Formulas and rules

Place value

digit x 10^n

Each digit is worth ten times more as you move one place to the left.

The number line

Numbers are ordered on a line: moving right adds, moving left subtracts.

Negative numbers

-a + b = b - a

Numbers below zero are negative. Adding a negative moves left, subtracting a negative moves right.

Order of operations (BIDMAS)

Brackets, Indices, Division/Multiplication, Addition/Subtraction

Work through brackets first, then powers, then multiplication and division left to right, then addition and subtraction.

Rounding

Look at the digit after the place you round to: 5 or more rounds up, less than 5 rounds down.

Powers and roots

a^n = a x a x ... (n times) ; sqrt(a^2) = a

A power is repeated multiplication. A square root undoes a square.

Scientific notation

a x 10^n with 1 <= a < 10

Write very large or very small numbers as a number between 1 and 10 times a power of ten.

Addition

Addition combines amounts. Line up digits by place value and carry when a column passes 9.

Subtraction

Subtraction finds the difference. Borrow from the next column when needed.

Multiplication

a x b = rows x columns

Multiplication is repeated addition, shown as a rectangular grid.

Division

dividend / divisor = quotient

Division shares a total into equal groups. It undoes multiplication.

Divisibility rules

By 2: even last digit. By 3: digit sum divisible by 3. By 5: last digit 0 or 5. By 9: digit sum divisible by 9. By 10: last digit 0.

Factors and multiples

A factor divides a number exactly. A multiple is found by multiplying a number by a whole number.

Prime numbers

A prime has exactly two factors: 1 and itself. Primes start 2, 3, 5, 7, 11, 13...

Prime factorization

n = 2^a x 3^b x 5^c x ...

Every whole number over 1 is a product of primes.

HCF and LCM

The HCF is the largest shared factor. The LCM is the smallest shared multiple.

Decimals

Decimals show parts of a whole using tenths, hundredths and thousandths. Line up the decimal points to add.

Ratio and proportion

a : b ; total / (a + b) = one part

A ratio compares amounts. To share in a ratio, find one part then multiply.

What is a fraction

numerator / denominator

A fraction is equal parts of a whole. The denominator is the number of parts, the numerator the number taken.

Equivalent fractions

a/b = (a x k)/(b x k)

Multiplying or dividing top and bottom by the same number keeps the value.

Simplifying fractions

Divide the numerator and denominator by their HCF.

Comparing fractions

With the same denominator, the larger numerator is bigger. With the same numerator, the smaller denominator is bigger.

Adding and subtracting fractions

a/b + c/d = (ad + bc) / bd

Find a common denominator, then add the numerators. Never add denominators.

Multiplying fractions

a/b x c/d = ac / bd

Multiply numerators together and denominators together.

Dividing fractions

a/b / c/d = a/b x d/c

Keep, flip, multiply: keep the first fraction, flip the second and multiply.

Mixed numbers and improper fractions

a b/c = (a x c + b) / c

A mixed number has a whole part and a fraction. Convert to an improper fraction to calculate.

Fraction, decimal and percentage

1/2 = 0.5 = 50%

Divide the numerator by the denominator for a decimal; multiply by 100 for a percentage.

Percentage of a number

p% of n = (p / 100) x n

A percentage is a fraction out of 100.

Percentage increase and decrease

change% = (new - old) / old x 100

Compare the change to the original amount.

Reverse percentage

original = final / multiplier

To find the original price after a change, divide by the multiplier.

Compound interest

A = P (1 + r)^n

Interest is added to the total each period, so it grows exponentially.

Algebraic notation

2x means 2 times x. x^2 means x squared. Like terms can be collected.

Expanding brackets

a(b + c) = ab + ac

Multiply the term outside by every term inside.

Factorising

ab + ac = a(b + c)

Take the highest common factor outside the brackets.

Solving linear equations

ax + b = c -> x = (c - b) / a

Do the same operation to both sides to keep the balance and isolate x.

Inequalities

Solve like an equation, but reverse the sign when multiplying or dividing by a negative.

