Real Numbers

In-Depth Class 10 Real Numbers Notes for Exam Success

Class 10 Real Numbers Notes provide a strong foundation in mathematics, covering essential topics like divisibility, HCF, LCM, prime factorization, and Euclid’s lemma. These notes are designed to help students grasp fundamental concepts with clear explanations and solved examples. Understanding real numbers is crucial for solving complex algebraic problems and for laying the groundwork for higher mathematics. The notes include practice questions, step-by-step solutions, and shortcuts that make learning easier and more effective. Whether it is calculating HCF and LCM of large numbers or exploring the properties of rational and irrational numbers, these notes ensure students are well-prepared for exams. Go through the NCERT textbook and solve the NCERT questions with the help of the NCERT Solutions for class 10 Maths. With these Real Numbers Notes, students can quickly revise concepts, improve problem-solving speed, and gain confidence. These notes are suitable for students preparing for Class 10 board exams, competitive tests, or anyone looking to strengthen their number theory skills.

What Are Real Numbers?

Real numbers comprise the complete set of rational and irrational numbers combined. A rational number can be expressed as p/q where p and q are integers and q ≠ 0, while an irrational number cannot be expressed in this form. This comprehensive number system forms the foundation of higher mathematics and appears extensively in competitive examinations like NTSE.

Every real number occupies a unique position on the number line, making them essential for measuring quantities, distances, and solving practical problems. The classification and properties of real numbers determine how we perform arithmetic operations and solve equations.

Classification of Numbers

Prime and Composite Numbers

A prime number has exactly two distinct factors: 1 and itself. The smallest prime number is 2, which is also the only even prime. Examples include 2, 3, 5, 7, 11, and 13. Understanding that 1 is neither prime nor composite is crucial—this special status stems from the requirement that primes must have exactly two different factors.

Composite numbers possess at least three factors. The smallest composite number is 4. Numbers like 6, 12, and 15 are composite because they can be divided evenly by numbers other than 1 and themselves.

Twin primes are pairs of prime numbers that differ by 2, such as (3, 5), (11, 13), and (17, 19). Co-prime numbers share no common factors except 1, meaning their highest common factor (HCF) equals 1. For instance, 12 and 35 are co-prime despite both being composite.

Even, Odd, and Consecutive Numbers

Even numbers are divisible by 2 and can be expressed as 2k for some integer k. Odd numbers cannot be divided evenly by 2 and take the form 2k + 1. Any integer sequence where each number differs from the next by 1 represents consecutive numbers, such as 50, 51, 52, 53.

Euclid's Division Lemma and Algorithm

The Division Lemma

For any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that:

a = bq + r, where 0 ≤ r < b

This fundamental principle states that when dividing a by b, you obtain a quotient and a remainder smaller than the divisor. If b divides a evenly, then r = 0.

An important property: if a = bq + r, then every common divisor of a and b is also a common divisor of b and r. This property makes Euclid's algorithm efficient for finding the HCF of two numbers.

Finding HCF Using Euclid's Algorithm

To find the HCF of two numbers c and d where c > d:

  1. Apply the division lemma to find q and r such that c = dq + r (0 ≤ r < d)
  2. If r = 0, then d is the HCF
  3. If r ≠ 0, apply the division lemma to d and r
  4. Continue until the remainder becomes zero—the divisor at this stage is the HCF

For example, to find HCF(575, 15):

  • 575 = 15 × 38 + 5
  • 15 = 5 × 3 + 0

Since the remainder is 0, the HCF is 5.

The Fundamental Theorem of Arithmetic

This theorem states that every composite number can be expressed as a product of prime numbers, and this factorization is unique except for the order of factors. This uniqueness principle is fundamental to number theory.

For instance, 140 can be factorized as:

  • 140 = 2 × 2 × 5 × 7 = 2² × 5 × 7

Regardless of the factorization method used, the same prime factors with the same exponents appear.

Applications in Finding HCF and LCM

HCF (Highest Common Factor): Take each common prime factor with the smallest exponent appearing in all factorizations.

LCM (Least Common Multiple): Take all prime factors with the greatest exponent appearing in any factorization.

The relationship HCF(a, b) × LCM(a, b) = a × b holds for any two positive integers.

