Number Theory and Computation

Section 1Number sets

Number sets are groups of numbers with shared features. Natural numbers are counting numbers like 1, 2, 3. Whole numbers include 0, integers include negatives, and rational numbers can be written as a fraction. Irrational numbers, like the square root of 2, cannot be written exactly as a fraction.

Section 2Factors, multiples, and primes

A factor divides exactly into a number, with no remainder. A multiple is what you get when you multiply a number by whole numbers. Easy to mix up - 6 is a factor of 24, but 24 is a multiple of 6. A prime number has exactly two factors: 1 and itself.

Section 3HCF and LCM

The highest common factor, or HCF, is the largest number that divides exactly into two or more numbers. The lowest common multiple, or LCM, is the smallest number that two or more numbers divide into exactly. Here's the trick: HCF is for sharing into equal groups; LCM is for matching cycles or repeated events.

Section 4Prime factorisation

Prime factorisation breaks a number into prime-number building blocks. For example, 60 = 2 x 2 x 3 x 5. Once numbers are written as prime factors, you can find HCF by taking the common prime factors and LCM by taking enough prime factors to build every number.

Section 5Powers and roots

A power tells you repeated multiplication, so 5 squared means 5 x 5. A root reverses a power: the square root of 49 is 7 because 7 squared is 49. Watch out: brackets matter, so (-4)^2 = 16, but -4^2 means the negative of 4 squared, which is -16.

Section 6Number bases

Base ten uses place values 1, 10, 100, 1000, and so on. Base two, or binary, uses place values 1, 2, 4, 8, 16, and so on, so 1011 base two means 8 + 0 + 2 + 1 = 11 base ten. You see this idea behind digital systems, from phones in Kingston classrooms to traffic-light controllers.