Much of mathematics is taught, researched and published in English, and many of its words are easy to recognise on paper but surprisingly hard to say aloud or use in a sentence. These terms help students and teachers read English textbooks and exam questions, follow lectures and online courses, read expressions such as x² or f(x) correctly, and write clear solutions and proofs.
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Numbers and Operations
sum
noun
Meaning
The result you get when you add two or more numbers together; for example, the sum of 4 and 6 is 10.
Where you use it
Very common in textbooks and word problems: 'find the sum of', 'the sum of the angles', 'the sum of a series'. In everyday British English, 'sums' also means simple arithmetic exercises, as in 'doing sums'.
Examples
The sum of the interior angles of a triangle is 180 degrees.
Find two consecutive numbers whose sum is 41.
Usage note
'Sum' and 'some' sound exactly the same in English; the context tells you which one is meant.
Common phrases
the sum of · find the sum · partial sum · sum of a series
The result of subtracting one number from another; for example, the difference between 10 and 4 is 6.
Where you use it
Used in arithmetic, algebra and word problems: 'the difference between two numbers', 'the difference of two squares'. When the order matters, textbooks usually subtract the second number from the first, or say 'the positive difference'.
Examples
The difference between their ages is twelve years.
Factorise x² − 9 using the difference of two squares.
Usage note
'Common difference' is the fixed gap between the terms of an arithmetic sequence.
Common phrases
the difference between · difference of two squares · find the difference · common difference
The result you get when you multiply numbers or expressions together; for example, the product of 3 and 5 is 15.
Where you use it
A basic word in textbooks and exams: 'the product of two numbers', 'write as a product of prime factors'. At university you also meet the dot product, the cross product and the product rule in calculus.
Examples
Write 60 as a product of its prime factors.
The product of two negative numbers is always positive.
Usage note
In everyday English a 'product' is something a company sells; in maths it always means the result of multiplication.
Common phrases
the product of · product of prime factors · product rule · dot product
The result of dividing one number by another; for example, when 20 is divided by 4, the quotient is 5.
Where you use it
More formal than 'answer'. In long division the parts are the dividend (the number being divided), the divisor, the quotient and the remainder. Calculus students also meet the quotient rule for differentiating fractions.
Examples
When you divide 47 by 5, the quotient is 9 and the remainder is 2.
Use the quotient rule to differentiate this fraction.
Usage note
Pronounced 'KWOH-shunt': the middle sound is 'sh', not 'tee-ent'.
Common phrases
quotient and remainder · quotient rule · the quotient of
A number that shows part of a whole, written as one number over another, such as ¾, where 3 is the numerator and 4 is the denominator.
Where you use it
You simplify, add, compare and convert fractions. Learn the spoken forms: ½ is 'a half' or 'one half', ⅓ 'one third', ¾ 'three quarters' (US also 'three fourths') and 5/8 'five eighths'. Longer ones are read with 'over': 'x over y'.
Examples
Convert the fraction to a decimal before you compare the two prices.
Always give your final answer as a fraction in its simplest form.
Usage note
Turkish says the denominator first ('dörtte üç'), while English says the numerator first ('three quarters').
Common phrases
simplify a fraction · proper fraction · improper fraction · equivalent fractions · convert a fraction
A whole number that can be positive, negative or zero, such as −3, 0 or 25, with no fractional or decimal part.
Where you use it
Common in number theory, algebra and exam questions: 'n is a positive integer', 'find all integer solutions'. The set of integers is written ℤ. Schools also say 'whole number', but that sometimes means only 0, 1, 2, 3…, so 'integer' is more precise.
Examples
Let n be a positive integer greater than 1.
The answer must be an integer, because you can't hire half a bus.
Usage note
The 'g' is soft, like the 'j' in 'jam': 'IN-ti-jer', not 'in-te-ger' with a hard g.
A whole number greater than 1 whose only factors are 1 and itself, such as 2, 3, 5, 7 and 11.
Where you use it
Often shortened to 'prime': '7 is a prime', 'the primes less than 20'. Whole numbers greater than 1 that are not prime are called composite numbers. Breaking a number into primes is called prime factorisation.
Examples
Is 91 a prime number? No, it's 7 times 13.
Two is the only even prime number.
Usage note
1 is not a prime number: a prime must have exactly two different factors.
Common phrases
prime number · prime factor · prime factorisation · composite number
To add a number to itself a given number of times; for example, if you multiply 4 by 3, you get 12.
Where you use it
You multiply one number by another. When reading aloud, 4 × 3 is 'four times three' or 'four multiplied by three'. Related words are multiplication (the operation), multiple (a result of multiplying) and the multiplication table.
Examples
Multiply both sides of the equation by 2 to clear the fraction.
If you multiply a negative number by a negative number, the result is positive.
Usage note
Use 'by' after multiply: 'multiply 4 by 3', not 'multiply 4 with 3'. When reading a calculation aloud, the shorter 'times' is more common.
Common phrases
multiply by · multiply both sides · multiply out the brackets · multiplied together
To find how many times one number fits into another, or to split a quantity into equal parts; 12 shared equally between 3 people gives 4 each.
Where you use it
You divide one number by another, or divide something into parts. Related words are division, divisor, divisible ('12 is divisible by 3') and remainder. In informal speech teachers also say '3 goes into 12 four times'.
Examples
Divide the total by the number of people to find each person's share.
You can't divide by zero, so x cannot be 2 in this fraction.
Usage note
Turkish '3'e bölmek' uses a case ending, but English uses 'by': 'divide by 3', not 'divide to 3'.
Common phrases
divide by · divide into · divisible by · long division
An amount expressed as a number out of 100 and written with the sign %; for example, 25% means 25 out of every 100.
Where you use it
You calculate, find, express or convert a percentage. 'Percent' follows a number ('15 percent'), while 'percentage' is used without one ('a high percentage'). A change from 10% to 12% is a rise of 2 percentage points, but a 20 percent increase.
Examples
What percentage of the students passed the exam?
Express 18 out of 24 as a percentage.
Usage note
Turkish puts the % sign before the number (%25), but English puts it after (25%).
Common phrases
calculate a percentage · express as a percentage · percentage increase · percentage decrease · percentage point
A number less than zero, written with a minus sign in front of it, such as −4 or −0.5.
Where you use it
Used for temperatures, debts, heights below sea level and coordinates. −4 can be read 'minus four' or 'negative four'; American teachers often prefer 'negative' for the number itself and 'minus' for subtraction.
Examples
The square of a negative number is always positive.
On the number line, negative numbers are to the left of zero.
Usage note
Zero is neither positive nor negative.
Common phrases
negative number · negative sign · negative four · positive and negative numbers
A number that gives a particular value when it is multiplied by itself; the square root of 49 is 7, because 7 × 7 = 49.
Where you use it
You find, take or calculate a square root. The symbol √ is called the radical sign or root sign, and √9 is read 'the square root of nine' or simply 'root nine'. Every positive number has two square roots, but √ means the positive one.
Examples
The square root of 50 is between 7 and 8.
Take the square root of both sides, and remember the plus or minus.
Usage note
A 'cube root' (∛) is the number that gives a value when it is multiplied by itself three times: the cube root of 8 is 2.
Common phrases
the square root of · take the square root · square root sign · cube root
The small raised number that shows how many times a base is multiplied by itself; in 2⁵, the exponent is 5 and the base is 2.
Where you use it
Common in algebra and science: 'the laws of exponents', 'a negative exponent', 'a fractional exponent'. 'Power' and, in British schools, 'index' (plural 'indices') are often used with the same meaning.
Examples
When you multiply powers with the same base, add the exponents.
A negative exponent means one over the power, so 2⁻³ is one eighth.
Usage note
The related adjective is 'exponential': exponential growth, exponential function.
Common phrases
laws of exponents · negative exponent · fractional exponent · base and exponent
The number above the line in a fraction, which shows how many parts you have; in 3/8, the numerator is 3.
