisor of b and r.
• If d divides both b and r then d divides bq + r = a, so every common divisor of b and r is a common divisor of a and b.
• Thus the sets of common divisors of (a,b) and (b,r) are identical; hence their greatest element (gcd) is the same.

Algorithm (step-by-step):

1. Arrange the two positive integers so that a ≥ b (<span style="white-space:nowrap;">if not, swap them).
2. Divide a by b to get quotient q and remainder r: a = bq + r, where 0 ≤ r < b.
3. If r = 0, then gcd(a, b) = b and stop.
4. Replace (a, b) by (b, r) and repeat from step 2.
5. The algorithm terminates because remainders strictly decrease and are non‑negative; the last nonzero remainder is the gcd.

Notes:

• For three or more numbers, gcd(a,b,c) = gcd(gcd(a,b), c).
• Special cases: gcd(a,0) = |a| (for a ≠ 0); gcd(0,0) is undefined or sometimes taken as 0.

When to use: Use this method whenever you need the greatest common divisor of two (or more) integers; it is efficient even for large integers and underlies many number‑theory algorithms.

Definition: A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself (e.g., 2, 3, 5, 7...). A composite number is a natural number greater than 1 that has more than two positive divisors (e.g., 4, 6, 8, 9...). The number 1 is neither prime nor composite.

Basic properties:

• 2 is the only even prime. All other even numbers are composite.
• If a number n has a divisor d > 1, then it also has a divisor n/d. To test primality it suffices to check divisors up to √n.
• Every composite number can be expressed as a product of primes (prime factorization).

Fundamental Theorem of Arithmetic: Every integer greater than 1 can be written uniquely (up to order) as a product of prime powers: n = p1^a1 × p2^a2 × ... × pk^ak, where p1, p2, ..., pk are distinct primes and a1, a2, ..., ak are positive integers.

Prime factorization uses: Finding HCF (GCD) and LCM. For two numbers expressed by prime powers, take minimum exponents for HCF and maximum exponents for LCM. For two numbers a and b, HCF(a,b) × LCM(a,b) = a × b.

Primality checks & simple divisibility tests:

• Divisible by 2: last digit even.
• Divisible by 3: sum of digits divisible by 3.
• Divisible by 5: last digit 0 or 5.
• Divisible by 7, 11, 13: use specific rules or test up to √n.

Methods to find primes: Sieve of Eratosthenes: mark multiples of each prime starting from 2 to eliminate composites, remaining unmarked numbers are primes.

Short proof idea (Euclid) that primes are infinite: Assume finite list of primes p1,...,pn. Consider N = p1p2...pn + 1. N is not divisible by any pi, so it has a new prime factor, contradiction. Hence infinitely many primes.

Applications / real-life relevance: Cryptography (RSA uses large primes), digital hashing, error detection, arranging objects in equal groups (divisibility), scheduling and packing (LCM/HCF problems), gears and teeth counts where coprime counts avoid repeated alignments.

Statement: Every integer greater than 1 can be written as a product of prime numbers, and this factorization is unique except for the order of the prime factors.

What it means: For any integer n > 1, there exist primes p1, p2, ..., pk and positive integers a1, a2, ..., ak such that

n = p1^a1 · p2^a2 · ... · pk^ak.

Existence (idea): Start dividing n by the smallest prime that divides it. The quotient is either 1 or a smaller integer; repeat the process. Since each division reduces the number, eventually you reach 1. The primes used along the way give a prime factorization.

Uniqueness (idea): If a number had two different prime factorizations, Euclid's lemma (a prime dividing a product divides one of the factors) forces the same primes to appear with the same exponents in both factorizations. Thus the factorization is unique up to the order of factors.

Notes: 1 is neither prime nor composite and has no prime factorization. The theorem applies only to integers > 1.

Practical algorithm (factor tree method):

• Pick any composite number n.
• Write n as a product of two factors (preferably using a small prime factor).
• If a factor is composite, split it further; if it's prime, stop splitting it.
• Continue until all leaves are primes; multiply them (with exponents) to get the prime-power representation.

Statement: If p is a prime number and p divides the product ab (written p | ab), then p divides a or p divides b (or both).

Why it matters: Euclid's Lemma is a key step in proving the Fundamental Theorem of Arithmetic (unique prime factorisation). It also has a useful generalisation: if p divides a1·a2·...·an then p divides at least one of the factors.

Proof (using gcd and Bezout's identity):

• Assume p is prime and p | ab.
• If p divides a, we are done. Otherwise gcd(a,p)=1 because p has no divisors other than 1 and itself.
• By Bezout's identity there exist integers x,y such that ax + py = 1.
• Multiply both sides by b: abx + pyb = b. The left side is divisible by p (abx is, because p | ab; pyb is obviously), so p divides b.
• Thus if p does not divide a, it must divide b. This proves the lemma.

