Real Numbers

Fundamental Theorem of Arithmetic

Every composite number can be written as a product of primes, and that factorisation is unique apart from the order of the prime factors. Once the primes are arranged in ascending order, there is exactly one way to write the number.

Prime factorisation

Writing a natural number as a product of powers of primes, for example 32760 = 2³ × 3² × 5 × 7 × 13. It is the basis of the prime factorisation method for finding HCF and LCM.

HCF (Highest Common Factor) by prime factorisation

Factorise each number into powers of primes; the HCF is the product of the smallest power of each prime factor common to the numbers. For 6 = 2 × 3 and 20 = 2² × 5, the only common prime is 2, so the HCF is 2.

LCM (Least Common Multiple) by prime factorisation

Factorise each number into powers of primes; the LCM is the product of the greatest power of every prime factor that appears in any of the numbers. For 6 = 2 × 3 and 20 = 2² × 5, the LCM is 2² × 3 × 5 = 60.

Product of the HCF and LCM of two positive integers

HCF(a, b) × LCM(a, b) = a × b for any two positive integers a and b. The relation fails for three numbers: 6, 72 and 120 have HCF 6 and LCM 360, whose product is not 6 × 72 × 120.

How do you find the LCM of two numbers when their HCF is already known?

Divide the product of the two numbers by their HCF, since HCF × LCM equals the product of the numbers. In the chapter's example, 96 = 2⁵ × 3 and 404 = 2² × 101 have HCF 4, so their LCM is (96 × 404) ÷ 4 = 9696.

How can you show that 4ⁿ never ends with the digit zero for any natural number n?

A number ending in 0 is divisible by 5, so 5 would have to appear in its prime factorisation. But 4ⁿ = 2²ⁿ has 2 as its only prime factor, and uniqueness of prime factorisation rules out any other prime, so no natural number n works.

Two people go round a track taking different times per round. Why does the LCM of their times tell you when they next meet at the start?

Each person is back at the start only at whole multiples of their own round time. They are there together only at a time that is a multiple of both times, and the first such moment is the least common multiple.

Irrational number

A number that is not expressible as a ratio p/q of integers p and q with q ≠ 0. Examples include √2, √3, √15, π and the non-repeating decimal 0.10110111011110...

Theorem: if a prime divides a², it divides a

For a prime p and a positive integer a, if a² is divisible by p then a is also divisible by p. It follows from the Fundamental Theorem of Arithmetic, since a² has exactly the same prime factors as a.

Proof by contradiction

A method that assumes the opposite of the statement to be proved and shows that this leads to something impossible. The contradiction means the assumption was wrong, so the original statement must be true.

Co-prime numbers

Two integers that have no common factor other than 1. In irrationality proofs, a fraction a/b is first reduced to lowest terms so that a and b are coprime.

What are the steps in proving that √2 is irrational?

Assume √2 = a/b with a and b coprime integers. Squaring gives 2b² = a², so 2 divides a² and therefore a; write a = 2c. Then b² = 2c², so 2 divides b as well. Now a and b share the factor 2, contradicting that they are coprime, so √2 is irrational.

In the proof that √2 is irrational, why does it matter that a and b are taken to be coprime?

Any fraction can first be reduced so its numerator and denominator have no common factor other than 1. The proof then shows 2 must divide both a and b, which is impossible for coprime numbers, and that clash is the contradiction that disproves the assumption.

How do you show that a number such as 5 − √3 is irrational?

Assume it is rational, equal to a/b with a and b coprime and b ≠ 0. Rearrange to isolate the surd: √3 = 5 − a/b = (5b − a)/b. The right-hand side is rational because a and b are integers, so √3 would be rational, contradicting the fact that √3 is irrational.

Sums and products involving an irrational number

Adding a rational number to an irrational one, or subtracting one from the other, always gives an irrational result; so does multiplying or dividing an irrational number by a non-zero rational. That is why 5 − √3 and 3√2 are irrational.

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