Where you use it
Used whenever fractions are simplified, added or compared. Teachers say 'the top' informally, but 'numerator' is expected in written answers. To get an equivalent fraction you multiply the numerator and the denominator by the same number.
Examples
If the numerator is larger than the denominator, the fraction is greater than one.
Factorise the numerator first and see what cancels.
Usage note
A memory trick: the Denominator is Down below, so the numerator is on top.
Common phrases
numerator and denominator · factorise the numerator · the numerator of the fraction
The number below the line in a fraction, which shows how many equal parts the whole is divided into; in 3/8, the denominator is 8.
Where you use it
To add or subtract fractions you need a common denominator, ideally the lowest common denominator. A denominator can never be zero, which is why some values are excluded from a function's domain.
Examples
Find a common denominator before you add 2/3 and 3/4.
The denominator is zero when x = 5, so the function is undefined there.
Usage note
'Common denominator' is also used in everyday English for something that several people or things share.
Common phrases
common denominator · lowest common denominator · rationalise the denominator
A number or expression that divides another exactly, with no remainder; the factors of 12 are 1, 2, 3, 4, 6 and 12.
Where you use it
Very common in arithmetic and algebra: 'list the factors', 'prime factors', 'a common factor'. It is also a verb: to 'factor' (US) or 'factorise' (UK) an expression means to write it as a product, as in x² − 9 = (x − 3)(x + 3).
Examples
Is 6 a factor of 84? Yes, because 84 divided by 6 is 14.
Take out the common factor of 3 before you factorise any further.
Usage note
'Factor' and 'multiple' are opposites: 3 is a factor of 12, and 12 is a multiple of 3.
Common phrases
common factor · prime factor · factor of · highest common factor · take out a factor
To replace a number with a simpler value close to it, to a given place value or number of decimal places; 3.476 rounded to one decimal place is 3.5.
Where you use it
Exam questions say 'round to the nearest whole number', 'round to two decimal places' or 'round to three significant figures'. You can also 'round up' or 'round down', and the result is a 'rounded' value.
Examples
Round your answer to the nearest centimetre.
Don't round too early, or your final answer will be slightly off.
Usage note
British English also uses 'round off', and 'correct to 2 decimal places' means the same as 'rounded to 2 decimal places'.
Common phrases
round to the nearest · round up · round down · round to two decimal places · rounding error
The largest whole number that divides two or more numbers exactly; the greatest common divisor of 12 and 18 is 6.
Where you use it
Abbreviated GCD. School textbooks also call it the greatest common factor (GCF, US) or the highest common factor (HCF, UK). It is used to simplify fractions and is found by listing factors, by prime factorisation or with the Euclidean algorithm.
Examples
Divide both numbers by their greatest common divisor to simplify 24/36.
If the greatest common divisor of two numbers is 1, they are called coprime.
Usage note
In Turkish schools the GCD is called EBOB and the LCM is called EKOK; the two are usually taught together.
Common phrases
greatest common divisor of · find the GCD · highest common factor · Euclidean algorithm
The smallest positive number that is a multiple of two or more numbers; the least common multiple of 4 and 6 is 12.
Where you use it
Abbreviated LCM; British textbooks also say 'lowest common multiple'. It is used to find a common denominator and to solve problems about events that repeat at different intervals, such as buses or flashing lights.
Examples
One bus leaves every 12 minutes and another every 18; their least common multiple, 36, tells you when they leave together again.
Use the least common multiple of the denominators as the common denominator.
Usage note
For two positive whole numbers, GCD × LCM equals their product: for 4 and 6, 2 × 12 = 24.
Common phrases
least common multiple of · lowest common multiple · find the LCM · LCM and GCD
The distance of a number from zero on the number line, which is never negative; the absolute value of both −7 and 7 is 7.
Where you use it
Written with vertical bars, |x|, and read 'the absolute value of x'; British textbooks also call it the 'modulus' ('mod x'). Equations and inequalities with absolute values are usually solved by splitting them into two cases.
Examples
Solve the absolute value equation |x − 3| = 5 by looking at two cases.
The absolute value of the difference gives the distance between two numbers.
Common phrases
absolute value of x · absolute value equation · absolute value bars · absolute value inequality
Has the same value as; the word used to read the = sign aloud, as in 'three plus four equals seven'.
Where you use it
The standard way to read '='. You can also say 'is equal to', 'is' or, in informal classroom speech, 'makes'. In written solutions, each new line usually starts with an equals sign placed under the one before.
Examples
Five times six equals thirty.
If 2x equals 14, then x equals 7.
Usage note
Say 'equals' or 'is equal to', not 'is equals' or 'is equal' without 'to'. The sign itself is the 'equals sign' (UK) or 'equal sign' (US).
Common phrases
equals sign · is equal to · x equals · equals zero
Multiplied by itself; used to read a power of 2 aloud, so x² is 'x squared' and 5² = 25 is 'five squared is twenty-five'.
Where you use it
Heard every day in class and in formulas such as a² + b² = c² ('a squared plus b squared equals c squared'). Units like m² are usually read 'square metres'. The verb is 'square': 'square both sides'.
Examples
The area is the side length squared, so a side of 7 cm gives 49 square centimetres.
Remember that x squared equals 16 has two solutions, 4 and −4.
Usage note
(a + b)² is read 'a plus b, all squared' or 'the square of a plus b'; 'a plus b squared' would mean a + b².
Common phrases
x squared · a plus b, all squared · square both sides · squared units
Raised to a given exponent; used to read powers aloud, so 2⁵ is 'two to the power of five', which means 2 × 2 × 2 × 2 × 2 = 32.
Where you use it
The full, formal reading of a power. In fast speech people shorten it: 'two to the fifth', 'ten to the minus three' (10⁻³) or 'x to the n' (xⁿ). For powers 2 and 3, 'squared' and 'cubed' are normally used instead.
Examples
Ten to the power of six is one million.
Any non-zero number to the power of zero equals one.
Usage note
Say 'two to the power of five', not 'two power five', and never translate Turkish 'üzeri' as 'over': 'two over five' means the fraction 2/5.
Common phrases
raise to the power of · two to the power of five · x to the n · ten to the minus three
The words used to read the division sign ÷ or a slash aloud; for example, 20 ÷ 5 = 4 is read 'twenty divided by five equals four'.
Where you use it
The standard spoken form in arithmetic. In algebra and with fractions, 'over' is more common: a/b is 'a over b'. For rates and units, 'per' is used: km/h is 'kilometres per hour'.
Examples
Thirty-six divided by four is nine.
The mean is the total of the values divided by the number of values.
The dot that separates the whole-number part of a number from its decimal part, as in 3.75; English uses a point where Turkish uses a comma.
Where you use it
Numbers are read with 'point', and the digits after it are said one by one: 3.75 is 'three point seven five', not 'three point seventy-five'. Questions often say 'give your answer to two decimal places', meaning two digits after the point.
Examples
Move the decimal point two places to the right to turn 0.45 into a percentage.
Pi starts three point one four one five nine, and its digits after the decimal point go on forever without repeating.
Usage note
English writes 1,500 for one thousand five hundred and 1.5 for one and a half; Turkish does the opposite.
Common phrases
decimal places · to two decimal places · move the decimal point · after the decimal point
The pair of curved symbols ( ) used to group parts of an expression so that they are calculated first, as in 3(x + 2).
Where you use it
US English says 'parentheses' (singular 'parenthesis'); British English usually says 'brackets' or 'round brackets'. You open, close, expand and multiply out parentheses. The order of operations is remembered as PEMDAS (US) or BIDMAS/BODMAS (UK).
Examples
Work out the part in parentheses first, then multiply.
You forgot to close the parentheses, so the calculator gave a different answer.
Usage note
In US English 'brackets' usually means square brackets [ ], and { } are called 'braces' or 'curly brackets'.
