Mathematical symbols and terminology can be confusing and can be a barrier to learning and understanding basic numeracy.

This page complements our numeracy skills pages and provides a quick glossary of common mathematical symbols and terminology with concise definitions.

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## Common Mathematical Symbols

### + Addition, Plus, Positive

**The addition symbol + is usually used to indicate that two or more numbers should be added together, for example, 2 + 2.**

The + symbol can also be used to indicate a positive number although this is less common, for example, +2. Our page on Positive and Negative Numbers explains that a number without a sign is considered to be positive, so the plus is not usually necessary.

See our page on Addition for more.

### − Subtraction, Minus, Negative

**This symbol has two main uses in mathematics:**

- − is used when one or more numbers are to be subtracted, for example, 2 − 2.
- The − symbol is also commonly used to show a minus or negative number, such as −2.

See our page on Subtraction for more.

### × or * or . Multiplication

**These symbols have the same meaning; commonly × is used to mean multiplication when handwritten or used on a calculator 2 × 2, for example.**

The symbol * is used in spreadsheets and other computer applications to indicate a multiplication, although * does have other more complex meanings in mathematics.

Less commonly, multiplication may also be symbolised by a dot . or indeed by no symbol at all. For example, if you see a number written outside brackets with no operator (symbol or sign), then it should be multiplied by the contents of the brackets: 2(3+2) is the same as 2×(3+2).

See our page on Multiplication for more.

### ÷ or / Division

**These symbols are both used to mean division in mathematics. ÷ is used commonly in handwritten calculations and on calculators, for example, 2 ÷ 2.**

/ is used in spreadsheets and other computer applications.

See our page on Division for more.

### = Equals

**The = equals symbol is used to show that the values on either side of it are the same. It is most commonly used to show the result of a calculation, for example 2 + 2 = 4, or in equations, such as 2 + 3 = 10 − 5.**

You may also come across other related symbols, although these are less common:

**≠**means not equal. For example, 2 + 2**≠**5 - 2. In computer applications (like Excel) the symbols <> mean not equal.**≡**means identical to. This is similar to, but not exactly the same as, equals. Therefore, if in doubt, stick to =.**≈**means approximately equal to, or almost equal to. The two sides of a relationship indicated by this symbol will**not**be accurate enough to manipulate mathematically.

### < Less Than and > Greater Than

This symbol **<** means less than, for example 2 < 4 means that 2 is less than 4.

This symbol **>** means greater than, for example 4 > 2.

**≤ ≥** These symbols mean ‘less than or equal to’ and ‘greater than or equal to’ and are commonly used in algebra. In computer applications <= and >= are used.

**≪ ≫** These symbols are less common and mean much less than, or much greater than.

### ± Plus or Minus

**This symbol ± means ‘plus or minus’. It is used to indicate, for example, confidence intervals around a number.**

The answer is said to be ‘plus or minus’ another number, or in other words, within a range around the given answer.

For example, 5 ± 2 could in practice be any number from 3 to 7.

### ∑ Sum

**The ∑ symbol means sum.**

∑ is the Greek capital sigma character. It is used commonly in algebraic functions, and you may also notice it in Excel - the AutoSum button has a sigma as its icon.

### ° Degree

**Degrees ° are used in several different ways.**

**As a measure of rotation**- the angle between the sides of a shape or the rotation of a circle. A circle is 360° and a right angle is 90°. See our section on Geometry for more.**A measure of temperature.**Degrees Celsius or Centigrade are used in most of the world (with the exception of the USA). Water freezes at 0°C and boils at 100°C. In the USA Fahrenheit is used. On the Fahrenheit scale water freezes at 32°F and boils at 212°F. See our page: Systems of Measurement for more information.

### ∠ Angle

**The angle symbol ∠ is used as shorthand in geometry (the study of shapes) for describing an angle.**

The expression ∠ABC is used to describe the angle at point B (between points A and C). Similarly, ∠BAC would be used to describe the angle of point A (between points B and C). For more on angles and other geometric terms see our pages on Geometry.

### √ Square Root

**√ is the symbol for square root. A square root is the number that, when multiplied by itself, gives the original number.**

For example, the square root of 4 is 2, because 2 x 2 = 4. The square root of 9 is 3, because 3 x 3 = 9.

See our page: Special Numbers and Concepts for more on square roots.

^{n} Power

**A superscripted integer (any whole number n) is the symbol used for the power of a number.**

For example,3^{2}, means 3 to the power of 2, which is the same as 3 squared (3 x 3).

4^{3} means 4 to the power of 3 or 4 cubed, that is 4 × 4 × 4.

See our pages on Calculating Area and Calculating Volume for examples of when squared and cubed numbers are used.

**Powers are also used as a shorthand way to write large and small numbers.**

Large numbers

10^{6} is 1,000,000 (one million).

10^{9} is 1,000,000,000 (one billion).

10^{12} is 1,000,000,000,000 (one trillion).

