Mathematical notation

Mathematical notation is an abstract writing system used mainly among researchers, and mathematicians in particular. These notations are supposed to be already known, and usually the definitions of these notations are not repeated in scientific texts. For example, the letter i (or j, particularly in engineering fields) is reserved for the imaginary unit; the name exp is reserved for exponentiation, and the letter $\pi$ is reserved for the period of the exponentiation, divided by 2i. In similar way, the character $\mathbb {\emptyset }$ denotes the empty set (which has no elements at all), $\mathbb {N}$ denotes the set of natural numbers, $\mathbb {R}$ denotes the set of real numbers, $\mathbb {C}$ denotes the set of complex numbers, and so on.

Overview

The basic concepts of this slang include grouping, that allows to combine several objects in one. Usually, the grouping is denoted with parenthesis. Also, the parenthesis are used to indicate the argument of operations; especially, if some operations $A,B,C$ from some set (called group) can be applied sequentially, one by one, in raw, for example, $A{\Big (}B{\big (}C(z){\big )}{\Big )}$.

All things mathematicians deal with are called objects, and each object is supposed to belong to some set of objects, which is either already defined, or allows some independent definition. The possibility of such independent definition is especially important in order to exclude from the consideration such things as set of all possible sets which easy lead to paradoxes, at very beginning.

Mathematicians like to give names to all objects they deal with. Some ot these names are so established, that they are supposed to known a priori, for example, the equality, basic arithmetical operations, natural numbers, numbers e and $\pi$, etvetera. Such names form the basics of the mathematical notations.

Equvalence $=$

Equivalence is renoted with symbol $=$. this symbol means that in some specific consideration, mathematicians makes no need to distinguish objects that appear at the left hand side of this character from the object that appear in the right hand side. In the most of cases, the equivalence is used to compare numbers.

Exceptions

This rule has an exception. Sometimes, one may say "the integrand in the right hand side of the equation (3,14) becomes zero at..." However, in this case, one means not the mathematical object, but the way it is expressed in specific formula, typed in a book or in a paper.

True and False

In the simplest case, the mathematical expression may have only one of two values, true or false. These are basic mathematical expressions and basic mathematical notations. By default, are expressions in a scientific text are supposed to have value true. In some programming languages, at initialization, all the logical variables are set to false. For variables that may have only one of these two values, operations of boolean algebra are defined: and, or, not for these operations, the following notations are in use:

Sets

In order to express operations with sets, the following symbols are used:

• $=$ which means that the sets are [[equal
• $\subset$ which subset

One of ways to define the set is following.

One begins with a curly parenthesis $\{$, which is part of the group operation.

Then one writes some letter or character, which denotes an element of the set

Then one writes the character $\in$ and specifies some known (standard) set, indicating the kind of elements that can be considered; for example, will it be from a set of words, from a set of animals, numbers, etc.

Then one writes the character : , and lists, separating with commas, all the properties which are specific for all elements of namely this set.

Such a definition should finish with the closing of the grouping, id est, a closing curly parenthesis $\}$.

Examples

With mathematical notations above, the upper part of the complex halfplane can be defined in such a way:

$\{z\in \mathbb {C} :\Im (z)\geq 0\}$.

Similarly, the lower part of the complex halfplane can be defined as

$\{z\in \mathbb {C} :\Im (z)\leq 0\}$.

According to these definitons, real numbers belong to the upper part of the complex halfplane and also to the lower part of the complex halfplane.

Quantifiers

• $\forall$ for all
• $\exists$ there exists

Numbers

Basic numbers are 0 and 1. With respect to these numbers, the arithmetic operations are defined. They are denoted with

$++$ (which has ony one argument)

$+$

$*$

$\exp$

and so on. At least for integers, all next operation in this raw can be constructed as recurrence of operations from previous rows. Most of conventional calculus is based on the operations summation, multiplication, exponentiation and the inverse functions. Some of numbers have own single-character names: 0,1,2,3,4,5,6,7,8,9. For example,

$1=0++$

$2=1++$

and so on. Most of larger numbers have no single-character mames; the integer numbers are denoted using the positional numeral system. The inverse operations (if exist) of basic arithmetic operations are denoted, correspondingly, with symbols

$--$

$-$

$/$

$\log$ and $~^{*}{\sqrt {~}}$

and so on.