Common phrases
in parentheses · open parentheses · close parentheses · expand the brackets
Multiplied by itself three times; used to read a power of 3 aloud, so x³ is 'x cubed' and 2³ = 8 is 'two cubed is eight'.
Where you use it
Used when reading expressions and formulas, such as the volume of a cube, s³ ('s cubed'). Units like cm³ are usually read 'cubic centimetres'. Higher powers use 'to the power of', or 'to the fourth', 'to the fifth' and so on.
Examples
Three cubed is twenty-seven, so the cube root of 27 is 3.
The volume of a cube with side a is a cubed.
Common phrases
x cubed · cubic centimetres · cube root · a plus b, all cubed
The usual spoken form of f(x), the output of a function f when the input is x; for example, if f(x) = x + 5, then 'f of 2' is 7.
Where you use it
Teachers and students say it all the time: 'f of x equals x squared', 'g of three', and 'f of g of x' for f(g(x)). The derivative f′(x) is read 'f prime of x' and the inverse f⁻¹(x) is read 'f inverse of x'.
Examples
If f of x equals 3x minus 1, what is f of 4?
Careful: the question asks for f of x plus h, not f of x plus f of h.
Usage note
f(x) does not mean f times x; the brackets show the input of the function.
Common phrases
f of x equals · f prime of x · f inverse of x · f of g of x
A mathematical statement with an equals sign, showing that two expressions have the same value, such as 2x + 3 = 11.
Where you use it
The central word of algebra. You write, set up, solve, rearrange and check an equation; its two parts are the left-hand side and the right-hand side. In word problems, teachers often ask you to 'form an equation' from the text.
Examples
Set up an equation for the total cost and solve it for x.
This equation has two solutions, so don't stop after finding the first one.
Usage note
An equation must contain an equals sign; 3x + 5 on its own is an expression, not an equation.
Common phrases
solve an equation · set up an equation · linear equation · both sides of the equation · rearrange an equation
A letter or symbol, such as x or y, that stands for a number which can change or which you do not know yet.
Where you use it
Used in algebra, functions and statistics. You introduce, define, isolate and substitute a variable. Teachers distinguish the independent variable (the input, often x) from the dependent variable (the output, often y).
Examples
Let's use the variable t for time in seconds.
Isolate the variable on one side before you divide.
Usage note
In everyday English 'variable' is also an adjective meaning 'changing': variable weather, variable costs.
Common phrases
independent variable · dependent variable · isolate the variable · define a variable · solve for the variable
To find the value or values of an unknown that make an equation or inequality true, or to find the answer to a problem.
Where you use it
You solve an equation, a system of equations, an inequality or a problem. 'Solve for x' means find x, or rearrange so that x is on its own. The noun is 'solution': an equation can have one, many or no solutions.
Examples
Solve for x: 3x − 4 = 11.
We solved the system by substitution and then checked both answers.
Usage note
You solve a problem or an equation, but for a test item English usually says 'answer the question', not 'solve the question'.
Common phrases
solve an equation · solve for x · solve a problem · solve by factorising · solve simultaneously
To write a fraction or expression in its shortest and clearest form without changing its value, for example writing 6/8 as 3/4.
Where you use it
A very common instruction in textbooks and exams: 'Simplify fully', 'Give your answer in its simplest form'. You simplify by collecting like terms, cancelling common factors and using the laws of exponents.
Examples
Simplify the expression 3x + 5x − 2 before you substitute x = 4.
Your answer is right, but you still need to simplify 12/18 to 2/3.
Common phrases
simplify fully · simplify the expression · simplify a fraction · simplest form
A combination of numbers, variables and operation signs, such as 4x² − 3x + 7, that has no equals sign and can be simplified or evaluated.
Where you use it
You simplify, expand, factorise and evaluate an expression. An expression is made of terms: in 4x² − 3x + 7 the terms are 4x², −3x and 7. Exam questions often say 'Write an expression for the perimeter'.
Examples
Write an expression for the total cost of n tickets at £12 each.
Evaluate the expression when x = −2.
Usage note
An expression has no equals sign and an equation has one: you solve an equation, but you simplify an expression.
Common phrases
algebraic expression · simplify an expression · evaluate an expression · expand an expression
The number that multiplies a variable in a term; in 5x² − 3x, the coefficient of x² is 5 and the coefficient of x is −3.
Where you use it
Used in algebra, polynomials and statistics. You find, compare or equate coefficients. A term with no number in front, such as x, has a coefficient of 1, and a number with no variable is called the constant term.
Examples
What is the coefficient of x in 7 − 4x?
Compare the coefficients on both sides to find a and b.
Usage note
Pronounced 'koh-ih-FISH-unt', with the stress on 'fish'.
Common phrases
coefficient of x · leading coefficient · compare coefficients · correlation coefficient
A statement that compares two values using <, >, ≤ or ≥, such as 2x + 1 > 7, showing that one side is larger or smaller than the other.
Where you use it
You solve an inequality and show its solution on a number line or in interval notation. Read the signs as 'is less than', 'is greater than', 'is less than or equal to' and 'is greater than or equal to'.
Examples
Solve the inequality 3x − 5 < 10 and show the answer on a number line.
When you divide both sides of an inequality by a negative number, reverse the sign.
Usage note
In everyday English 'inequality' also means social or economic unfairness, as in 'income inequality'.
Common phrases
solve an inequality · linear inequality · reverse the inequality sign · strict inequality
An expression made of terms with whole-number powers of a variable, added or subtracted, such as 3x⁴ − 2x² + x − 7.
Where you use it
Polynomials are described by their degree, the highest power: linear (1), quadratic (2), cubic (3), quartic (4). You factorise, divide, graph and find the roots of a polynomial. It is also an adjective: polynomial equation, polynomial time.
Examples
What is the degree of this polynomial?
Use the factor theorem to check whether x − 2 divides the polynomial.
Usage note
The stress is on the third syllable: 'pol-ee-NOH-mee-ul'.
Common phrases
degree of a polynomial · cubic polynomial · polynomial division · roots of a polynomial
An equation in which the highest power of the variable is 2, written in the form ax² + bx + c = 0, where a is not zero.
Where you use it
Solved by factorising, completing the square or the quadratic formula, x = (−b ± √(b² − 4ac)) / 2a. The discriminant, b² − 4ac, tells you whether there are two, one or no real roots. People often just say 'a quadratic'.
Examples
This quadratic equation doesn't factorise nicely, so use the formula.
The discriminant is negative, so the quadratic equation has no real roots.
Usage note
'Quadratic' comes from the Latin word for 'square', not 'four': it refers to x², not x⁴.
Common phrases
solve a quadratic equation · quadratic formula · complete the square · roots of a quadratic
A value of the variable that makes an equation or polynomial equal to zero; the roots of x² − 5x + 6 = 0 are 2 and 3.
Where you use it
Also called a 'solution' or, for functions, a 'zero'. On a graph, the real roots are where the curve crosses the x-axis. The word also appears in 'square root', 'cube root' and 'nth root'.
Examples
Find the roots of the equation and check them by substitution.
A quadratic can have at most two real roots.
Usage note
A 'repeated root' or 'double root' occurs when the same value is a root twice, as in (x − 3)² = 0.
Common phrases
real roots · repeated root · find the roots · nth root · sum of the roots
To put a particular number in place of a variable in a formula or expression to see what result it gives; an informal word for 'substitute'.
Where you use it
Very common in spoken classroom English: 'plug in x = 2', 'plug the values into the formula', 'plug it back in to check'. In formal writing 'substitute' is preferred: 'Substituting x = 2 gives…'.
Examples
Plug in x = 3 and see if both sides are equal.
Once you have the radius, just plug it into the area formula.
Usage note
'Plug in' is informal; in exam answers and proofs, write 'substitute' or 'substituting'.
Common phrases
plug in a value · plug into the formula · plug it back in · plug in and check
To remove each other because their combined effect is zero, as when +3x and −3x are added, or when the same factor is above and below a fraction line.