Decimal Representation of Rational Numbers

Terminating Decimals

A rational number p/q (in lowest terms) has a terminating decimal expansion if and only if the prime factorization of q is of the form 2^m × 5^n, where m and n are non-negative integers.

Example: 13/3125 = 13/(2⁰ × 5⁵) has a terminating decimal expansion.

Non-Terminating Repeating Decimals

If the denominator q (in lowest terms) has prime factors other than 2 or 5, the rational number has a non-terminating repeating decimal expansion.

Example: 5/3 = 1.66666... (repeating)

Irrational Numbers

Irrational numbers cannot be expressed as p/q where p and q are integers. Their decimal expansions are non-terminating and non-repeating. Common examples include √2, √3, √5, π, and e.

Proving Irrationality

To prove √2 is irrational, assume it's rational: √2 = a/b (where a and b are co-prime). Squaring both sides gives 2 = a²/b², so a² = 2b². This means a² is even, hence a is even. Let a = 2c, then 4c² = 2b², so b² = 2c², making b even. But this contradicts our assumption that a and b are co-prime. Therefore, √2 is irrational.

Important property: If a is rational and b is irrational, then a + b and a × b (where a ≠ 0) are both irrational.

Divisibility Rules

Understanding divisibility rules accelerates problem-solving:

  • Divisibility by 2: Last digit is even
  • Divisibility by 3: Sum of digits is divisible by 3
  • Divisibility by 4: Last two digits form a number divisible by 4
  • Divisibility by 5: Last digit is 0 or 5
  • Divisibility by 6: Number is divisible by both 2 and 3
  • Divisibility by 9: Sum of digits is divisible by 9
  • Divisibility by 11: Difference between sum of digits in odd positions and sum of digits in even positions is 0 or divisible by 11

For divisibility by 7: Double the units digit and subtract from the remaining number. If the result is divisible by 7, so is the original number.

Essential Formulas for Real Numbers

Concept Formula/Property Explanation
Euclid's Division Lemma a = bq + r, where 0 ≤ r < b For integers a and b, unique quotient q and remainder r exist
HCF × LCM Relationship HCF(a,b) × LCM(a,b) = a × b Product of HCF and LCM equals product of the numbers
Terminating Decimal Condition q = 2^m × 5^n Rational p/q terminates if q has only factors 2 and 5
Odd Integer Forms 6q + 1, 6q + 3, 6q + 5 Any odd positive integer takes one of these forms
Square Forms (mod 3) 3m or 3m + 1 Any perfect square is of form 3m or 3m + 1
Cube Forms (mod 9) 9m, 9m + 1, or 9m + 8 Any perfect cube takes one of these three forms

Practical Applications

Real numbers and their properties enable solving diverse problems:

  1. Finding optimal arrangements: Using HCF to determine maximum equal groups
  2. Meeting point problems: Using LCM to calculate when cyclic events coincide
  3. Prime factorization: Breaking down numbers for cryptography and computer science
  4. Decimal conversions: Understanding when fractions terminate or repeat

Mastery of real numbers provides the foundation for algebra, calculus, and advanced mathematics, making these concepts indispensable for academic success and competitive examinations.

Class 10 Real Numbers Notes & Questions with Solved Problems

The totality of rational numbers and irrational numbers is called the Real Numbers. That is, all numbers which are either rational or irrational are classified as real numbers. A real number which is not rational is called irrational.

Divisibility Relation

If a is divisible by b, we write as b | a ("b divides a").

  • b is a factor of a (e.g. 2 is a factor of 10).
  • b is a divisor of a (e.g. 3 is a divisor of 24 since 24 = 3 × 8).
  • a is a multiple of b (e.g. 15 is a multiple of 5).

A non-zero integer a is said to divide another integer b if there exists an integer c such that b = a × c. Here, a is the divisor, c is the quotient, and b is the dividend.

Classification of Numbers

  • Even Numbers: Natural numbers divisible by 2. Examples: 2, 4, 6, 8, ...
  • Odd Numbers: Natural numbers not divisible by 2. Examples: 1, 3, 5, 7, ...
  • Consecutive Numbers: Series of numbers each differing by one. Example: 50, 51, 52, 53.