10^{100} written long-hand would be 1 with 100 0's (one Googol).

Small numbers

10^{-3} is 0.001 (one thousandth)

10^{-6} is 0.000001 (one millionth)

Powers can also be written using the **^** symbol.

10^6 = 10^{6} = 1,000,000 (one million).

### . Decimal Point

. is the decimal point symbol, often referred to simply as ‘point’. See our page on ** Decimals** for examples of its use.

### , Thousands Separator

**A comma can be used to split large numbers and make them easier to read.**

A thousand can be written as 1,000 as well as 1000 and a million as 1,000,000 or 1000000. The comma splits larger numbers into blocks of three digits.

In most English speaking countries the , does not have any mathematical function, it is simply used to make numbers easier to read.

In some other countries, especially in Europe, the comma may be used instead of a decimal point and indeed, a decimal point may be used in place of a comma as a visual separator. This is explained in more detail on our Introduction to Numbers page.

### [ ], ( ) Brackets, Parentheses

**Brackets ( ) are used to determine the order of a calculation as dictated by the BODMAS rule.**

Parts of a calculation included within brackets are calculated first, for example

**5 + 3 × 2 = 11****(5 + 3) × 2 = 16**

### % Percentage

**The % symbol means percentage, or the number out of 100.**

Learn all about percentages on our page: Introduction to Percentages

### π Pi

π or Pi is the Greek character for the ‘p’ sound. It occurs frequently in mathematics and is a mathematical constant. Pi is a circle's circumference divided by its diameter and has the value 3.141592653. It is an irrational number, which means that its decimal places continue to infinity.

### ∞ Infinity

**The ∞ symbol signifies infinity, the concept that numbers go on for ever.**

However large a number you have, you can always have a larger one, because you can always add one to it.

Infinity is not a number, but the *idea* of numbers going on for ever. You cannot add one to infinity, any more than you can add one to a person, or to love or hate.

### \(\bar x\) (x-bar) Mean

**\(\bar x\) is the mean of all the possible values of x.**

You will mostly come across this symbol in statistics.

See our page on Averages for more information.

### ! Factorial

**! is the symbol for factorial.**

n! is the product (multiplication) of all the numbers from n down to 1, inclusive, i.e. n × (n−1) × (n−2) × … × 2 × 1.

For example:

5! = 5 × 4 × 3 × 2 × 1 = 120

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

### | Pipe

Pipe '|' is also also referred to as vertical bar, vbar, pike and has many uses in mathematics, physics and computing.

Most commonly in basic mathematics, it used to denote **absolute value** or **modulus** of a real number, where \(\vert x \vert\)is the *absolute value or modulus of \(x\)*.

Mathematically, this is defined as

$$\vert x \vert = \biggl\{\begin{eqnarray} -x, x \lt 0 \\ x, x \ge 0 \end{eqnarray}$$

Simply, \(\vert x \vert\) is the non-negative value of \(x\). For example, the modulus of 6 is 6 and the modulus of −6 is also 6.

It is also used in probability, where P(Z|Y)denotes the probability of X given Y.

### ∝ Proportional

**∝**** means ‘is proportional to**’, and is used to show something that varies in relation to something else.

For example, if x = 2y, then x ∝ y.

### ∴ Therefore

∴ is a useful shorthand form of ‘therefore’, used throughout maths and science.

### ∵ Because

∵ is a useful shorthand form of ‘because’, not to be confused with ‘therefore’.

## Mathematical Terminology (A-Z)

Amplitude

When an object or point moves in a cyclic pattern, or is subject to vibration or oscillation (e.g. a pendulum), the *amplitude* is the maximum distance it moves from its centre point. See an introduction to geometry for more.

Apothem

The line connecting the centre of a regular polygon with one of its sides. The line is perpendicular (at a right angle) to the side.

Area

Geometric area is defined as the space occupied by a flat shape or the surface of an object. Area is measured in square units, such as square metres (m^{2}). For more, see our page on area, surface area and volume.

Asymptote

An asymptote is a straight line or axis that is specifically related to a curved line. As the curved line extends (tends) to infinity, it approaches, but never touches, its asymptote (that is, the distance between the curve and the asymptote tends to zero). It occurs in geometry and trigonometry.

Axis

A line of reference about which an object, point or line is drawn, rotated or measured. In a symmetrical shape, an axis is usually a line of symmetry.

Coefficient

A coefficient is a number or quantity multiplying another quantity. It is usually placed before a *variable*. In the expression 6*x*, 6 is the coefficient and *x* is the variable.

Circumference

The circumference is the length of the distance around the edge of a circle. It is a type of *perimeter* that is unique to circular shapes. For more, see our page on curved shapes.

Data

Data are a collection of values, information or characteristics, which are often numerical in nature. They may be collected by scientific experiment or other observational means. They may be *quantitative* or *qualitative* variables. A datum is a single value of a single variable. See our page on Types of Data for more.