Where you use it
Common in spoken explanations: 'the x terms cancel out', 'the 2s cancel'. You can also cancel a common factor. Teachers often warn that factors cancel but terms do not: in (x + 2)/2, the 2s do not cancel.
Examples
When you add the two equations, the y terms cancel out.
The factor (x − 1) cancels out from the top and bottom of the fraction.
Common phrases
the terms cancel out · cancel each other out · cancel a common factor · cancel out the fractions
A rule that takes each input from a set and gives exactly one output, for example f(x) = 2x + 1, which doubles a number and adds one.
Where you use it
One of the most used words from school to university. You define, evaluate, graph and differentiate a function. People say 'y is a function of x' and talk about linear, quadratic, exponential and trigonometric functions.
Examples
Evaluate the function at x = 3 and plot the point.
Is this graph a function? Check whether any vertical line crosses it twice.
Common phrases
define a function · evaluate a function · linear function · a function of x · graph of a function
A drawing that shows the relationship between quantities, especially all the points (x, y) of a function shown on a pair of axes.
Where you use it
You draw, sketch, plot and read a graph; 'graph' is also a verb: 'graph the function'. To 'sketch' is to show the general shape and key points, while to 'plot' is to mark exact points. In graph theory, a graph is a set of points joined by lines.
Examples
Sketch the graph of y = x² − 4 and label where it crosses the x-axis.
According to the graph, the car reached its top speed after ten seconds.
Common phrases
sketch a graph · plot a graph · read off the graph · graph paper · the graph of a function
A number that measures how steep a line is: the change in y divided by the change in x between any two points on the line.
Where you use it
In y = mx + b the slope is m. People say 'rise over run' to remember the formula. A positive slope goes up from left to right and a negative slope goes down. British English usually says 'gradient'.
Examples
The line through (1, 2) and (3, 8) has a slope of 3.
Parallel lines have the same slope.
Usage note
US textbooks usually write the equation of a line as y = mx + b, while UK textbooks write y = mx + c.
Common phrases
slope of a line · positive slope · negative slope · slope-intercept form · rise over run
One of the reference lines on a graph, usually the horizontal x-axis and the vertical y-axis, used to locate points by their coordinates.
Where you use it
You draw, label and scale the axes. A graph 'crosses' or 'cuts' the x-axis at its x-intercepts. Other common uses are the 'axis of symmetry' of a parabola and the 'axis of rotation' in geometry.
Examples
Label each axis with the variable and its unit.
The parabola crosses the x-axis at −2 and 2.
Usage note
The plural is 'axes', pronounced /ˈæksiːz/ ('AK-seez'), unlike 'axes' /ˈæksɪz/, the plural of 'axe'.
Common phrases
x-axis · y-axis · label the axes · axis of symmetry · cross the axis
The set of all input values for which a function is defined; for f(x) = 1/x, the domain is all real numbers except 0.
Where you use it
Usually paired with 'range', the set of possible output values. You state, find or restrict the domain. Values are left out when they make a denominator zero or put a negative number under a square root.
Examples
State the domain and range of the function.
The domain of √(x − 3) is x ≥ 3.
Usage note
Outside maths, 'domain' also means a field of knowledge or an internet address, as in 'domain name'.
Common phrases
domain and range · find the domain · restrict the domain · domain of a function
Related to a straight line; a linear equation or function has no power of the variable higher than 1, so its graph is a straight line.
Where you use it
Common phrases are linear equation, linear function, linear graph and linear relationship. At university, 'linear algebra' studies vectors and matrices. 'Non-linear' describes curves and anything else that is not linear.
Examples
Solve the linear equation 4x + 9 = 1.
The cost is linear in the number of hours, so the graph is a straight line.
Common phrases
linear equation · linear function · linear graph · linear algebra · non-linear
A point where a graph crosses one of the axes; the y-intercept of y = 2x + 3 is 3, because the line crosses the y-axis at (0, 3).
Where you use it
Usually used as 'x-intercept' and 'y-intercept'. To find the y-intercept, set x = 0; to find the x-intercepts, set y = 0. The form y = mx + b is called the slope-intercept form.
Examples
Set x equal to zero to find the y-intercept.
The graph has two x-intercepts, at −1 and 4.
Usage note
As a noun the stress is on the first syllable (IN-ter-cept); as a verb meaning 'to stop something on its way', it is on the last (in-ter-CEPT).
Common phrases
x-intercept · y-intercept · find the intercepts · slope-intercept form
A function that reverses the effect of another function, so that if f(2) = 7, the inverse of f takes 7 back to 2.
Where you use it
Written f⁻¹(x) and read 'f inverse of x'. You find it by swapping x and y and solving for y. Only one-to-one functions have an inverse, and the graph of f⁻¹ is the reflection of the graph of f in the line y = x.
Examples
Find the inverse function of f(x) = 2x − 5.
The natural logarithm is the inverse function of eˣ.
Usage note
f⁻¹(x) is not 1/f(x): here the −1 means 'inverse', not 'to the power of minus one'.
Common phrases
find the inverse function · f inverse · inverse of a function · one-to-one function
The amount of turn between two lines or rays that meet at a point, measured in degrees or radians.
Where you use it
You measure, calculate, bisect and label angles. The main types are acute (less than 90°), right (exactly 90°), obtuse (between 90° and 180°) and reflex (more than 180°). Angles are often named with three letters, as in 'angle ABC'.
Examples
Measure the angle with a protractor and round to the nearest degree.
Angle ABC is obtuse, so this can't be a right triangle.
Usage note
Don't confuse 'angle' /ˈæŋɡəl/ with 'angel' /ˈeɪndʒəl/; they are spelled and pronounced differently.
Common phrases
measure an angle · acute angle · obtuse angle · interior angle · angle of elevation
A flat shape with three straight sides and three angles; the angles inside any triangle always add up to 180 degrees.
Where you use it
Triangles are classified by their sides as equilateral (all equal), isosceles (two equal) or scalene (none equal), and by their angles as right, acute or obtuse. They are named by their vertices, as in 'triangle ABC'.
Examples
Triangle ABC is isosceles, with AB equal to AC.
Split the shape into two triangles and add their areas.
Usage note
A triangle with a 90° angle is a 'right triangle' in US English and a 'right-angled triangle' in British English.
Common phrases
right triangle · isosceles triangle · equilateral triangle · area of a triangle · similar triangles
The size of a flat surface, measured in square units such as cm² or m²; a rectangle 4 m long and 3 m wide has an area of 12 m².
Where you use it
You find, calculate or work out the area of a shape. Learn the standard formulas: rectangle length × width, triangle ½ × base × height, circle πr². 'Surface area' is the total area of all the faces of a solid.
Examples
The area of a circle with a radius of 5 cm is about 78.5 square centimetres.
Find the area of the shaded region.
Usage note
Area is measured in square units (cm²) and perimeter in ordinary units (cm); mixing them up is a common exam mistake.
Common phrases
area of a circle · surface area · find the area · shaded area · square units
The distance from the centre of a circle or sphere to any point on its edge; it is half the diameter.
Where you use it
Used in formulas such as the area of a circle, πr², and its circumference, 2πr. You find, measure or are given the radius. In geometry problems the letter r almost always stands for the radius.
Examples
A circle has a radius of 6 cm. Find its circumference.
If you double the radius, the area becomes four times larger.
Usage note
The plural is 'radii' /ˈreɪdiaɪ/ in formal writing, though 'radiuses' is also heard.
Common phrases
radius of a circle · radius of the sphere · radius r · given radius
The total length of the boundary of a flat shape; for a rectangle 5 cm long and 3 cm wide, the perimeter is 16 cm.
Where you use it
You find or calculate the perimeter by adding the lengths of all the sides. The perimeter of a circle is called its 'circumference'. Perimeter is a length, so it is measured in cm or m, not in square units.
