Salick Academy

Number Theory and Computation

40 min readNo calculator
By the end of this topic you should be able to
  • Classify numbers into the number sets
  • Find the HCF and LCM using prime factorisation
  • Apply the order of operations correctly
  • Compute with fractions
  • Round to decimal places and significant figures
  • Write numbers in standard form

The number sets

Each set sits inside the next, and questions ask you to name which one a number belongs to.

Set Contains Example
Natural numbers N\mathbb{N} counting numbers 1,2,3,…1, 2, 3, \ldots
Whole numbers W\mathbb{W} the naturals and zero 0,1,2,…0, 1, 2, \ldots
Integers Z\mathbb{Z} whole numbers, positive and negative …,−2,−1,0,1,…\ldots, -2, -1, 0, 1, \ldots
Rational numbers Q\mathbb{Q} anything writable as a fraction 34\tfrac34, −5-5, 0.3‾0.\overline{3}
Irrational numbers cannot be written as a fraction 2\sqrt2, π\pi
Real numbers R\mathbb{R} rationals and irrationals together all of the above

A decimal that stops, or that repeats forever in a pattern, is rational. One that goes on forever without repeating is irrational.

Factors, multiples, HCF and LCM

Write each number as a product of primes and the rest follows.

24=23×336=22×3224 = 2^3 \times 3 \qquad\qquad 36 = 2^2 \times 3^2

The order of operations

Brackets, Order (indices), Division and Multiplication, Addition and Subtraction.

Division and multiplication rank equally — work left to right across them. Same for addition and subtraction.

3+4×22−(6−2)=3+4×4−4=3+16−4=153 + 4 \times 2^2 - (6-2) = 3 + 4 \times 4 - 4 = 3 + 16 - 4 = 15

Fractions

  • Adding or subtracting — get a common denominator first. 23+14=812+312=1112\tfrac23 + \tfrac14 = \tfrac{8}{12} + \tfrac{3}{12} = \tfrac{11}{12}.
  • Multiplying — multiply the tops, multiply the bottoms, then cancel.
  • Dividing — turn the second fraction upside down and multiply.

Rounding: decimal places and significant figures

Decimal places — count places after the point. Significant figures — count from the first non-zero digit.

For 0.003050.00305: the leading zeros are placeholders and are not significant, but the zero between the 3 and the 5 is, because it sits between two non-zero digits. So it has 3 significant figures.

Look only at the digit immediately after your cut-off. If it is 5 or more, round up; otherwise leave the digit alone. So 3.74193.7419 to 2 decimal places is 3.743.74 — the 1 decides it, not the 9 further along.

Standard form

45 800=4.58×10445\,800 = 4.58 \times 10^4 and 0.000472=4.72×10−40.000472 = 4.72 \times 10^{-4}.

Numbers bigger than 1 take a positive index; numbers smaller than 1 take a negative one. The index is how many places the decimal point moves.