Alternative explanation (prime-factor view): In the prime factorisation of ab every prime factor comes from the factors a or b. If p appears in the prime factors of ab, it must appear in the factorisation of a or b. Hence p divides a or p divides b.

Generalisation: If gcd(a,c)=1 and a | bc, then a | c. This follows by writing a as product of primes and applying Euclid's Lemma to each prime factor.

Important remark (why primality is needed): The lemma fails if the divisor is composite. Example: 6 divides 2·3 = 6, but 6 divides neither 2 nor 3.

Definitions

HCF (Highest Common Factor), also called GCD (Greatest Common Divisor), of two or more integers is the largest positive integer that divides each of them exactly. LCM (Lowest Common Multiple) of two or more integers is the smallest positive integer that is a multiple of each of them.

Why they matter

HCF is useful for simplifying fractions and dividing objects into maximum equal groups. LCM is used to find when repeating events coincide (schedules, cycles), or to find a common denominator for adding/subtracting fractions.

Methods to find HCF and LCM

• Prime factorization method: Express each number as product of primes. For HCF multiply common primes with their lowest powers. For LCM multiply all primes that appear with their highest powers.
• Division (short division) method: Repeatedly divide numbers by common prime divisors until no common divisor > 1 remains. Product of divisors gives HCF (for the divisor method focused on HCF) or combine appropriately for LCM when using combined division tables.
• Euclid's algorithm (efficient for HCF/GCD): For two numbers a and b (a > b), compute remainder r = a mod b, then replace a by b and b by r, repeat until remainder is 0. Last nonzero remainder is HCF. Then use HCF to get LCM by LCM = |a*b| / HCF.

Relationship between HCF and LCM (for two numbers)

For two nonzero integers a and b: HCF(a, b) × LCM(a, b) = |a × b|.

Special notes

• HCF of numbers that are pairwise coprime may still be >1 only if a common factor exists in all; otherwise HCF = 1.
• For more than two numbers, you can iteratively use HCF and LCM: HCF(a,b,c) = HCF(HCF(a,b), c); LCM(a,b,c) = LCM(LCM(a,b), c).

Rational numbers are numbers that can be expressed as a fraction p/q where p and q are integers and q ≠ 0. Every rational number has a decimal representation obtained by dividing p by q. There are two possible types of decimal expansions for rational numbers:

• Terminating decimals: The decimal stops after a finite number of digits (for example 0.75).
• Repeating (recurring) decimals: After some point a block of digits repeats indefinitely (for example 0.333..., 0.1818...).

Key criterion (in lowest terms p/q): the decimal expansion of p/q terminates iff the denominator q has no prime factors other than 2 and/or 5. If q has any other prime factor (e.g., 3, 7, 11), the decimal expansion is repeating.

How to get the decimal from a fraction: perform long division of numerator by denominator. If remainders become 0 at some step, the division terminates. If a remainder repeats, the digits between two occurrences of the same remainder form the repeating block.

How to convert repeating decimals back to fractions (methods):

• Pure repeating (only repeating block): if x = 0.(R) where R is an r-digit repeating block, then x = integer(R)/(10^r - 1). Example: 0.(3) = 3/9 = 1/3.
• Mixed decimal (non-repeating part followed by a repeating block): if x has n non-repeating digits followed by r repeating digits, then x = (integer formed by first n+r digits - integer formed by first n digits)/(10^{n+r} - 10^n).

Length of the repeating block (repetend): For a reduced fraction p/q where q' is q with all factors 2 and 5 removed, the length of the repetend is the smallest positive integer k such that 10^k ≡ 1 (mod q'). Thus the repetend length depends on the multiplicative order of 10 modulo q'.

All rational numbers have either terminating or eventually periodic decimal expansions; irrational numbers have non-terminating, non-repeating decimals.

Definition: An irrational number is a real number that cannot be expressed as a ratio of two integers (i.e., it cannot be written as p/q where p and q are integers and q ≠ 0). In decimal form an irrational number has a non-terminating, non-repeating expansion.

Key points:

• Irrational numbers together with rational numbers make up the set of real numbers: R = Q ∪ (R\Q).
• Common examples: π = 3.14159..., e = 2.71828..., √2 = 1.4142135..., √3 = 1.73205....
• Decimal property: irrational decimals never terminate and never repeat a fixed pattern of digits.
• Density: Between any two distinct real numbers there are infinitely many irrational numbers (and also infinitely many rational numbers).

Simple proof idea (√2 is irrational): Suppose √2 = p/q in lowest terms with integers p,q. Then p² = 2q², so p² is even ⇒ p is even. Write p = 2k; then 4k² = 2q² ⇒ q² = 2k² ⇒ q is even. Both p and q are even, contradicting that p/q was in lowest terms. So √2 is not rational.

