Real Numbers

Every whole number above 1 is a product of primes in exactly one way; that gives HCF, LCM and proofs that numbers like √2 are irrational.

Key ideas

• Fundamental Theorem of Arithmetic: every composite number can be written as a product of primes in exactly one way, apart from the order of the factors.
• HCF: multiply the smallest power of each prime that appears in all the numbers.
• LCM: multiply the greatest power of each prime that appears in any of the numbers.
• For two numbers, HCF × LCM equals the product of the numbers. This does not hold for three numbers in general.
• A rational number can be written as p/q with integers p and q, q ≠ 0. An irrational number cannot.
• If a prime p divides a², then p divides a. This is the key step in proving √2, √3 and √5 irrational.
• Rational + irrational is irrational, and a non-zero rational × irrational is irrational.

Remember

Watch out

• Using HCF × LCM = product for three numbers.
• Taking the greatest power for the HCF (it is the smallest).
• Writing 1 as a prime factor (1 is not prime).

Worked example