# Math Notation

A collection of math syntax and notation. The pages demonstrate the usage of symbols, expressions, and operators to represent mathematical concepts. See the order of operations to see how mathematicians disambiguate how expressions are evaluated.

## Arithmetic

In mathematics the plus symbol represents addition between two numbers. Addition is the process of combining two things together.

In mathematics the minus symbol represents the subtraction operator. The expression on the right is subtracted from the expression on the left.

A dot between two numbers denotes multiplication in mathematics. The formal name for the dot symbol is an interpunct.

In mathematics there are multiple different ways to represent division. The obelus is often used in elementary mathematics, then later the horizontal bar is adopted for convenience, and the forward slash is used because of the prevelence of computers and calculators.

In mathematics parentheses are used to group expressions to be evaluated together and organize the order of operations. Everything within the parantheses is evaluated, then the rest of the calculation is carried out.

Two expressions placed next to each other and separated by parenthesis are assumed to be multiplied together.

In mathematics when two variables are next to each other the implied operation is multiplication. This makes many equations and formulas more simple.

A fraction is represented using a horizontal bar between two expressions. The expression on top is called the numerator and the expression on bottom is called the denominator.

The decimal point is used to separate the whole part of a number from the decimal part. The whole part is represented using number digits to the left of the decimal point and the decimal part is represented using digits to the right of the decimal point.

The syntax for a ratio is two numbers separated by the colon symbol.

## Algebra

A function is written as the function's name followed by parentheses containing the input to the function. If the function has multiple inputs they are separated by a comma.

The notation for taking the absolute value is two vertical lines on either side of the expression being evaluated.

The exclamation mark is used to represent the factorial of a number in mathematics. A factorial is a unary operator.

In mathematics superscript text is used to indicate the exponent of some number.

Logarithms are often abreviated as "log" followed by a subscript number representing the base and the number that the logarithm is being applied to.

A radical is used to represent fractional exponents. By itself it is used to represent the square root of an expression, but it also is used to represent higher roots as well.

The radical symbol by itself is used to denote taking the square root of a number.

The summation operator is represented by the ∑ (capital sigma) and represents the operation of summing a sequence of expressions together.

The capital Greek letter Pi is used to represent the product operator in mathematics. The operator has three parts: an initial value, an end value, and the expression being evaluated.

## Logic

Two stacked horizontal lines respresents the equals symbol in mathematics. The two expressions on either side are equal, or the same, when the equal sign is placed in between them.

The symbol > represents the logical expression that the left side is greater than the right side.

The symbol < represents the logic expression that the left side is less than the right side.

The symbol for a less than or equal to is ≤. It combines the less than symbol < and the equals symbol = together.

The symbol for not equal is a equal sign with a diagonal line through it.

The symbol for approximately equal is a squiggly equals sign. It is used to show that two numbers are roughly equal, but not exactly equal.

## Geometry

A point in the cartesian coordinate system is denoted by two numbers in parentheses separated by a comma. The first number represents the distance from origin in the x-direction and teh second number represents the distance from the origin in the y-direction.

A point in the polar coordinate system is denoted by two numbers in parentheses separated by a comma. The first number represents the radius r (distance from the origin) and an angle θ (the greek letter theta) relative to the origin.

The notation for a symbol is a small symbol written in text, sometimes followed by three letters that correspond to a figure.

When referring to a circle in a diagram the symbol ⊙ (odot) is used followed by the variable associated with the center point of the circle. For example, ⊙P would mean the circle whose center is located at the point P.

Complementary angles can visually be denoted as two angles who sum to a perpendicular or square angle.

Congruent angles are denoted with tick marks across the angle.

A line is written as two letter with a line overtop, where the letters represent the end points of the line.

Parallel lines are denoted by the parallel symbol placed between the notation of the two lines. A line is denoted by the start and end letter with a line over top.

A perpendicular angle is visually denoted by drawing a square at the vertex of the angle. The measured angle is equal to π/2 radians or 90°.

The symbol for two perpendicular lines is a horizontal line with another line drawn perpendicular to it.

Supplementary angles can visually be denoted as two angles who sum to 180 degrees or PI degrees.

A triangle is denoted using the triangle symbol followed by three letters that represent the points of the triangle.

## Trigonometry

The notation for a symbol is a small symbol written in text, sometimes followed by three letters that correspond to a figure.

A line is written as two letter with a line overtop, where the letters represent the end points of the line.

A perpendicular angle is visually denoted by drawing a square at the vertex of the angle. The measured angle is equal to π/2 radians or 90°.

The symbol for two perpendicular lines is a horizontal line with another line drawn perpendicular to it.

A triangle is denoted using the triangle symbol followed by three letters that represent the points of the triangle.

## Probability

In probability a random variable is denoted using capital latin letters usually X, Y, and Z or A, and B respectively.