Prime, Composite, Co-prime, Perfect & More

  • Prime Number: A natural number with exactly two distinct positive divisors: 1 and itself (e.g. 2, 3, 5, 7,...).
  • Composite Number: A natural number with at least three positive divisors (e.g. 4, 6, 12...).
  • Twin-prime: Pair of primes differing by 2 (e.g. (3,5), (5,7)).
  • Co-prime: Two numbers having no common factor other than 1 (e.g. 8, 15).
  • Perfect Number: A number equal to the sum of its proper divisors (e.g. 6, 28).

Euclid's Division Lemma & Algorithm

Euclid's Division Lemma:
Given any two positive integers a and b, there exist unique integers q and r such that:

a = bq + r, with 0 ≤ r < b

Euclid's Division Algorithm:
Used to find the HCF (GCD) of two positive integers by repeated use of the Division Lemma.
Steps:

  1. Divide a by b, get quotient q and remainder r; a = bq + r,0 ≤ r < b.
  2. If r = 0, b is the HCF. Else, set a = b, b = r and repeat.

Example: HCF of 128 and 240:
240 = 128 × 1 + 112
128 = 112 × 1 + 16
112 = 16 × 7 + 0
HCF: 16

Fundamental Theorem of Arithmetic

Every composite number can be expressed (factorized) as a product of prime numbers, and this factorization is unique (except for the order of factors).

  • Practical Example of Prime Factorization

140 = 2 × 2 × 5 × 7
The order may differ, but the prime set remains unique.

Rational & Irrational Numbers

  • Rational Number

A number of the form p/q, where p, q are integers and q ≠ 0.
Examples: 4 (=4/1), 1/3, 0.25 (=1/4), recurring decimals (0.333... = 1/3)

  • Irrational Number

A number which is not rational, but can be represented on the number line.
Examples: √2, π, e.

Any non-terminating non-recurring decimal is irrational.

If x is rational and y is irrational, then x+y and x−y are irrational.

Divisibility Rules (2 to 11)

  • By 2: Last digit even.
  • By 3: Sum of digits is multiple of 3.
  • By 4: Last two digits form a multiple of 4.
  • By 5: Last digit 0 or 5.
  • By 6: Divisible by 2 and 3.
  • By 7: Double the last digit, subtract from rest; divisible by 7 means original number is too.
  • By 8: Last three digits divisible by 8.
  • By 9: Sum of digits multiple of 9.
  • By 10: Last digit is 0.
  • By 11: Difference of sums of alternate digits is 0 or multiple of 11.

[Insert diagram: Examples for divisibility by different numbers]

Decimal Representation of Rational Numbers

Theorem: Let x = p/q be a rational number in lowest terms.

  • If q’s prime factorization is only of the form 2m × 5n, x has a terminating decimal expansion.
  • Otherwise, x has a non-terminating, repeating decimal expansion.

Recurring decimals are rational; non-terminating, non-recurring decimals are irrational.

Worked Examples

  1. Show that any positive odd integer is of the form 6q+1, 6q+3, or 6q+5.
    Solution: By division, an odd integer divided by 6 gives a remainder of 1, 3, or 5.
  2. Use Euclid's division lemma to show cube of any positive integer is of the form 9m, 9m+1, or 9m+8.
  3. Show that only one among n, n+2, n+4 is divisible by 3 for any positive integer n.
    Solution: By dividing n by 3 and checking remainders, only one of the numbers is divisible by 3 in each case.

Key Theorems

  • If p is a prime number and p divides a², then p divides a.
  • A rational number has a terminating decimal if the denominator has no prime factor other than 2 or 5.
  • √2, √3, √5, ... are irrational numbers (with proof by contradiction).

Practice Exercises

  1. Check if 6n can ever end with the digit 0 for any natural number n.
    Solution: No, as prime factorisation of 6n contains only 2s and 3s, never a 5.
  2. An army contingent of 616 is to march behind a band of 32; find max columns.
    Solution: HCF(616, 32) = 8
  3. Prove that √3 is irrational.
    Solution: Suppose √3 = a/b. Then a²=3b²; both a, b divisible by 3, contradiction unless both zero. So, irrational.

