✖

Mathematical symbols

Mathematics is full of all sorts of symbols in which the ordinary mortal can easily get lost. This article attempts to cover and explain at least the basic symbols that you may encounter at every mathematical step.

List of symbols

  • +, −, · , /Basic symbols for common operations of addition, subtraction, multiplication and division. The asterisk is often used as a multiplication symbol: * or the common dot ".". It is difficult to say what is the correct notation - standards, typographic norms and conventions are not identical. Most often, if the software allows it, a central dot is used: · . For hyphenation, either the aforementioned slash or a colon : or a colon with a line in the middle: ÷. The slash is probably the most commonly used.

  • (), [], {}: three kinds of parentheses. They have different uses. Round brackets are used as parentheses to determine the priority of operators and to join some expressions. For example, if you were to write:

    $$ 1+2/3+4 $$

    you might not be sure what you mean. This is because it means one plus two-thirds, plus four. If you wanted to make it one big fraction, you would use round brackets:

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

    Round brackets are used, for example, as coordinates in some coordinate system. If you wanted to write a point with coordinates x = 10 and y = 12, you would write it as [10, 12].

  • x2 The superscript is used for powers. The preceding expression x2 can be written out as x · x. We call the superscript expression an exponent.

  • $\sqrt{}$The : sign for the square root. It denotes the inverse function to the exponent. For the square root:

    $$ \sqrt{x}\cdot\sqrt{x}=x $$

  • % is the sign for percent and is also used in mathematics to calculate percents. There's nothing more complicated than that.

  • |x|The : vertical lines denote the absolute value. The absolute value makes a negative number a positive number. So for example: |−5| = 5. Vertical lines can also indicate the distance between two points in geometry. If we have the points A and B, then we denote the distance between the points by |AB|.

  • !: the exclamation mark indicates the factorial. The factorial returns the product of all natural numbers that are less than or equal to the given number. The exclamation point 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, or "Pi", is a Ludolph number. It is most often used in goniometry and drawing, because Pi is used to calculate, for example, the diameter of a circle. The character π is a letter of the Greek alphabet. The approximate value is 3.1415... Because it is an irrational number, it cannot be quantified as a whole.

  • e is another important mathematical constant, the Euler number. The constant is named after Leonhard Euler, who was an important Swiss mathematician. Euler's number is most often used for logarithms - the natural logarithm has Euler's number as its base. Like Pi, Euler's number is an irrational number and cannot be quantified. An approximate value is 2,71...

  • D(f) and H(f) denotes the definitional range and the range of values for functions. The definitional domain is the set of elements that we can choose as arguments to a function. The value scope is the set of elements that a function value can take.

  • ${1 \choose 2}$ is a combinational number. It looks like a fraction without a fractional line, but with parentheses around it. The parentheses are normally there, we don't just write two numbers with nothing. The combinational number is used to write combinations shorter.

  • $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$ are standard sets that I work with in math. From the left, these are: natural numbers, integers, rational numbers, real numbers, and complex numbers.

  • ⊆, ∈, × are operators used in set theory. From left: subset, symbol for containing an element in a set, Cartesian product.

  • $f^{\prime}$ denotes (more precisely, the comma) the derivative of a function f.

  • $\wedge, \vee, \Rightarrow, \Leftrightarrow, \neg$ are symbols used in propositional logic. From left: conjunction, disjunction, implication, equivalence, and negation.

  • $\sum_{i=1}^n a_i$ is the symbol for sum and denotes the sum of some sequence ai. This is already a pretty scary symbol, but there's nothing too complicated about it. The sum starts with the value i = 1 and incrementally adds the values of the sequence ai and increments (increases) the value of the variable i by one until the value of i equals the superscript, n. This causes the sequence to be incrementally added:

    $$ \sum_{i=1}^n a_i = a_1+a_2+\ldots+a_n $$

    Suppose we have a sequence ai = i, that is, a sequence 1, 2, 3, … How would we write the sum of the first five members of the sequence?

    $$ \sum_{i=1}^5a_i=1+2+3+4+5=15 $$

  • $\prod_{i=1}^{n}a_i$ is a symbol called the product, and it is equivalent to the previous sum, only it is not addition that works as an inner operation, but multiplication. The product returns the product of the given members of the sequence (the sum returns the sum). So the product of the first five members of the previous sequence ai = i would look like this:

    $$ \prod_{i=1}^5 a_i = 1\cdot2\cdot3\cdot4\cdot5=120 $$