Properties (useful at Class 10 level):

• The sum or difference of two irrational numbers may be rational or irrational (e.g., √2 + (1−√2)=1 is rational; √2+√3 is irrational).
• Product or quotient of two irrational numbers may be rational or irrational (e.g., √2·√2=2 is rational; √2·√3 is irrational).
• If r is a nonzero rational number and x is irrational, then r·x is irrational. (If r·x were rational then x would be rational.)
• Adding a rational number to an irrational number gives an irrational number (except in the special case when the rational cancels the irrational part).

How to recognize some irrationals quickly: √n is irrational whenever n is a positive integer that is not a perfect square. Numbers that arise from geometry (diagonals, circumferences) or certain limits (π, e) are typically irrational.

Real-life connections: The number π appears in formulas for circumference and area of circles (C = 2πr, A = πr²). √2 appears as the diagonal length of a unit square (important in construction and geometry). e appears in continuous growth models such as continuously compounded interest and some natural processes.

What are real numbers? Real numbers are all numbers that can be placed as points on a continuous number line. They include natural numbers, whole numbers, integers, rational numbers (fractions and terminating/repeating decimals) and irrational numbers (non‑terminating, non‑repeating decimals).

Classification

• Natural numbers: 1, 2, 3, ...
• Whole numbers: 0, 1, 2, ...
• Integers: ..., -2, -1, 0, 1, 2, ...
• Rational numbers: numbers expressible as p/q (q ≠ 0). Their decimal expansion is either terminating or repeating.
• Irrational numbers: numbers that cannot be written as p/q. Their decimal expansion is non‑terminating and non‑repeating (examples: √2, π).

Representation on the number line

• Every real number corresponds to exactly one point on the number line and every point on the number line corresponds to exactly one real number.
• To locate a rational number p/q: perform the division p ÷ q to get its decimal expansion and mark that point. If the simplified denominator has only prime factors 2 and/or 5, the decimal terminates; otherwise it repeats.
• To locate an irrational number (for example √2): use successive approximations—find increasingly narrow intervals that contain the number (interval nesting) or construct geometrically (√2 is the diagonal of a unit square) and project its length onto the number line. Decimal approximations (1.414, 1.4142, 1.41421, ...) give increasingly accurate locations.

Important properties

• Density: Between any two distinct real numbers there exist infinitely many rational numbers and infinitely many irrational numbers.
• Ordering and distance: Given real numbers a and b, a point representing a is to the left of b if a < b; the distance between them is |a - b|.
• Archimedean property (useful for locating and approximating): For any real x, there exists a natural number n such that n > x. Equivalently, for any ε > 0 there exists n in N with 1/n < ε.

Why this matters

The number line gives a geometric picture of numbers and their order, and it is the basis for measuring distances, comparing magnitudes, and understanding continuity and approximation. Placing rationals and irrationals on the same line shows the completeness of real numbers: there are no “gaps” left unrepresented.

Overview: The chapter Real Numbers introduces tools (Euclid's division lemma, Euclid's algorithm, prime factorization and the Fundamental Theorem of Arithmetic) that are used to solve practical problems involving highest common factor (HCF/GCD), least common multiple (LCM), classification of numbers (rational/irrational), and repeating decimals. "Applications and Problem Solving" means using these tools to model and solve everyday partitioning, scheduling, remainder and representation problems.

Key methods:

• Euclid's division lemma: For integers a and b (b > 0) there exist unique integers q and r such that a = bq + r, 0 ≤ r < b. This is the basis of the Euclid (GCD) algorithm.
• Euclid's algorithm: Repeatedly apply the division lemma to find the HCF/GCD quickly. If a = bq + r, then gcd(a,b) = gcd(b,r) and continue until remainder 0.
• Prime factorization and Fundamental Theorem of Arithmetic: Every integer > 1 can be written uniquely (up to order) as product of primes. Use prime powers to compute HCF and LCM: HCF uses minimum exponents, LCM uses maximum exponents.
• Converting repeating decimals to rational numbers: Use algebraic manipulation: if x has repeating part, form equations multiplying by powers of 10 to eliminate repeat and solve for x.

Typical applications:

• Grouping or partition problems (use HCF): e.g., divide objects into the largest possible identical groups, or place items in equal rows with maximum elements per row.
• Scheduling / synchronization problems (use LCM): e.g., find when two repeating events coincide (buses, bells, lights).
• Remainder problems: use modular reasoning and sometimes LCM/HCF to find numbers leaving specified remainders on division by different divisors.
• Decimal representation: classify numbers as terminating/recurring (rational) or non-terminating non-repeating (irrational); convert recurring decimals into fractions for exact answers.

Strategy for problem solving:

• Model the situation with integers (counts, minutes, objects).
• Decide whether HCF (equal largest groups), LCM (first common occurrence) or prime-factor techniques are needed.
• Apply Euclid's algorithm for large numbers or prime-factorization for transparency.
• Check answers in the context (units, positive integers, remainders range).