Diameter

Diameter is a term used in geometry to define a straight line that passes through the centre of a circle or sphere, touching the circumference or surface at both ends. The diameter is twice the *radius*.

Extrapolate

Extrapolate is a term used in data analysis. It refers to the extension of a graph, curve, or range of values into a range for which no data exists, inferring the values of unknown data from trends in the known data.

Factor

A factor is a number that we multiply by another number. A factor divides into another number a whole number of times. Most numbers have an even number of factors. A *square number* has an odd number of factors. A *prime number* has two factors – itself and 1. A *prime factor* is a factor that is a prime number. For example, the prime factors of 21 are 3 and 7 (because 3 × 7 = 21, and 3 and 7 are prime numbers).

Mean, Median and Mode

The *mean* (average) of a set of data is calculated by adding all numbers in the data set and then dividing by the number of values in the set. When the data set is ordered from least to greatest, the *median* is the middle value. The mode is the number that occurs most times.

Operation

A mathematical operation is step or stage in a calculation, or a mathematical ‘action’. The basic arithmetical operations are addition, subtraction, multiplication and division. The order in which operations are carried out in a calculation is important. The order of operations is known as BODMAS.

Mathematical operations are often referred to as ‘sums’. Strictly speaking, a ‘sum’ is an addition operation. At SYN we refer to operations and calculations, but in everyday language you may often hear the general term ‘sums’, which is incorrect.

Perimeter

The perimeter of a 2-dimensional shape is the continuous line (or the length of the line) that defines the outline of the shape. The perimeter of a circular shape is specifically called its *circumference*. Our page on Perimeter explains this in more detail.

Proportion

Proportion is a relative of ratio. Ratios compare one part to another part, and proportions compare one part to the whole. For example, ‘3 in every 10 adults in England are overweight’. Proportion is related to fractions.

Pythagoras

Pythagoras was a Greek philosopher, credited with a number of important mathematical and scientific discoveries, arguably the most significant of which has become known as Pythagoras’ Theorem.

It is an important rule that applies only to right-angled triangles. It says that ‘the square on the hypotenuse is equal to the sum of the squares on the other two sides.’

Quantitative and Qualitative

*Quantitative data* are numeric variables or values that can be expressed numerically, i.e. how much, how many, how often, and are obtained by counting or measurement.

*Qualitative data* are type variables that do not have a numerical value and may be expressed descriptively, i.e. by using a name or symbol, and are obtained by observation.

See our page on types of data for more.

Radian

The radian is the SI unit for angular measurement. One radian is equivalent to the angle subtended at the centre of a circle by an arc equal in length to the radius. One radian is just under 57.3 degrees. A full rotation (360 degrees) is 2π radians.

Radius

The term radius is used in the context of circles and other curved shapes. It is the distance between the centre point of a circle, sphere or arc, to its outer edge, surface or *circumference*. The *diameter* is twice the radius. For more, see our page on curved shapes.

Range

In statistics, the range of a given data set is the difference between the largest and smallest values.

Ratio

Ratio is a mathematical term used for comparing the size of one part to another part. Ratios are usually shown as two or more numbers separated with a colon, for example, 7:5, 1:8 or 5:2:1.

Standard Deviation

The standard deviation of a data set measures how far the data differ from the mean value, i.e. it is a measure of the variation or spread of a set of values. Where the spread of the data is low and all values are close to the mean, then the standard deviation will be low. A high standard deviation indicates that the data are spread out over a wider range

Term

A term is a single mathematical expression. It can be a single number, a single variable (e.g. *x*), or several constants and variables multiplied together (e.g. 3*x*2). Terms are usually separated by addition or subtraction operations. A term may include addition or subtraction operations, but only in brackets, e.g. 3(2 -x3).

Variable

A variable is a *factor* in a mathematical expression, arithmetic relationship or scientific experiment that is subject to change. An experiment usually has three kinds of variables: independent, dependent, and controlled. In the expression 6*x*, 6 is the *coefficient* and *x* is the variable.

Variance

Variance is a statistical measurement that indicates the spread *between members* in a data set. It measures how far each member in the set is from the mean and therefore from every other member in the set.

Vector

Vectors describe mathematical quantities that have both magnitude and direction. Vectors occur in many maths and physics applications, e.g. the study of motion, where velocity, acceleration, force, displacement and momentum are all vector quantities.

Volume

Volume is the three-dimensional space occupied by a solid or hollow shape. It is quantified by the cubic measurement of the space enclosed by its surfaces. Volume is measured in cubic units, e.g. m^{3}.

Further Reading from Skills You Need

**The Skills You Need Guide to Numeracy**

This four-part guide takes you through the basics of numeracy from arithmetic to algebra, with stops in between at fractions, decimals, geometry and statistics.

Whether you want to brush up on your basics, or help your children with their learning, this is the book for you.

Continue to:

Real World Maths

See also: Systems of Measurement

Ordering Mathematical Operations (BODMAS)

Graphs and Charts