Examples
The garden has a perimeter of 48 metres, so we need 48 metres of fencing.
Two rectangles can have the same perimeter but different areas.
Usage note
The stress is on the second syllable: 'puh-RIM-it-er'.
Common phrases
find the perimeter · perimeter of a rectangle · perimeter and area · total perimeter
Always the same distance apart and never meeting, however far they are extended; used of lines, planes and the sides of shapes.
Where you use it
One line is parallel to another, and the symbol is ∥, as in AB ∥ CD. Parallel lines have equal slopes. Related words include 'perpendicular' (meeting at a right angle) and 'parallelogram'.
Examples
Line AB is parallel to line CD, so the alternate angles are equal.
Find the equation of the line parallel to y = 2x + 1 that passes through (0, 5).
Usage note
Spelling trap: one 'r', a double 'l' in the middle and a single 'l' at the end: p-a-r-a-l-l-e-l.
Common phrases
parallel to · parallel lines · parallel sides · parallel and perpendicular
A corner point where two or more lines or edges meet, such as a corner of a triangle or a cube; also the turning point of a parabola.
Where you use it
A triangle has three vertices and a cube has eight. Angles are named by their vertex: in angle ABC the vertex is B. For the parabola y = ax² + bx + c, the vertex is at x = −b/2a.
Examples
Label each vertex of the triangle with a capital letter.
The vertex of the parabola is at (2, −1), so its minimum value is −1.
Usage note
In maths texts the plural is 'vertices' /ˈvɜːrtɪsiːz/ ('VER-ti-seez'), not 'vertexes'.
Common phrases
vertex of a triangle · vertex of a parabola · opposite vertex · vertices of a cube
An angle of exactly 90 degrees, like the corner of a square, usually marked on diagrams with a small square.
Where you use it
Lines that meet at a right angle are perpendicular. Common phrases are 'at right angles to', 'a right-angled triangle' (UK) and 'a right triangle' (US). Two right angles make a straight angle of 180°.
Examples
The wall meets the floor at a right angle.
Mark the right angle on the diagram before you use Pythagoras.
Usage note
'Right' here means 'upright, correct', not the direction; it has nothing to do with left and right.
Common phrases
at a right angle · at right angles to · right-angled triangle · mark the right angle
The rule that in a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
Where you use it
Used to find a missing side of a right triangle and the distance between two points. You apply or use the Pythagorean theorem; in class people simply say 'use Pythagoras'. Whole-number sets such as 3, 4, 5 are called Pythagorean triples.
Examples
By the Pythagorean theorem, the diagonal of a 6 by 8 rectangle is 10.
You can only use the Pythagorean theorem if the triangle has a right angle.
Usage note
US English says 'the Pythagorean theorem'; British English usually says 'Pythagoras' theorem' or 'Pythagoras's theorem'.
Common phrases
apply the Pythagorean theorem · Pythagorean triple · use Pythagoras · by Pythagoras
The amount of space a solid object takes up or a container holds, measured in cubic units such as cm³ or m³.
Where you use it
You find or calculate the volume of cubes, cuboids, cylinders, cones and spheres. Key formulas: cuboid length × width × height, cylinder πr²h, sphere ⁴⁄₃πr³. For liquids, volume is often given in litres (1 litre = 1000 cm³).
Examples
Find the volume of a cylinder with radius 3 cm and height 10 cm.
The tank has a volume of 2 cubic metres, which is 2,000 litres.
Usage note
In everyday English 'volume' also means loudness ('turn the volume down') and one book in a series.
Common phrases
volume of a cylinder · find the volume · cubic units · surface area and volume
A straight line from one side of a circle or sphere to the other through its centre, or the length of this line, which is twice the radius.
Where you use it
Often given in real-life problems about pipes, wheels and pizzas, so you have to halve it to get the radius. The circumference of a circle is π times the diameter (πd).
Examples
The wheel has a diameter of 70 cm, so its radius is 35 cm.
Be careful: the question gives the diameter, not the radius.
Usage note
The stress is on the second syllable: 'dye-AM-it-er'.
Common phrases
diameter of a circle · in diameter · a diameter of · diameter and radius
Exactly the same shape and size, so that one figure fits perfectly on top of the other, possibly after being turned or flipped over.
Where you use it
Triangles are proved congruent with conditions such as SSS, SAS, ASA and RHS. The symbol is ≅: △ABC ≅ △DEF. In number theory, 'a is congruent to b modulo n' means a and b leave the same remainder when divided by n.
Examples
Show that triangles ABC and DEF are congruent.
Since the triangles are congruent, the corresponding sides are equal.
Usage note
Congruent shapes are also the same size; similar shapes only have the same shape.
Common phrases
congruent triangles · congruent to · prove congruent · congruent modulo n
Having exactly the same shape but not necessarily the same size, so that corresponding angles are equal and corresponding sides are in the same ratio.
Where you use it
Similar figures are linked by a scale factor. If the scale factor is k, lengths are multiplied by k, areas by k² and volumes by k³. The symbol is ~, as in △ABC ~ △DEF.
Examples
The two triangles are similar, so the ratio of their sides is 2 to 5.
Similar triangles let you find the height of a tree from the length of its shadow.
Usage note
In everyday English 'similar' just means 'alike'; in geometry it has this exact technical meaning.
Common phrases
similar triangles · similar to · similar figures · scale factor
In a right triangle, the ratio of the side opposite an angle to the hypotenuse; it is written sin, so sin 30° = 0.5.
Where you use it
One of the three basic trigonometric ratios, remembered with SOH CAH TOA (sine = opposite over hypotenuse). It is also a function: the sine curve repeats every 360°, or 2π radians. 'sin x' is read 'sine x'.
Examples
Use the sine of 40° to find how high the ladder reaches.
The sine rule works for any triangle, not only right triangles.
Usage note
'Sine' is pronounced exactly like 'sign' /saɪn/, not 'see-neh'.
Common phrases
sine of an angle · sine rule · sine curve · inverse sine
The longest side of a right triangle, which is the side opposite the right angle.
Where you use it
Appears in the Pythagorean theorem and in trigonometry, where the other two sides are named 'opposite' and 'adjacent' in relation to a given angle. You find, calculate or label the hypotenuse.
Examples
The two shorter sides are 5 and 12, so the hypotenuse is 13.
Label the hypotenuse first, then the opposite and adjacent sides.
Usage note
Pronounced 'hy-POT-en-yooz' in British English and 'hy-POT-en-oos' in American English, with the stress on 'pot'.
Common phrases
length of the hypotenuse · find the hypotenuse · opposite over hypotenuse · adjacent over hypotenuse
In a right triangle, the ratio of the side next to an angle (the adjacent side) to the hypotenuse; it is written cos, so cos 60° = 0.5.
Where you use it
The 'CAH' in SOH CAH TOA: cosine = adjacent over hypotenuse. The cosine rule, c² = a² + b² − 2ab cos C, works in any triangle. 'cos x' is read 'coz x' or 'cosine x'.
Examples
Use the cosine rule to find the third side of the triangle.
The cosine of 90 degrees is zero.
Usage note
Pronounced 'KOH-sine', with the second part said like 'sign'; the short form 'cos' sounds like 'coz'.
Common phrases
cosine rule · cosine of an angle · inverse cosine · cosine graph
In a right triangle, the ratio of the opposite side to the adjacent side, written tan; also a straight line that touches a curve at a point without cutting across it there.
Where you use it
Two meanings: in trigonometry, tan x = sin x / cos x (the 'TOA' in SOH CAH TOA); in geometry and calculus, 'the tangent to the curve at a point' has the same slope as the curve there.
Examples
The tangent of 45 degrees is exactly 1.
Find the equation of the tangent to y = x² at the point (1, 1).
Usage note
'Go off on a tangent' is an everyday idiom meaning to suddenly start talking about something unrelated.