Illustrative Problems

Find HCF of 825 and 175 by Euclid's algorithm:
825 = 175 × 4 + 125
175 = 125 × 1 + 50
125 = 50 × 2 + 25
50 = 25 × 2 + 0
HCF = 25

FUNDAMENTAL THEOREM OF ARITHMETIC

Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes, and this factorization is unique, except for the order in which the prime factors occur.

Example: Prime Factorization of 140 in Different Orders

Some Important Results:

  1. Let 'p' be a prime number and 'a' be a positive integer. If 'p' divides a², then 'p' divides 'a'.
  2. Let x be a rational number whose decimal expansion terminates. Then, x can be expressed in the form p/q, where p and q are co-primes, and prime factorization of q is of the form 2m × 5n, where m, n are non-negative integers.
  3. Let x = p/q be a rational number, such that the prime factorization of q is not of the form 2m × 5n where m, n are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating.

CLASSIFICATION OF REAL NUMBERS

Real Numbers are classified into rational and irrational numbers.

RATIONAL NUMBERS

A number which can be expressed in the form p/q where p and q are integers and q ≠ 0 is called a rational number.

Examples:

  • 4 is a rational number since 4 can be written as 4/1 where 4 and 1 are integers and the denominator 1 ≠ 0.
  • ¾, -2/5 are also rational numbers.
  • Recurring decimals are also rational numbers (0.333…, 0.111111…, 0.166666…)

Key Point: Between any two numbers, there can be infinite number of other rational numbers.

IRRATIONAL NUMBERS

Numbers which are not rational but which can be represented by points on the number line are called irrational numbers.

Examples:  etc.

Numbers like π, ε are also irrational numbers.

Key Point: Between any two numbers, there are infinite numbers of irrational numbers.

Another Way of Looking at Rational and Irrational Numbers

  • Any terminating or recurring decimal is a rational number.
  • Any non-terminating non-recurring decimal is an irrational number.

RULES FOR DIVISIBILITY

In a number of situations, we will need to find the factors of a given number. Some of the factors of a given number can, in a number of situations, be found very easily either by observation or by applying simple rules.

Divisibility by 2

A number divisible by 2 will have an even number as its last digit.

Examples: 128, 246, 2346

Divisibility by 3

A number is divisible by 3 if the sum of its digits is a multiple of 3.

Example: 9123 → 9 + 1 + 2 + 3 = 15 (multiple of 3) ✓

74549 → 7 + 4 + 5 + 4 + 9 = 29 (not a multiple of 3) ✗

Divisibility by 4

A number is divisible by 4 if the number formed with its last two digits is divisible by 4.

Example: 178564 → last two digits: 64 (divisible by 4) ✓

476854 → last two digits: 54 (not divisible by 4) ✗

Divisibility by 5

A number is divisible by 5 if its last digit is 5 or zero.

Examples: 15, 40, 125, 3450

Divisibility by 6

A number is divisible by 6 if it is divisible both by 2 and 3.

Examples: 18, 42, 96

Divisibility by 7

Rule: Take the units digit of the number, double it and subtract this figure from the remaining part of the number. If the result so obtained is divisible by 7, then the original number is divisible by 7.

Example 1: 595

  • Units digit: 5, doubled: 10
  • Remaining part: 59
  • 59 - 10 = 49 (divisible by 7) ✓

Example 2: 967

  • Units digit: 7, doubled: 14
  • Remaining part: 96
  • 96 - 14 = 82 (not divisible by 7) ✗

Divisibility by 8

A number is divisible by 8, if the number formed by the last 3 digits of the number is divisible by 8.

Example: 3816 → last three digits: 816 (divisible by 8) ✓

Divisibility by 9

A number is divisible by 9 if the sum of its digits is a multiple of 9.

Example: 6318 → 6 + 3 + 1 + 8 = 18 (multiple of 9) ✓

Divisibility by 10

A number divisible by 10 should end in zero.

Divisibility by 11

A number is divisible by 11 if the difference between the sum of digits in odd places and the sum of the digits in even places is equal to zero or a multiple of 11.

