Real Numbers

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order of the primes. For example, 32760 can be factorised as 2³ × 3² × 5 × 7 × 13. This theorem is crucial for understanding the properties of numbers and their divisibility.

To find the HCF, multiply the smallest power of each common prime factor in the numbers. For LCM, multiply the greatest power of each prime factor involved. For example, for 6 (2¹ × 3¹) and 20 (2² × 5¹), HCF is 2¹ = 2 and LCM is 2² × 3¹ × 5¹ = 60. This method is efficient and directly uses the prime factorisation of numbers.

The product of HCF and LCM of two numbers equals the product of the numbers because HCF captures the common prime factors and LCM includes all prime factors. For example, for numbers 6 and 20, HCF is 2 and LCM is 60, and 6 × 20 = 2 × 60 = 120. This relationship simplifies calculations involving HCF and LCM.

No, a number in the form of 4^n cannot end with the digit zero because its prime factorisation only includes the prime 2 (4^n = (2)²ⁿ). For a number to end with zero, its prime factorisation must include both 2 and 5. Since 5 is missing, 4^n will never end with zero.

A number is irrational if it cannot be expressed as a fraction p/q where p and q are integers and q ≠ 0. Examples include √2 and π. These numbers have non-terminating, non-repeating decimal expansions, distinguishing them from rational numbers.

Assuming √2 is rational leads to a contradiction. If √2 = a/b (a, b coprime), then 2b² = a² implies a² is even, so a is even. Let a = 2c, then b² = 2c², making b even. But if both a and b are even, they're not coprime, contradicting our assumption. Hence, √2 is irrational.

Euclid's division algorithm states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. This algorithm is foundational for finding the HCF of two numbers through repeated division.

The theorem ensures that every number has a unique prime factorisation. When proving √p (p prime) is irrational, we assume it's rational (√p = a/b), leading to p dividing a² and hence a. Substituting a = pc leads to b² = p c², implying p divides b, contradicting a and b being coprime. Thus, √p is irrational.

Both expressions can be factorised: 7 × 11 × 13 + 13 = 13(7 × 11 + 1) and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5(7 × 6 × 4 × 3 × 2 × 1 + 1). Since both are products of numbers greater than 1, they are composite, illustrating how addition can turn prime products into composites.

The decimal expansion of a rational number p/q is terminating if the prime factorisation of q (after simplifying) has only 2s and/or 5s. Otherwise, it's non-terminating repeating. For example, 1/2 = 0.5 (terminating), while 1/3 = 0.333... (repeating). This helps in identifying the nature of rational numbers quickly.

For three numbers, the HCF is the product of the smallest power of each common prime factor. For example, for 6 (2 × 3), 72 (2³ × 3²), and 120 (2³ × 3 × 5), the HCF is 2¹ × 3¹ = 6. This method ensures the highest common divisor is found efficiently.

Rational numbers can be expressed as fractions p/q where p and q are integers and q ≠ 0, with terminating or repeating decimal expansions. Irrational numbers cannot be expressed as such fractions and have non-terminating, non-repeating decimal expansions, like √2 or π.

No, the sum of a rational and an irrational number is always irrational. For example, 1 (rational) + √2 (irrational) = 1 + √2, which is irrational. This is because if the sum were rational, subtracting the rational part would leave an irrational number equal to a rational number, which is impossible.

The product is irrational because if it were rational, dividing by the non-zero rational number would yield an irrational number equal to a rational number, leading to a contradiction. For example, 2 (rational) × √3 (irrational) = 2√3, which is irrational.

Check if the number can be expressed as a fraction p/q with integers p, q (q ≠ 0). If yes, it's rational; otherwise, it's irrational. For roots, if the number under the root isn't a perfect square, cube, etc., it's likely irrational, like √5 or ³√7.

Prime numbers are the building blocks in the theorem, as every composite number is uniquely expressed as a product of primes. For example, 12 = 2² × 3. This uniqueness helps in various mathematical proofs and applications, such as simplifying fractions or finding HCF and LCM.

The factor tree breaks down a number into its prime factors by repeatedly dividing it into smaller factors until only primes remain. For example, 32760 can be broken down to 2³ × 3² × 5 × 7 × 13. This visual method simplifies understanding and ensures accurate factorisation.

While the theorem states that the factorisation is unique, the order of prime factors doesn't matter. For example, 30 = 2 × 3 × 5 or 5 × 2 × 3. The uniqueness lies in the combination of primes and their powers, not their sequence, making the theorem flexible yet precise.

Yes, if one number is a multiple of the other. For example, the HCF of 6 and 12 is 6, because 6 is a factor of 12. In such cases, the smaller number is the HCF, and the larger number is the LCM, simplifying calculations.

The LCM is the smallest number that both numbers divide into without leaving a remainder, so it must be at least as large as the larger number. For example, the LCM of 4 and 6 is 12, which is greater than both. This ensures all multiples are covered.

Real numbers, including both rational and irrational, are used in measurements, finance, engineering, and science. For example, √2 is used in calculating diagonal distances, and π is essential in geometry and physics. Understanding their properties ensures accurate and practical applications.

A rational number p/q has a terminating decimal if the denominator q, after simplifying, has no prime factors other than 2 or 5. Otherwise, it has a repeating decimal. For example, 3/8 = 0.375 (terminating) as 8 = 2³, while 5/6 ≈ 0.833... (repeating) as 6 = 2 × 3.

Multiply the prime factors together to see if the product equals the original number. For example, 12 = 2² × 3, and 2² × 3 = 4 × 3 = 12. This verification ensures the factorisation is correct and adheres to the Fundamental Theorem of Arithmetic.