Common phrases
tangent to the curve · tangent line · tangent of an angle · go off on a tangent
A unit for measuring angles: the angle at the centre of a circle made by an arc as long as the radius. A full turn is 2π radians, or 360°.
Where you use it
Standard in calculus and higher maths, where formulas such as the derivative of sin x assume the angle is in radians. You convert between degrees and radians using π radians = 180°. Remember to switch your calculator to radian mode (RAD) when needed.
Examples
Convert 60 degrees to radians: the answer is π/3.
My answer was wrong because the calculator was set to degrees instead of radians.
Usage note
One radian is about 57.3 degrees.
Common phrases
in radians · convert to radians · radian measure · radian mode
A function that gives the rate of change of another function at each point, which is the slope of its graph there; the derivative of x² is 2x.
Where you use it
You find, take or calculate the derivative and use it to find tangents, maximum and minimum points or velocity. It is written f′(x), read 'f prime of x', or dy/dx, read 'dee y by dee x' (UK) or 'dee y dee x' (US).
Examples
Find the derivative of f(x) = 3x² + 2x and evaluate it at x = 1.
Set the derivative equal to zero to find the maximum and minimum points.
Usage note
In finance a 'derivative' is a type of financial contract; in maths it always refers to a rate of change.
Common phrases
take the derivative · first derivative · second derivative · partial derivative · derivative with respect to x
The result of integration: either a family of functions whose derivative is the given function, or a number giving the area under a curve between two limits.
Where you use it
An indefinite integral, such as ∫2x dx = x² + C, includes a constant of integration C; a definite integral has lower and upper limits and gives a number. ∫f(x) dx is read 'the integral of f of x dx'.
Examples
Evaluate the definite integral of x² from 0 to 3.
Don't forget the plus C at the end of an indefinite integral.
Usage note
As an adjective, 'integral' also means 'essential' in everyday English: 'an integral part of the course'.
Common phrases
definite integral · indefinite integral · evaluate the integral · constant of integration
The value that a function or sequence gets closer and closer to as the input approaches a certain number or grows without bound.
Where you use it
The basic idea behind derivatives and integrals. 'lim x→0 sin x / x = 1' is read 'the limit as x approaches zero of sine x over x equals one'. A limit may exist, fail to exist or be infinite. Definite integrals also have upper and lower limits.
Examples
Find the limit as x approaches 2 of (x² − 4)/(x − 2).
The limit doesn't exist because the left-hand and right-hand limits are different.
Usage note
Formal writing says 'as x approaches 2' or, in British English, 'as x tends to 2'; 'as x goes to 2' is more informal.
Common phrases
the limit as x approaches · find the limit · left-hand limit · the limit does not exist · take the limit
To find the derivative of a function, that is, a formula for its rate of change; if you differentiate x³, you get 3x².
Where you use it
You differentiate a function 'with respect to' a variable: 'differentiate y with respect to x'. The process is called 'differentiation', and its main rules are the power rule, the product rule, the quotient rule and the chain rule.
Examples
Differentiate f(x) = 5x⁴ − 2x with respect to x.
If you differentiate position with respect to time, you get velocity.
Usage note
In everyday English 'differentiate' means to see or show the difference between things: 'differentiate between two ideas'.
Common phrases
differentiate with respect to x · differentiate twice · implicit differentiation · differentiate a function
To find the integral of a function: either a function whose derivative is the given one, or the area under its graph between two limits.
Where you use it
You integrate a function 'with respect to' a variable, often 'between the limits 0 and 1' or 'from 0 to 1'. The process is called 'integration', and common methods include substitution and integration by parts.
Examples
Integrate 6x² with respect to x and add the constant.
To find the area, integrate the function from 1 to 4.
Usage note
In everyday English 'integrate' means to combine things or to become part of a group: 'integrate into a new school'.
Common phrases
integrate with respect to x · integrate from a to b · integration by parts · integrate by substitution
The rule for differentiating a function that is inside another function: the derivative of f(g(x)) is f′(g(x)) × g′(x).
Where you use it
Used for composite functions such as (3x + 1)⁵ or sin(x²). Teachers put it as 'differentiate the outside, keep the inside, then multiply by the derivative of the inside'. In Leibniz notation it is dy/dx = dy/du × du/dx.
Examples
Use the chain rule to differentiate (3x + 1)⁵; the answer is 15(3x + 1)⁴.
You forgot the chain rule: the derivative of sin(2x) is 2cos(2x), not cos(2x).
Common phrases
apply the chain rule · use the chain rule · by the chain rule · chain rule and product rule
Having no breaks, jumps or holes; a function is continuous at a point if its limit there equals its value, so its graph can be drawn without lifting the pen.
Where you use it
You say a function is 'continuous at x = a' or 'continuous on an interval'. The opposite is 'discontinuous', and the point where it happens is a 'discontinuity'. Every differentiable function is continuous, but not every continuous function is differentiable.
Examples
Show that the function is continuous at x = 2.
The function |x| is continuous everywhere, but it has no derivative at zero.
Usage note
In statistics, 'continuous data' can take any value in a range (like height), unlike 'discrete data' (like the number of children).
Common phrases
continuous at a point · continuous function · continuous on an interval · piecewise continuous
A well-defined collection of distinct objects, called elements, usually written inside curly brackets, such as {2, 4, 6}.
Where you use it
The basic language of modern mathematics. Common phrases are 'the set of real numbers', 'the empty set' (∅), 'the solution set' and 'set notation'. Sets are named with capital letters, and ∈ is read 'is an element of'.
Examples
Let A be the set of even numbers less than 10.
Write the solution set of the inequality in interval notation.
Usage note
'Set' is also a common verb in maths: 'set the derivative equal to zero', 'set up an equation'.
Common phrases
the empty set · solution set · set notation · the set of real numbers · finite set
A set whose elements all belong to another set; {1, 2} is a subset of {1, 2, 3}, written {1, 2} ⊆ {1, 2, 3}.
Where you use it
A ⊆ B is read 'A is a subset of B'. A 'proper subset' is one that is not equal to the whole set. The empty set is a subset of every set, and a set with n elements has 2ⁿ subsets.
Examples
List every subset of {a, b, c}; there should be eight.
The integers are a subset of the rational numbers.
Common phrases
subset of · proper subset · number of subsets · is a subset of
One of the objects that belong to a set; 3 is an element of {1, 3, 5}, written 3 ∈ {1, 3, 5}.
Where you use it
x ∈ A is read 'x is an element of A', 'x belongs to A' or simply 'x is in A'; x ∉ A means 'x is not an element of A'. 'Member' is used with the same meaning, and the numbers in a matrix are also called its elements or entries.
Examples
How many elements does the set {2, 4, 6, 8} have?
Let x be an element of A ∩ B.
Usage note
In chemistry an 'element' is a substance such as oxygen; in maths it always means a member of a set.
Common phrases
element of a set · number of elements · is an element of · belongs to
The set of all elements that are in at least one of two or more sets; the union of {1, 2} and {2, 3} is {1, 2, 3}.
Where you use it
Written A ∪ B and read 'A union B'. On a Venn diagram it is the whole area covered by both circles. In logic and probability it matches the word 'or': P(A ∪ B) is the probability of A or B.
Examples
Shade the union of A and B on the Venn diagram.
The union of the even and odd integers is the set of all integers.
Common phrases
union of A and B · A union B · union and intersection · disjoint union
The set of elements that belong to all of the given sets at the same time; the intersection of {1, 2, 3} and {2, 3, 4} is {2, 3}.
Where you use it
Written A ∩ B and read 'A intersect B' or 'the intersection of A and B'. It matches the word 'and' in logic. In geometry, the 'point of intersection' is where two lines or curves meet.
Examples
If the intersection of A and B is empty, the sets are disjoint.
Find the point of intersection of the two lines.