Example 1: 132

  • Sum of digits in odd places: 1 + 2 = 3
  • Sum of digits in even places: 3
  • Difference: 3 - 3 = 0 ✓

Example 2: 89394811

  • Sum of digits in odd places: 8 + 3 + 4 + 1 = 16
  • Sum of digits in even places: 9 + 9 + 8 + 1 = 27
  • Difference: 27 - 16 = 11 (multiple of 11) ✓

Divisibility by 19

Rule: Take the units digit of the number, double it and add this figure to the remaining part of the number. If the result so obtained is divisible by 19, then the original number is divisible by 19.

Example: 665

  • Units digit: 5, doubled: 10
  • Remaining part: 66
  • 66 + 10 = 76 (divisible by 19) ✓

DECIMAL REPRESENTATION OF RATIONAL NUMBERS

Theorem 1

Let x = p/q be a rational number such that q ≠ 0 and prime factorization of q is of the form 2m × 5n where m, n are non-negative integers then x has a decimal representation which terminates.

Example:

Theorem 2

Let x = p/q be a rational number such that q ≠ 0 and prime factorization of q is not of the form 2m × 5n, where m, n are non-negative integers, then x has a decimal expansion which is non-terminating repeating.

Example:

Rational Number Form of prime factorisation of the denominator Decimal expansion of rational number
x = p/q, where p and q are coprime and q ≠ 0 q = 2m5n where n and m are non-negative integers terminating
x = p/q, where p and q are coprime and q ≠ 0 q ≠ 2m5n where n and m are non-negative integers non-terminating

EUCLID'S DIVISION LEMMA

Euclid's Division Lemma

For any two positive integers a and b, a > b there exist unique integers q and r such that:

a = bq + r (0 ≤ r < b)

Key Relationships:

  • HCF(a, b) × LCM(a, b) = a × b
  • HCF of two or more prime numbers is always 1
  • LCM of two or more prime numbers is equal to their products

SOLVED EXAMPLES

Example 1: Is 7 × 11 × 13 + 11 a composite number?

Solution:

11(7×13 + 1) = 11(91 + 1) = 11 × 92 = 1012

It is a composite number which can be factorized into primes.

Example 2: Find the LCM and HCF of 84, 90 and 120 by applying the prime factorization method.

Solution:

84 = 2² × 3 × 7

90 = 2 × 3² × 5

120 = 2³ × 3 × 5

Prime factors Least exponent
2 1
3 1
5 0
7 0

HCF = 2¹ × 3¹ = 6

Common prime factors Greatest exponent
2 3
3 2
5 1
7 1

LCM = 2³ × 3² × 5¹ × 7¹ = 8 × 9 × 5 × 7 = 2520

Example 3: Given that HCF (306, 657) = 9. Find the LCM (306, 657).

Solution:

HCF (306, 657) × LCM (306, 657) = 306 × 657

⇒ 9 × LCM (306, 657) = 306 × 657

⇒ LCM (306, 657) = (306 × 657)/9 = 34 × 657

⇒ LCM (306, 657) = 22338

Example 4: Prove that √2 is an irrational number.

Solution:

Let assume on the contrary that √2 is a rational number.

Then, there exists positive integer a and b such that √2 = a/b

where, a and b are co primes i.e. their HCF is 1.

⇒ (√2)² = (a/b)²

⇒ 2 = a²/b²

⇒ a² = 2b²

⇒ a² is multiple of 2

⇒ a is a multiple of 2 ....(i)

⇒ a = 2c for some integer c.

⇒ a² = 4c²

⇒ 2b² = 4c²

⇒ b² = 2c²

⇒ b² is a multiple of 2

⇒ b is a multiple of 2 ....(ii)

From (i) and (ii), a and b have at least 2 as a common factor. But this contradicts the fact that a and b are co-prime. This means that √2 is an irrational number.

PRACTICE PROBLEMS

Let's Try - Questions

  1. What can you say about the prime factorization of the denominators of the following rationales:
    • (i) 36.12345
    • (ii) 36.5678 (repeating)
  2. For a pair of integers 151, 16, find the quotient q and the remainder r when the larger number a is divided by the smaller number b and verify that a = bq + r and 0 ≤ r < b.
  3. Prove that the square of any positive integer of the form 5q + 1 is of the same form.
  4. 144 cartons of coke cans and 90 cartons of Pepsi cans are to be stacked is a canteen. If each stack is of same height and is to contains cartons of the same drink, what would be the greatest number of cartons each stack would have?
  5. Show that n² - n is divisible by 2 for every positive integer n.
  6. Check whether 6n can end with the digit 0 for any natural number.
  7. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
  8. Prove that 3 - √5 is an irrational number.
  9. Without actually performing the long division, state whether 13/3125 has terminating decimal expansion or not.