A probability distribution is denoted like a function. Often a capital P is used for the function name, and the capital letter X is used for the argument to the function.

Conditional probability is denoted using a vertical bar between the two variables

A joint probability distribution is denoted like a function often using P as the function name. The capital letters X and Y are often used to represent the random variables of the distribution and are arguments to the function.

The number of possible ways to choose r combinations from n total items is denoted using two parentheses with the n value above the r value. A subscript p or c is used to denote whether it is a combination or permutation.

The number of possible ways to choose r permutations from n total items is denoted using two parentheses with the n value above the r value. A subscript p or c is used to denote whether it is a combination or permutation.

## Numbers

Euler's Number is represented in mathematics with the letter e and has some unique exponential properties.

The greek letter pi is a number in mathematics that describes the ratio a circle's circumference divided by its diamater.

The symbol tau is used in mathematics to represent the ratio of the circumeference of the circle divided by its radius.

Two dimensional space is denoted using the symbol for the set of real numbers followed by a superscript two.

Three dimensional space is denoted using the symbol for the set of real numbers followed by a superscript three.

A complex number is represented in two parts. The first is the real number part and the second is the imaginary number part.

The decimal point is used to separate the whole part of a number from the decimal part. The whole part is represented using number digits to the left of the decimal point and the decimal part is represented using digits to the right of the decimal point.

The letter i is used in mathematics to represent the square root of negative one.

## Set Theory

An empty set is respresented as a zero with a diagonal line through it, or as an empty pair of curly braces.

A set is written with curly braces containing the elements of the set separated by commas.

The syntax for "there exists" is a backwards captial E. It is often used in conjuction with a variable with certain properties.

The ∀ (for all) symbol is used in math to describe the meaning of one or more variables in a statement.

The element of symbol describes membership to a set. When reading an equation the symbol can be read as "in" or "belongs to".

A proper superset is denoted by the superset symbol which looks like a U rotated ninety degrees to the left.

A proper subset is denoted by the symbol ⊂

The colon symbol is used in math to denote a condition for a statement. The symbol can be read as "such that" in a math expression.

The superset operator in set theory is denoted using the superset symbol which looks like a U turned ninety degrees counter clockwise with a horizontal line underneath.

The subset operator is denoted using a U shapes symbol rotated ninety degrees to the righ with a horizontal line underneath.

The cap symbol is used in mathematics to represent the intersect operation for two sets.

The minus symbol is used in set theory to represent the difference operator for two sets. The operation removes all elements found in one set from another and returns the resulting set.

The cup symbol is used in mathematics to represent the union operation for two sets. The union operator returns a set containing the elements from both sets.

## Boolean Logic

The "and" operator is denoted with the carrot symbol, which looks like a equilateral triangle with no bottom side. The operator evaluates to true if both the left and right expressions are true, otherwise evaluates to false.

Two stacked horizontal lines respresents the equals symbol in mathematics. The two expressions on either side are equal, or the same, when the equal sign is placed in between them.

The logical or symbol describes when either one thing is true, the other thing is true, or both is true.

The equivalent operator is used to express that two expressions are equivalent, although not necassarily equal.

The boolean operator for implication is denoted using a double arrow pointing to the right.

The negation symbol is used to reperesent the unary operator for negation, which inverts the value of the expression it is applied to.

Xor is the boolean operator that describes the operation of exclusive or.

## Linear Algebra

A variable that represents a vector often has an arrow over the top to indicate that it is a vector.

To vertical bars on either side of a vector represent taking the magnitude or length of that vector

In linear algebra the cross product operator is the times operator that is used in elementary mathematics.

In Linear Algebra, a m by n matrix is denoted as a grid of numbers with two brackets on either side. The variable m corresponds to the number of rows, and the variable n corresponds to the number of columns.

The syntax for a determinant is to vertical bars on either side of the matrix.

The syntax for a scalar product is to angle brackets on either side of two vectors separated by a comma.

## Calculus

The syntax for a limit is a the abbreviation "lim" followed an expression. Underneath the letters "lim" is the value the variable approaches within the expression denoted as the variable with an arrow to the value it is approaching.

The antiderivative of a function is denoted using the capital Latin letter F.

The first derivative of a function is denoted by a apostrophe after the function name. Alternatively the partial symbol can be used to represent the derivative with respect to a variable.

The integral operator is written using the integral symbol ∫ and has four parts: the expression being integrated, the differential, and start value and an end value.

The nabla symbol is used to represent taking the gradient of a function.

## Computing

When entering an equation into a calculator or the computer the forward slash is used to represent division.

When computing an expression which contains the exponent operator, the caret symbol is often used in place of what normally would be superscript text.

When entering an expression in a computer multiplication is represented using the asterisk symbol.

In computing subtraction is denoted using the hyphen symbol.