Introduction
Real Numbers form the foundation of higher mathematics. In this chapter, you will learn about the Fundamental Theorem of Arithmetic, which states that every composite number can be expressed as a product of prime numbers in a unique way. You will also explore how to use this theorem to find the HCF and LCM of numbers and to prove the irrationality of certain numbers such as the square roots of 2, 3, and 5.
Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states that every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. This means that for any composite number, there is one and only one way to write it as a product of primes (ignoring the order). For example, 36 = 2 x 2 x 3 x 3 = 2^2 x 3^2. No matter how you factorise 36, you will always get the same prime factors.
Key Points
- •Every composite number has a unique prime factorisation
- •Prime factorisation is used to find HCF and LCM
- •HCF is the product of the smallest powers of common prime factors
- •LCM is the product of the greatest powers of all prime factors
- •HCF(a,b) x LCM(a,b) = a x b for any two positive integers a and b
Worked Example
Find the HCF and LCM of 12, 15 and 21. 12 = 2^2 x 3 15 = 3 x 5 21 = 3 x 7 HCF = 3 (smallest power of common prime factor) LCM = 2^2 x 3 x 5 x 7 = 420 (greatest power of all prime factors)
Watch Out
Always write the complete prime factorisation before finding HCF and LCM. Many students lose marks by skipping this step.
HCF and LCM Using Prime Factorisation
To find the HCF (Highest Common Factor) using prime factorisation, first express each number as a product of prime factors. Then identify the common prime factors and take the lowest power of each common factor. Their product gives the HCF. For LCM (Lowest Common Multiple), take all prime factors that appear in any of the numbers and use the highest power of each. The product gives the LCM.
Key Points
- •HCF uses the LOWEST power of COMMON factors only
- •LCM uses the HIGHEST power of ALL factors
- •For two numbers a and b: HCF x LCM = a x b
- •This relationship does NOT extend directly to three numbers
- •HCF of co-prime numbers is always 1
Worked Example
Find HCF and LCM of 6 and 20. 6 = 2 x 3 20 = 2^2 x 5 HCF = 2 (common factor with lowest power) LCM = 2^2 x 3 x 5 = 60 Verification: HCF x LCM = 2 x 60 = 120 = 6 x 20. Verified!
Watch Out
In exams, always verify your answer using HCF x LCM = product of the two numbers.
Irrational Numbers and Proofs
An irrational number is a number that cannot be expressed as p/q where p and q are integers and q is not zero. The proof of irrationality uses the method of contradiction. To prove that a number like the square root of 2 is irrational, we assume it is rational (i.e., it can be written as p/q in simplest form), and then show this leads to a contradiction — both p and q turn out to be even, which contradicts our assumption that p/q was in its simplest form.
Key Points
- •Irrational numbers cannot be expressed as p/q (q not equal to 0)
- •Proofs use the method of contradiction (assume rational, derive contradiction)
- •Square roots of all prime numbers are irrational
- •The sum or difference of a rational and an irrational number is irrational
- •The product or quotient of a non-zero rational with an irrational number is irrational
Worked Example
Prove that the square root of 3 is irrational. Assume the square root of 3 is rational. Then the square root of 3 = p/q where p, q are co-prime integers (HCF = 1). Squaring: 3 = p^2/q^2, so p^2 = 3q^2. This means p^2 is divisible by 3, so p is divisible by 3. Let p = 3m. Then (3m)^2 = 3q^2, so 9m^2 = 3q^2, so q^2 = 3m^2. This means q^2 is divisible by 3, so q is divisible by 3. But both p and q are divisible by 3, contradicting our assumption that they are co-prime. Hence, the square root of 3 is irrational.
Watch Out
In board exams, the irrationality proof for sqrt(2), sqrt(3), and sqrt(5) is very common for 3-mark questions. Memorise the structure.
Revisiting Rational Numbers and Their Decimal Expansions
A rational number p/q (in simplest form) has a terminating decimal expansion if the prime factorisation of q is of the form 2^n x 5^m, where n and m are non-negative integers. If the denominator has any prime factor other than 2 or 5, the decimal expansion will be non-terminating repeating.
Key Points
- •Terminating decimals: denominator has only 2 and 5 as prime factors
- •Non-terminating repeating decimals: denominator has prime factors other than 2 or 5
- •Every terminating or repeating decimal represents a rational number
- •Non-terminating non-repeating decimals are irrational
Worked Example
Is 13/3125 a terminating decimal? 3125 = 5^5 Since the denominator has only 5 as a prime factor (form 2^0 x 5^5), the decimal expansion is terminating. 13/3125 = 13/(5^5) = 13 x 2^5 / (2^5 x 5^5) = 416/100000 = 0.00416
Watch Out
First simplify the fraction to its lowest terms before checking the denominator. A common mistake is checking the denominator without simplifying.
Quick Summary
- ✓The Fundamental Theorem of Arithmetic: Every composite number has a unique prime factorisation
- ✓HCF = product of smallest powers of common prime factors
- ✓LCM = product of greatest powers of all prime factors
- ✓For two numbers: HCF(a,b) x LCM(a,b) = a x b
- ✓Irrationality proofs use the method of contradiction
- ✓Square roots of prime numbers are always irrational
- ✓Terminating decimal: denominator (in simplest form) = 2^n x 5^m
- ✓Non-terminating repeating decimal: denominator has factors other than 2 and 5
Key Formulas
HCF(a,b) x LCM(a,b) = a x b
HCF = Product of smallest powers of common prime factors
LCM = Product of greatest powers of all prime factors
Terminating decimal condition: q = 2^n x 5^m (in p/q simplest form)
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