Answers:

  • Question 2: q = 9 and r = 7
  • Question 4: 18
  • Question 7: 8

Frequently Asked Questions

Rational numbers are values that can be expressed as a fraction p/q where both p and q are integers and q is not zero. These numbers include all integers, finite decimals, and repeating decimals. For example, 5 can be written as 5/1, making it rational. Similarly, 0.75 equals 3/4, and 0.333... equals 1/3, both rational.

Irrational numbers cannot be expressed as such fractions. Their decimal representations continue infinitely without repeating any pattern. Classic examples include √2 (approximately 1.414213...), π (approximately 3.141592...), and √5.

To identify them, examine the decimal expansion. If it terminates (like 0.25) or repeats (like 0.666...), the number is rational. If the decimal continues forever without establishing a repeating pattern, it's irrational. For square roots specifically, if you're taking the square root of a perfect square (4, 9, 16, 25, etc.), the result is rational. However, the square root of any non-perfect square is irrational.

A practical test involves attempting to express the number as a fraction. If you can find integers p and q that exactly represent the value, it's rational. Mathematical proofs, like the contradiction method used to prove √2 is irrational, provide definitive answers. Understanding this distinction is fundamental because it affects how we perform calculations, how we represent measurements, and how we solve equations in advanced mathematics. Both types together form the complete set of real numbers that we use to measure continuous quantities in the physical world.

Euclid's Division Algorithm is an efficient systematic method for finding the Highest Common Factor (HCF) of two positive integers. The algorithm is based on Euclid's Division Lemma, which states that for any two positive integers a and b (where a > b), there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < b.

The algorithm works through repeated application of this lemma. Start by dividing the larger number by the smaller number. Then take the divisor and the remainder from this division and repeat the process. Continue this iterative procedure until you obtain a remainder of zero. The divisor at this final step is the HCF of the original two numbers.

For example, to find HCF(240, 128): First, 240 = 128 × 1 + 112. Next, 128 = 112 × 1 + 16. Then, 112 = 16 × 7 + 0. Since the remainder is now zero, the HCF is 16.

This method is remarkably efficient because each division significantly reduces the size of the numbers involved, typically cutting them by at least half with each iteration. This makes it far faster than listing all factors of both numbers and finding the greatest common one, especially for large numbers. The algorithm's efficiency stems from a mathematical property: any common divisor of a and b must also be a common divisor of b and r (the remainder). This preservation of common divisors through each step guarantees that the final non-zero remainder is indeed the HCF. This ancient algorithm, documented over 2,300 years ago, remains one of the oldest algorithms still in widespread use today, forming the basis for modern computational methods in number theory and cryptography.

The Fundamental Theorem of Arithmetic states that every composite number (integer greater than 1) can be expressed as a product of prime numbers, and this prime factorization is unique except for the order in which the prime factors are written. This theorem is foundational to number theory and has profound implications throughout mathematics.

For example, the number 60 can be factorized as 2 × 2 × 3 × 5, or written more compactly as 2² × 3 × 5. No matter what method you use to break down 60 into primes, you will always arrive at exactly two 2s, one 3, and one 5. The uniqueness guarantee means that prime numbers truly serve as the "building blocks" or "atoms" of all integers, just as chemical elements are the building blocks of compounds.

This theorem's importance extends across multiple areas. First, it provides a systematic way to find the HCF and LCM of numbers. To find the HCF, take each common prime factor with the smallest exponent; for LCM, take all prime factors with the greatest exponent. Second, it forms the theoretical foundation for understanding divisibility properties and solving Diophantine equations. Third, it's essential in cryptography, particularly in RSA encryption, which relies on the difficulty of factoring large composite numbers into their prime components.

The theorem also helps prove other important mathematical results. For instance, it can be used to demonstrate that √2 is irrational and to establish properties about perfect squares and cubes. Without this fundamental principle, much of modern number theory, algebra, and computer science would lack a coherent foundation. The theorem's simplicity belies its power—it takes an infinite collection of integers and provides them with a clear, unambiguous structure based on primes, making it one of the most elegant and useful results in all of mathematics.