Systems of linear equations

Two equations, two unknowns: eliminate one variable by adding or substituting.

Quadratic formula

x = (-b +/- sqrt(b^2 - 4ac)) / 2a

Solves ax^2 + bx + c = 0. The discriminant b^2 - 4ac tells how many roots.

Difference of two squares

a^2 - b^2 = (a - b)(a + b)

A square subtracted from a square factorises into a product of a difference and a sum.

Sequences

arithmetic: u_n = a + (n-1)d ; geometric: u_n = a x r^(n-1)

Arithmetic sequences add a constant, geometric ones multiply by a constant.

Perimeter and area

perimeter = sum of sides; rectangle area = l x w

Perimeter is the distance around a shape, area is the surface inside it.

Area of a triangle

area = (base x height) / 2

Half the base times the perpendicular height.

Circle area and circumference

area = pi r^2 ; circumference = 2 pi r

Pi is roughly 3.14. The diameter is twice the radius.

Angle rules

triangle = 180 deg; straight line = 180 deg; full turn = 360 deg

Angles in a triangle add to 180. On a straight line they add to 180. Angles at a point add to 360.

Pythagoras theorem

a^2 + b^2 = c^2

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Thales theorem

AM/AB = AN/AC = MN/BC

A line parallel to one side of a triangle cuts the other two sides proportionally.

Volume

box = l x w x h; cylinder = pi r^2 h

Volume measures the space inside a solid, in cubic units.

Symmetry

A shape has a line of symmetry when it folds exactly onto itself.

Coordinates

A point (x, y) gives the horizontal then vertical position. x comes before y.

Vectors

u + v = (x1 + x2, y1 + y2) ; |u| = sqrt(x^2 + y^2)

Add vectors component by component. The length is found with Pythagoras.

Mean

mean = sum of values / number of values

The mean averages the data.

Median, mode and range

Median is the middle value when ordered. Mode is the most frequent. Range is highest minus lowest.

Probability

P = favourable outcomes / total outcomes

Probability runs from 0 (impossible) to 1 (certain).

Combined probability

independent: P(A and B) = P(A) x P(B)

For independent events, multiply the probabilities.

Quartiles and spread

IQR = Q3 - Q1

Quartiles split ordered data into four parts; the IQR measures spread.

Function notation

f(x) = ...

A function gives exactly one output for each input.

Linear and affine functions

y = mx + c

m is the slope (steepness), c is the y-intercept.

Slope between two points

m = (y2 - y1) / (x2 - x1)

The slope measures the change in y for each unit of x.

Quadratic functions

y = ax^2 + bx + c

The graph is a parabola. If a > 0 it opens up, if a < 0 it opens down.

Exponential and logarithm

y = a^x ; log_a(a^x) = x

A logarithm answers "what power gives this number".

SOH CAH TOA

sin = O/H ; cos = A/H ; tan = O/A

In a right triangle these ratios link an angle to two sides.

Exact trigonometric values

sin 30 = 1/2, cos 30 = sqrt3/2, tan 45 = 1

Common angles 0, 30, 45, 60, 90 have exact values worth memorising.

Pythagorean identity

sin^2 x + cos^2 x = 1

This identity holds for every angle.

Sine and cosine rules

a/sin A = b/sin B ; c^2 = a^2 + b^2 - 2ab cos C

Use the sine rule for pairs of side and angle, the cosine rule for three sides or two sides and the included angle.

Derivative of a power

d/dx (x^n) = n x^(n-1)

Bring the power down and reduce it by one.

Sum, product and quotient rules

(uv)' = u'v + uv' ; (u/v)' = (u'v - uv') / v^2

Differentiate sums term by term; use the product and quotient rules for combinations.

Tangent line and stationary points

y = f(a) + f'(a)(x - a)

The derivative at a point is the slope of the tangent. Stationary points have f'(x) = 0.

Integrals

integral of x^n = x^(n+1)/(n+1) + C

Integration undoes differentiation and measures the area under a curve.