Mathematical Operations (Cambridge O Level Maths)

Revision Note

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Author

Dan

Last updated

23 October 2023

Mathematical Symbols

What mathematical symbols do I need to know?

Starting with the basic four operations

$+$ Addition, plus, sum, total

$-$ Subtraction, minus, difference, take away

$\times$ Multiplication, product, times

$\div$ Division, quotient, divide, share

Equals signs

$=$ Equal to e.g. $3 x + 7 = 19$

$\neq$ Not equal to e.g. $2 - 5 \neq 5 - 2$

$\approx$ Approximately equal to e.g. $π \approx 3.14$

$\equiv$ Identical to e.g. $12 x + 6 \equiv 3 (4 x + 2)$

Inequality signs

$>$ Greater than e.g. $5 > - 2$

$<$ Less than $\frac{1}{2} < \frac{2}{3}$

$\geq$ Greater than or equal to

$\leq$ Less than or equal to

Other helpful symbols

$()$ Brackets: used to group symbols e.g. $(2 x + 4) - 3 (x + 7)$

$3^{x}$ Powers (also called indices, order or exponents): repeated multiplication of the base number

e.g. $3^{4} = 3 \times 3 \times 3 \times 3 = 81$

$\sqrt{}$ Square root: opposite of squaring e.g. $9^{2} = 81$ $\Leftrightarrow$ $\sqrt{81} = 9$

$\pm$ Plus Minus: Used to allow for two distinct answers, often in quadratics

$a \pm b$ is shorthand for $a + b$ and $a - b$

e.g. $x = 3 \pm 2.5$ leading to $x = 5.5$ and $x = 0.5$

$π$ Pi: the ratio of the circumference of a circle to its diameter

Order of Operations (BIDMAS/BODMAS)

What is the order of operations (BIDMAS/BODMAS)?

If there are multiple operations then they must be done in the following order:

Brackets: ( )

Perform the calculation(s) inside any brackets first

Indices or Order: ˄, ², ³, √, etc

These include powers, roots, reciprocals

Divisions or Multiplications: × or ÷

If there are more than one of these then work them out left to right

Additions or Subtractions: + or -

If there are more than one of these then work them out left to right

The acronyms BIDMAS and BODMAS can help you remember

What do I do if the calculation involves fractions, powers or roots?

For fractions there are invisible brackets around the numerator and around the denominator

$\frac{2 + 5}{7 - 2}$ means (2+5)÷ (7-2)
Instead of using brackets we extend the fraction line to show exactly what is on the top and what is on the bottom

For powers there are invisible brackets around the power

$4^{3 + 1}$ means 4^(3+1)
Instead of using brackets we write the whole to the right of the number and raised

For roots there are invisible brackets under the root

$\sqrt{9 + 16}$ means √(9+16)
Instead of using brackets we extend the line on the root symbol to show exactly what is being rooted

Worked example

Work out $(5 - 3) + 2 \times 7^{2}$

First deal with anything inside BRACKETS

$2 + 2 \times 7^{2}$

Then any POWERS

$2 + 2 \times 49$

Followed by any MULTIPLICATIONS or DIVISIONS

$2 + 98$

Finally deal with any ADDITIONS or SUBTRACTIONS

$(5 - 3) + 2 \times 7^{2} = 100$

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