Determining whether a rational number produces a terminating or non-terminating decimal requires examining its denominator when the fraction is in its lowest terms (simplified form where numerator and denominator are co-prime). The key principle is straightforward: a rational number p/q has a terminating decimal expansion if and only if the prime factorization of q contains only the factors 2 and 5, expressed as q = 2m × 5n where m and n are non-negative integers.

This rule exists because our decimal system is base 10, and 10 = 2 × 5. When a denominator contains only these factors, we can always multiply both numerator and denominator by appropriate powers of 2 or 5 to make the denominator a power of 10, resulting in a terminating decimal.

For example, consider 13/3125. First, simplify if needed (this fraction is already in lowest terms). Then factorize the denominator: 3125 = 5^5. Since this contains only the prime factor 5, the decimal terminates. Indeed, 13/3125 = 0.00416.

Conversely, consider 5/12. Simplifying (already simplified), we factorize: 12 = 2² × 3. Because 3 appears in the factorization, this fraction produces a non-terminating repeating decimal: 5/12 = 0.41666...

The process is systematic: reduce the fraction to lowest terms, perform prime factorization on the denominator, and check whether any primes other than 2 or 5 appear. If they do, the decimal repeats indefinitely. If only 2s and 5s appear (or the denominator is 1), the decimal terminates. This understanding is practically valuable when working with fractions, allowing you to predict computational behavior and choose appropriate representations for different contexts, whether in pure mathematics, science, engineering, or financial calculations where precision and representation matter significantly.

Proving that √2 is irrational requires the proof by contradiction method, also called reductio ad absurdum. This elegant technique assumes the opposite of what you want to prove, then demonstrates that this assumption leads to a logical impossibility, thereby confirming your original claim.

Begin by assuming √2 is rational. By definition, this means √2 can be expressed as a/b where a and b are integers with no common factors (co-prime) and b ≠ 0. We can always reduce any fraction to this form, so this assumption is reasonable. Now square both sides: 2 = a²/b², which rearranges to 2b² = a². This equation tells us that a² is even (since it equals 2 times another integer). A crucial number theory fact states that if a² is even, then a itself must be even. Therefore, we can write a = 2k for some integer k.

Substituting back: 2b² = (2k)² = 4k², which simplifies to b² = 2k². This shows that b² is also even, which means b must be even too. Here's the contradiction: we've now proven that both a and b are even, meaning they share 2 as a common factor. But this directly contradicts our initial statement that a and b are co-prime (have no common factors). This logical impossibility can only arise from our initial assumption being false. Therefore, √2 cannot be rational, making it irrational.

This same proof technique extends to other square roots. Generally, if p is a prime number, then √p is irrational. The proof follows similar logic, using the fact that if p divides a², then p divides a. This method has been used since ancient Greek mathematics and demonstrates the power of logical reasoning. Understanding this proof technique is valuable beyond just this specific result—it teaches critical thinking, logical structure, and how to construct rigorous mathematical arguments that are applicable across various fields of study and problem-solving situations.

Finding the HCF (Highest Common Factor) and LCM (Least Common Multiple) of three or more numbers requires a systematic approach using prime factorization. This method is more reliable and efficient than trying to extend two-number algorithms directly.

For HCF of multiple numbers, first find the complete prime factorization of each number. Write each number as a product of prime factors with their exponents. Then, identify all prime factors that appear in every single number—these are the common prime factors. For each common prime, take the smallest exponent that appears across all factorizations. Multiply these common primes (with their minimum exponents) together to get the HCF.

For example, to find HCF(84, 90, 120): 84 = 2² × 3 × 7; 90 = 2 × 3² × 5; 120 = 2³ × 3 × 5. The common prime factors are 2 and 3. The minimum exponent for 2 is 1 (from 90), and for 3 is 1 (from both 84 and 120). Therefore, HCF = 2¹ × 3¹ = 6.

For LCM of multiple numbers, again start with prime factorizations. This time, identify all prime factors that appear in any of the numbers. For each prime, take the largest exponent that appears in any factorization. Multiply these primes (with their maximum exponents) together to get the LCM.