Common phrases
intersection of A and B · point of intersection · A intersect B · empty intersection
A phrase joining two statements that are either both true or both false, so that each one implies the other; written ⇔ or 'iff'.
Where you use it
Common in definitions and theorems: 'n is even if and only if n² is even'. To prove such a statement you prove both directions. In writing it is often shortened to 'iff', but it is still read aloud as 'if and only if'.
Examples
A triangle is equilateral if and only if all its angles are 60°.
To prove an if and only if statement, prove each direction separately.
Usage note
'p if and only if q' means the same as 'p is a necessary and sufficient condition for q'.
Common phrases
if and only if · iff · necessary and sufficient · prove both directions
The statement you get by swapping the 'if' part and the 'then' part of a conditional; the converse of 'if p, then q' is 'if q, then p'.
Where you use it
A theorem can be true while its converse is false, so the converse must be proved separately. Don't confuse it with the 'inverse' (if not p, then not q) or the 'contrapositive' (if not q, then not p), which is always equivalent to the original.
Examples
The converse of the Pythagorean theorem is also true.
'If a number is divisible by 4, it is even' is true, but its converse is false.
Usage note
In everyday English 'conversely' means 'in the opposite way': 'Conversely, if the price falls, demand rises.'
Common phrases
the converse of · the converse is false · prove the converse · converse and contrapositive
A logical argument that shows, step by step from accepted facts, that a mathematical statement is always true.
Where you use it
You write, give, check or complete a proof. Common methods are direct proof, proof by contradiction and proof by induction. Questions often say 'prove that…' or 'show that…', and the end of a proof is marked with QED or a small square.
Examples
The proof is short, but each step needs a clear reason.
Can you give a proof that the square root of 2 is irrational?
Usage note
'Proof' is the noun and 'prove' is the verb: 'I proved it' and 'I wrote a proof', never 'I proofed it'.
Common phrases
write a proof · direct proof · complete the proof · rigorous proof
An important mathematical statement that has been proved to be true, such as the Pythagorean theorem.
Where you use it
You state, prove and apply a theorem. In textbooks and papers theorems are numbered (Theorem 3.2), often with a lemma before them and a corollary after them. A statement that is believed but not yet proved is a conjecture.
Examples
By the theorem we proved last week, the two angles must be equal.
State the theorem carefully before you apply it.
Usage note
Pronounced 'THEER-um' in British English and 'THEE-uh-rum' or 'THEER-um' in American English; the 'th' is the sound in 'think'.
Common phrases
prove a theorem · state a theorem · apply a theorem · by the theorem · fundamental theorem
A fixed phrase used to introduce a variable or an assumption at the start of a solution or proof, as in 'Let x be the number of tickets sold'.
Where you use it
Found in every textbook and proof. Common forms are 'Let x be a real number', 'Let n be an even integer' and 'Let ABC be a triangle'. Similar openings include 'Let x = …', 'Suppose that…' and 'Assume that…'.
Examples
Let x be the width of the rectangle; then its length is x + 4.
Let n be any odd integer, so n = 2k + 1 for some integer k.
Usage note
Turkish puts 'olsun' at the end, but English starts with 'Let': 'Let x be an integer'. Use the base form 'be', not 'is'.
Common phrases
let x be · let n be an integer · let ABC be a triangle · let x equal
A single example that shows a general statement is false; for the claim 'all prime numbers are odd', the number 2 is a counterexample.
Where you use it
You find, give or construct a counterexample. Questions often say 'prove or disprove' or 'give a counterexample to show the statement is false'. One counterexample is enough to disprove a claim, but no number of examples can prove one.
Examples
Give a counterexample to the claim that x² > x for every real number x.
x = 0.5 is a counterexample, because 0.25 is less than 0.5.
Common phrases
give a counterexample · find a counterexample · counterexample to the claim · prove or disprove
A smaller statement, proved on its own, that is used as a step towards proving a larger theorem.
Where you use it
Common in university lecture notes and research papers: 'We first prove a lemma', 'By Lemma 2.1, …'. A corollary, by contrast, is a result that follows easily from a theorem. Some famous lemmas, such as Zorn's lemma, are major results in their own right.
Examples
The proof becomes much shorter if we first prove a lemma about even numbers.
By Lemma 3, the sequence is bounded.
Usage note
The plural is 'lemmas'; the older form 'lemmata' is sometimes seen in formal writing.
Common phrases
prove a lemma · by the lemma · lemma and corollary · technical lemma
A method of proof in which you assume the statement is false and show that this leads to something impossible, so the statement must be true.
Where you use it
Typical openings are 'Suppose, for contradiction, that…' and 'Assume the opposite'. The argument ends with 'This is a contradiction' or a symbol such as ↯. Classic examples prove that √2 is irrational and that there are infinitely many primes.
Examples
We use proof by contradiction: suppose √2 = p/q in lowest terms.
Proof by contradiction is often the easiest way to show that something does not exist.
Usage note
It is also known by its Latin name, 'reductio ad absurdum'.
Common phrases
use proof by contradiction · suppose for contradiction · reach a contradiction · assume the opposite
A method of proving a statement for all positive integers: show it is true for the first case, then show that if it holds for n, it also holds for n + 1.
Where you use it
The two parts are the 'base case' and the 'inductive step', in which you use the 'inductive hypothesis'. Exams often use it for formulas such as 1 + 2 + … + n = n(n + 1)/2. It is also called 'mathematical induction'.
Examples
Use proof by induction to show that 1 + 3 + 5 + … + (2n − 1) = n².
In a proof by induction, don't forget to check the base case.
Usage note
Mathematical induction is a rigorous proof method, unlike 'inductive reasoning' in science, which generalises from examples.
Common phrases
proof by induction · base case · inductive step · inductive hypothesis · mathematical induction
Short for the Latin 'quod erat demonstrandum', meaning 'which was to be shown'; written at the end of a proof to show that it is complete.
Where you use it
Read letter by letter: 'Q-E-D'. In modern textbooks and papers it is often replaced by a small square (□ or ∎) at the end of the proof. Students sometimes also say 'QED' as a joke when they win an argument.
Examples
Therefore n² is even, which is what we wanted to show. QED.
Our lecturer writes QED at the end of every proof instead of the little square.
Common phrases
Q.E.D. · end of proof · tombstone symbol · write QED
A phrase in proofs meaning that only one case is treated because the other cases work in exactly the same way, as in 'assume a ≤ b'.
Where you use it
Usually shortened to WLOG in lecture notes but read aloud in full. A typical sentence is 'Without loss of generality, assume x ≥ y'. It should only be used when the cases really are symmetric.
Examples
Without loss of generality, let a be the largest of the three numbers.
You can't say 'without loss of generality' here, because the two cases aren't symmetric.
Common phrases
without loss of generality, assume · WLOG · by symmetry · without loss of generality, let
A linking phrase used in proofs and solutions to show that the next statement is a logical result of what has just been shown.
Where you use it
A more formal alternative to 'so'. Similar linking words in mathematical writing are 'hence', 'therefore', 'thus' and 'consequently'. Use 'it follows from A that B' when you want to name the reason.
Examples
Since x² = 9 and x is positive, it follows that x = 3.
It follows from Lemma 2 that the function is increasing.
Common phrases
it follows that · it follows from · it then follows that · hence · therefore
A maths question written as a short story or real-life situation, which you must turn into numbers or equations before solving it.
Where you use it
Common in school maths, entrance exams and standardised tests. Teachers advise you to read carefully, mark what is given and what is asked, define a variable and form an equation. Also called a 'story problem' (US) or a 'worded problem' (UK).
Examples
Most students lost points on the word problems, not on the algebra itself.
In this word problem, the key information is in the last sentence.
Common phrases
solve a word problem · age word problems · set up a word problem · multi-step word problem
A single value that represents a group of numbers, usually their total divided by how many there are; the average of 4, 6 and 11 is 7.
