Mathematical symbols

Mathematics is full of all kinds of symbols in which an ordinary mortal can easily get lost. This article tries to cover and explain at least the basic symbols you may meet at every step in maths.

List of symbols

+, −, · , /: basic symbols for the everyday operations of addition, subtraction, multiplication and division. An asterisk * or an ordinary full stop “.” is also often used as the multiplication sign. It is hard to say which notation is right – standards, typographic norms and customs do not agree. If the software allows it, the most common choice is the centred dot: ·. For division you use either the slash mentioned above, or a colon : or a colon with a line in the middle: ÷. The slash is probably the most common.
(), [], {}: three kinds of brackets. They have different uses. Round brackets are used to set the priority of operators and to group expressions together. If you wrote, for example:

$1 + 2 / 3 + 4$

it might not be clear what you mean. As written, it means one plus two thirds, plus four. If you wanted one big fraction, you would use round brackets:

$(1 + 2) / (3 + 4)$

Square brackets are used, for example, to write intervals. To write a point with coordinates x = 10 and y = 12 you would use round brackets: (10, 12).
x² the superscript is used for powers. The previous expression x² can be written out as x · x. The expression in the superscript is called the exponent.
$\sqrt{}$ : the sign for the square root. It denotes the inverse function of squaring. For the square root we have:

$\sqrt{x} \cdot \sqrt{x} = x$
% is the sign for per cent and it is used in maths too, for calculating with percentages. There is nothing more complicated behind it.
|x|: vertical bars denote the absolute value. The absolute value turns a negative number into a positive one. So, for example: |−5| = 5. Vertical bars can also denote the distance between two points in geometry. If we have points A and B, we denote the distance between them |AB|.
!: the exclamation mark denotes the factorial. The factorial returns the product of all positive integers less than or equal to the given number. The exclamation mark is written after the expression whose factorial we want to calculate. Example: 5! = 5 · 4 · 3 · 2 · 1 = 120.
π is a typical mathematical constant. π, read “pi”, is used most often in trigonometry and in geometric constructions, because pi is used to calculate, for example, the circumference and area of a circle. The symbol π is a letter of the Greek alphabet. Its approximate value is 3.1415… Because it is an irrational number, it cannot be written out in full.
e is another important mathematical constant, Euler’s number. The constant is named after Leonhard Euler, a notable Swiss mathematician. Euler’s number is most often used with logarithms – the natural logarithm has Euler’s number as its base. Like pi, Euler’s number is an irrational number and cannot be written out in full either. Its approximate value is 2.71…
The domain and the range are two sets belonging to a function. The domain is the set of elements we can choose as the argument of the function. The range (image) is the set of elements that the function value can take.
$(\binom{1}{2})$ is a binomial coefficient. It looks like a fraction without the fraction line, but with brackets around it. The brackets are normally written there – we don’t write just two numbers with nothing around them. The binomial coefficient is used as a shorter way of writing combinations.
ℤ⁺, ℤ, ℚ, ℝ, ℂ are the standard sets used in mathematics. From the left they are: the positive integers, the integers, the rational numbers, the real numbers and the complex numbers.
⊆ , ∈ , × are operators used in the theory of sets. From the left: subset, the symbol for an element belonging to a set, the Cartesian product.
f′ denotes (more precisely, the prime mark does) the derivative of the function f.
∧ , ∨ , ⇒ , ⇔ , ¬ are symbols used in propositional logic. From the left: conjunction, disjunction, implication, equivalence and negation.
$\sum_{i = 1}^{n} a_{i}$ is the symbol for a sum and denotes the sum of some sequence a_i. This is quite a frightening symbol, but there is nothing very complicated about it. The sum starts with the value i = 1 and gradually adds the values of the sequence a_i and increments (increases) the value of the variable i by one, until the value of i equals the upper index, that is, n. So the sequence is gradually added up:

$\sum_{i = 1}^{n} a_{i} = a_{1} + a_{2} + \dots + a_{n}$

Say we have the sequence a_i = i, that is, the sequence 1, 2, 3, … How would we write the sum of the first five terms of the sequence?

$\sum_{i = 1}^{5} a_{i} = 1 + 2 + 3 + 4 + 5 = 15$
$\prod_{i = 1}^{n} a_{i}$ is the symbol called the product and it is the equivalent of the previous sum, except that the inner operation is not addition but multiplication. The product returns the product of the given terms of the sequence (the sum returns their sum). So the product of the first five terms of the previous sequence a_i = i would look like this:

$\prod_{i = 1}^{5} a_{i} = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120$

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