Using the same example for LCM(84, 90, 120): All primes appearing are 2, 3, 5, and 7. Maximum exponents: 2³ (from 120), 3² (from 90), 5¹ (from 90 or 120), and 7¹ (from 84). Therefore, LCM = 2³ × 3² × 5 × 7 = 8 × 9 × 5 × 7 = 2520.

This approach is particularly powerful because it scales efficiently to any number of integers, provides clear visualization of the mathematical structure, and helps understand relationships between numbers. Alternative methods exist, such as finding HCF or LCM of pairs successively, but prime factorization remains the most transparent and least error-prone technique for manual calculation and conceptual understanding.

Divisibility rules are shortcuts that allow you to determine whether one number divides another without performing actual division. These rules significantly accelerate mental arithmetic, problem-solving in competitive examinations, and everyday calculations, making them invaluable tools for students and professionals alike.

The divisibility rule for 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3. For example, consider 9123. Sum the digits: 9 + 1 + 2 + 3 = 15. Since 15 is divisible by 3, so is 9123. This works because of modular arithmetic properties in base 10.

For divisibility by 11, calculate the alternating sum: add digits in odd positions, add digits in even positions, then find the difference. If this difference is 0 or divisible by 11, the original number is divisible by 11. For 785345: odd positions (1st, 3rd, 5th) = 7 + 5 + 4 = 16; even positions (2nd, 4th, 6th) = 8 + 3 + 5 = 16. The difference is 0, so 785345 is divisible by 11.

The divisibility rule for 7 is less commonly known but highly useful: take the last digit, double it, and subtract from the remaining number. If the result is divisible by 7, so is the original. For 595: last digit is 5, doubled gives 10. Remaining number is 59. Calculate 59 - 10 = 49. Since 49 is divisible by 7, so is 595. For larger numbers, repeat this process until you reach a manageable number.

Practical applications include quickly factoring numbers, simplifying fractions, solving problems involving remainders, checking arithmetic in accounting and bookkeeping, and verifying solutions in algebra. In competitive examinations like NTSE, these rules save precious time. For instance, when asked if a large number is divisible by 9, simply summing its digits is far faster than long division. In real-world scenarios, merchants historically used these rules for quick calculations before electronic calculators. Understanding why these rules work—rooted in modular arithmetic and number theory—deepens mathematical intuition and provides tools applicable across various quantitative fields.

Real numbers, particularly through HCF and LCM concepts, provide powerful tools for solving numerous practical problems in daily life, from scheduling to resource distribution. Understanding these applications transforms abstract mathematical concepts into useful problem-solving techniques.

HCF applications typically involve finding the largest possible equal groups or the greatest uniform measurement. Consider a classic problem: arranging 144 cartons of one product and 90 cartons of another in stacks where each stack has the same number of cartons and contains only one type of product. Finding HCF(144, 90) = 18 tells us the maximum number of cartons per stack. This principle applies to cutting materials into equal pieces without waste, distributing items equally among groups, or finding the largest tile size that fits perfectly in a given space.

LCM applications generally involve finding when repeating cycles coincide or determining the smallest common multiple for scheduling. For example, if two buses leave a station simultaneously, with one completing a circuit every 18 minutes and another every 12 minutes, when will they next depart together? LCM(18, 12) = 36 minutes provides the answer. This principle applies to traffic light synchronization, planetary alignment calculations, shift scheduling for workers, and determining when periodic maintenance tasks align.

Combined HCF-LCM problems appear in optimization scenarios. If you know HCF(a, b) and LCM(a, b), you can find unknown values using the relationship: HCF(a, b) × LCM(a, b) = a × b. This proves useful in reverse-engineering problems or verifying solutions.

Educational applications include understanding musical rhythms (where different note lengths create patterns), computer science algorithms (especially in managing memory allocation and process scheduling), and project management (coordinating tasks with different durations). The army contingent marching problem from the document finding how many columns allow 616 and 32 members to march identically—demonstrates military logistics applications. Similarly, astronomy uses these concepts to predict celestial events, and manufacturing employs them for production line synchronization. Mastering these applications develops problem-solving skills transferable across disciplines, making abstract number theory tangibly relevant and immediately applicable.