Where you use it
Very common in word problems: average speed, average score, average age. In statistics the more precise word is 'mean'; the 'median' (middle value) and the 'mode' (most frequent value) are other kinds of average.
Examples
The average of five test scores is 72. What is their total?
Average speed is total distance divided by total time, not the average of the two speeds.
Usage note
The fixed phrase is 'on average': 'On average, students spend two hours on homework', not 'in average'.
Common phrases
on average · average speed · the average of · find the average · above average
A comparison of two or more quantities that shows how big one is compared with the other, written like 3:2 and read 'three to two'.
Where you use it
Very common in word problems: 'the ratio of boys to girls', 'share £60 in the ratio 2:3', 'simplify the ratio'. Ratios can also be written as fractions, and an equation stating that two ratios are equal is called a proportion.
Examples
The ratio of boys to girls in the class is 3 to 4.
Mix the paint in the ratio 2:1, two parts white to one part blue.
Usage note
The order matters: 'the ratio of boys to girls' (3:4) is not the same as 'the ratio of girls to boys' (4:3).
Common phrases
the ratio of a to b · in the ratio · simplify a ratio · ratio and proportion
Changing together with another quantity so that their ratio stays fixed; if y is proportional to x, doubling x also doubles y.
Where you use it
Common phrases are 'directly proportional to' (y = kx) and 'inversely proportional to' (y = k/x), where k is the constant of proportionality. The symbol ∝ is read 'is proportional to'.
Examples
The cost is directly proportional to the number of kilograms you buy.
For a fixed distance, the time taken is inversely proportional to the speed.
Common phrases
directly proportional to · inversely proportional to · constant of proportionality · proportional to
Not less than a given number, so the true value may be equal to it or larger; 'at least 5' means 5 or more, written ≥ 5.
Where you use it
A key phrase in word problems and probability, together with its opposite 'at most' (≤). Turning it into the right inequality is often the hardest part of a question, so check whether the boundary value is included.
Examples
Each team must have at least four players.
What is the probability of getting at least one six in three rolls?
Usage note
'More than 5' (> 5) does not include 5, but 'at least 5' (≥ 5) does.
Common phrases
at least one · at least twice · at least and at most · at least as large as
In the same order as the things just mentioned; 'a and b are 3 and 5, respectively' means that a is 3 and b is 5.
Where you use it
Very common in exam questions, textbooks and papers to match two lists in one sentence. It usually comes at the end of the clause, after a comma. If you miss it, you may swap the values in a problem.
Examples
Tom and Ana are 12 and 15 years old, respectively.
The sides AB and BC have lengths of 6 cm and 8 cm, respectively.
Usage note
Don't confuse 'respectively' (in that order) with 'respectfully' (politely, with respect).
Common phrases
3 and 5, respectively · x and y, respectively · the first and second, respectively
To find an approximate value quickly, usually by rounding the numbers first, so that you can check whether an exact answer is reasonable.
Where you use it
Questions say 'estimate the answer', 'give an estimate' or 'estimate by rounding each number to one significant figure'. The symbol ≈ means 'is approximately equal to'. Estimating is a useful check against calculator mistakes.
Examples
Estimate 49 × 21 first: it should be about 1,000.
Your estimate was 300, but the calculator says 3,000, so check where the decimal point is.
Usage note
The verb is pronounced 'ES-ti-mate' /ˈestɪmeɪt/, but the noun is 'ES-ti-mut' /ˈestɪmət/.
Common phrases
estimate the answer · rough estimate · give an estimate · estimate by rounding
Following one after another in order with no gaps; 7, 8 and 9 are consecutive integers, and 4, 6 and 8 are consecutive even numbers.
Where you use it
A favourite of word problems: 'the sum of three consecutive integers is 72'. Write them as n, n + 1, n + 2, or as n, n + 2, n + 4 for consecutive even or odd numbers. It is also used in everyday English: 'for the third consecutive year'.
Examples
The sum of three consecutive integers is 72. Find the numbers.
The product of two consecutive integers is always even.
Usage note
The stress is on the second syllable: 'kun-SEK-yuh-tiv'.
A ratio that compares two quantities measured in different units, such as kilometres per hour or litres per minute.
Where you use it
Used in speed, work and filling problems: 'at a rate of 60 km per hour', 'pipe A fills the tank at a rate of 5 litres per minute'. In calculus, the 'rate of change' is what the derivative measures. Other uses include interest rate and exchange rate.
Examples
A tap fills the tank at a rate of 12 litres per minute. How long does it take to fill 300 litres?
Working together, they paint at a combined rate of one room every three hours.
Usage note
Turkish 'oran' can mean both 'ratio' and 'rate'; in English, a rate usually contains 'per' and compares two different units.
Common phrases
at a rate of · rate of change · constant rate · rate per hour · interest rate
To calculate an answer or find the solution to a problem by thinking about it carefully; especially common in British classrooms and exam papers.
Where you use it
Exam instructions say 'Work out the value of x' or 'Work out the area'. In speech: 'Can you work it out?', 'I can't work out where I went wrong'. American English often uses 'figure out' for this. A 'worked example' is an example with a full solution.
Examples
Work out the total cost of 15 tickets at £8.50 each.
I worked out the answer, but I made a mistake in the last step.
Usage note
'Work out' also means to exercise at the gym and, for plans, to end well: 'Things worked out well.'
Common phrases
work out the answer · work out the value of · work it out · worked example
Ten questions from this glossary. Pick the right meaning for each term and see your score.
How to learn mathematics vocabulary
Read real texts from the field regularly: English textbooks, past exam papers with worked solutions, lecture notes and problem-solving forums. Notice how each step of a solution is put into words.
Practise reading expressions aloud: x² is 'x squared', f(x) is 'f of x', 3.5 is 'three point five' and ½ is 'one half'. Saying each step while you solve a problem makes the words automatic.
Learn the operation words in sets of four: add, subtract, multiply, divide; plus, minus, times, divided by; sum, difference, product, quotient.
Keep a list of the fixed phrases of solutions and proofs, such as 'let x be', 'suppose that', 'it follows that', 'hence' and 'therefore', and reuse them in your own written answers.
Watch the number format: English writes 3.5 with a decimal point and 1,000 with a comma for thousands, the opposite of Turkish.
Frequently asked questions
How can I learn English vocabulary for mathematics?
Read real texts from the field and pay attention to how the words are used: English textbooks, worked solutions, lecture notes and online course materials. Learn each term with the verbs and phrases around it (solve an equation, find the derivative, draw a graph) and try to explain your own solutions in English, first aloud and then in writing.
How do I read mathematical expressions aloud in English?
Most symbols have a standard spoken form: 2 + 3 = 5 is 'two plus three equals five', x² is 'x squared', x³ is 'x cubed', 2⁵ is 'two to the power of five' or 'two to the fifth', √9 is 'the square root of nine', f(x) is 'f of x' and a/b is 'a over b'. With brackets, speakers pause or say 'all' to show grouping: (a + b)² is 'a plus b, all squared'.
Why do English and Turkish write decimal numbers differently?
English-speaking countries use a point to separate the whole part from the decimal part (3.75, read 'three point seven five') and a comma to group thousands (12,500). Turkish uses the opposite: 3,75 and 12.500. When you read or write in English, switch to the English format, otherwise 1.500 may be understood as one and a half instead of one thousand five hundred.
Is it 'math' or 'maths'?
Both are short for 'mathematics'. 'Math' is American English and 'maths' is British English; both take a singular verb ('Maths is my favourite subject', 'Math is hard'). Choose one and use it consistently, usually the one that matches the spelling you use (color or colour).
Which mathematics terms should I learn first?
Start with the words you meet in every lesson: equation, variable, function, sum, product, fraction, solve, simplify and the ways to read symbols aloud. Then move to the core words of each area: geometry (angle, area, radius), calculus (derivative, integral, limit), and the phrases of proofs and word problems. This glossary follows the same order, from the most essential terms to the more specialist ones.
