math notation

Calculate

Obtain a numerical answer showing the relevant stages in the working.

Comment

Give a judgment based on a given statement or result of a calculation.

Compare

Give an account of the similarities between two (or more) items or situations, referring to both (all) of them throughout.

Compare and contrast

Give an account of similarities and differences between two (or more) items or situations, referring to both (all) of them throughout.

Construct

Display information in a diagrammatic or logical form.

Contrast

Give an account of the differences between two (or more) items or situations, referring to both (all) of them throughout.

Deduce

Reach a conclusion from the information given.

Demonstrate

Make clear by reasoning or evidence, illustrating with examples or practical application.

Describe

Give a detailed account.

Determine

Obtain the only possible answer.

Differentiate

Obtain the derivative of a function.

Distinguish

Make clear the differences between two or more concepts or items.

Draw

Represent by means of a labelled, accurate diagram or graph, using a pencil. A ruler (straight edge) should be used for straight lines. Diagrams should be drawn to scale. Graphs should have points correctly plotted (if appropriate) and joined in a straight line or smooth curve.

Estimate

Obtain an approximate value.

Explain

Give a detailed account including reasons or causes.

Find

Obtain an answer showing relevant stages in the working.

Hence

Use the preceding work to obtain the required result.

Hence or otherwise

It is suggested that the preceding work is used, but other methods could also receive credit.

Identify

Provide an answer from a number of possibilities.

Integrate

Obtain the integral of a function.

Interpret

Use knowledge and understanding to recognize trends and draw conclusions from given information.

Investigate

Observe, study, or make a detailed and systematic examination, in order to establish facts and reach new conclusions.

Justify

Give valid reasons or evidence to support an answer or conclusion.

Label

Add labels to a diagram.

List

Give a sequence of brief answers with no explanation.

Plot

Mark the position of points on a diagram.

Predict

Give an expected result.

Prove

Use a sequence of logical steps to obtain the required result in a formal way.

Show

Give the steps in a calculation or derivation.

Show that

Obtain the required result (possibly using information given) without the formality of proof. 'Show that' questions do not generally require the use of a calculator.

Sketch

Represent by means of a diagram or graph (labelled as appropriate). The sketch should give a general idea of the required shape or relationship, and should include relevant features.

Solve

Obtain the answer(s) using algebraic and/or numerical and/or graphical methods.

State

Give a specific name, value or other brief answer without explanation or calculation.

Suggest

Propose a solution, hypothesis or other possible answer.

Verify

Provide evidence that validates the result.

Write down

Obtain the answer(s), usually by extracting information. Little or no calculation is required. Working does not need to be shown.

ℕ

the set of positive integers and zero, {0, 1, 2, 3, ...}

ℤ

the set of integers, {0, ± 1, ± 2, ± 3, ...}

ℤ+

the set of positive integers, {1, 2, 3, ...}

ℚ

the set of rational numbers

ℚ+

the set of positive rational numbers, {x | x ∈ℚ, x > 0}

ℝ

the set of real numbers

ℝ+

the set of positive real numbers, {x | x ∈ℝ, x > 0}

{x1, x2, . . . }

the set with elements x1, x2, ...

n(A)

the number of elements in the finite set A

{x| }

the set of all x such that

∈

is an element of

∉

is not an element of

∅

the empty (null) set

U

the universal set

∪

union

∩

intersection

A′

the complement of the set A

a^1/2

a to the power 1/2, square root of a (if a ≥0 then a ≥0)

a^1/n

a to the power of 1/n, nth root of a (if a ≥0 then a^n ≥0)

a^−n = 1/a^n

a to the power of −n, reciprocal of a^n

|x|

the modulus or absolute value of x, that is { x for x ≥0, x ∈ℝ; -x for x < 0, x ∈ℝ }

≡

identity

≈

is approximately equal to

>

is greater than

≥

is greater than or equal to

<

is less than

≤

is less than or equal to

≯

is not greater than

≮

is not less than

⇒

implies

⇔

implies and is implied by

u_n

the nth term of a sequence or series

d

the common difference of an arithmetic sequence

r

the common ratio of a geometric sequence

S_n

the sum of the first n terms of a sequence, u1 + u2 + ... + un

S_∞

the sum to infinity of a sequence, u1 + u2 + ... + u1 + u2 + ... + un

∑_{i=1}^n

summation notation

n!

n(n −1)(n −2) . . . 3 × 2 × 1

C_r^n

n! / (r!(n −r)!)

∆

the discriminant of a quadratic equation, Δ = b^2 −4ac

f(x)

the image of x under the function f

f^−1

the inverse function of the function f

f ∘ g

the composite function of f and g

dy/dx

the derivative of y with respect to x

f ′(x)

the derivative of f (x) with respect to x

d^2y/dx^2

the second derivative of y with respect to x

f ′′(x)

the second derivative of f (x) with respect to x

∫y dx

the indefinite integral of y with respect to x

∫_a^b y dx

the definite integral of y with respect to x between the limits x = a and x = b

e^x

the exponential function of x

log_a x

the logarithm to the base a of x

ln x

the natural logarithm of x, log_e x

sin, cos, tan

the circular functions

A(x, y)

the point A in the plane with Cartesian coordinates x and y

[AB]

the line segment with end points A and B

AB

the length of [AB]

(AB)

the line containing points A and B

Â

the angle at A

CÂB

the angle between [CA] and [AB]

∆ABC

the triangle whose vertices are A, B and C

P(A)

probability of event A

P(A′)

probability of the event 'not A'

P(A|B)

probability of the event A given B

x1, x2, ...

observations