Mathematical notations

Mathematical notation is a collection of several abstract writing systems used in Mathematics and in other scientific and technical contexts. Mathematical notation, in its purest form, is used to express relationships between different mathematical objects. In a technical application, the same symbols and general ideas may be borrowed to describe the theoretical values of a physical quantity. In this manner, mathematical notation serves as a common language for expressing ideas and results across all scientific disciplines, regardless of spoken language.

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.

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

Similarly, the lower part of the complex halfplane can be defined as
 * $$ \{ z \in \mathbb{C} : \Im(z)\le 0 \} $$.

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
 * $$\exist$$ there exists

Numbers
Basic numbers are 0 and 1. Operation ++ is used to construct the set of natural numbers; it is postulated, that for any natural number \(x\), operation ++ is defined. The result is denoted as \(x++\) and it presumed to be also natural number. With respect to these numbers, the arithmetic operations are defined. They are denoted with
 * $$++$$ (which has ony one argument)
 * $$\times$$
 * $$\exp$$
 * $\mathrm{tet}$
 * $\mathrm{tet}$

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

and so on.

Keywords
Mathematica Mathematics Physics Science Philosophy