Mathematical notation: Difference between revisions

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'''Mathematical notations''' is kind of [[prefessional slang]] used mainly among [[researcher]]s, especially,
{{subpages}}
among [[mathematician]]s, and especially, in the writing language. These notaitons are supposed to be already known, and usually the definition of these notationst are not repeated in scientific texts. For example, letter i is reserved for the [[imaginary unit]]; name  [[exp]]  is reserved for the [[exponentiation]], and letter <math>\pi</math> is reserved for the [[periodic function|period]] of the exponentiation, divided by 2i. In similar way, the character
'''Mathematical notation''' is a collection of several abstract [[writing system|writing systems]] used in [[Mathematics]] and in other [[Science|scientific]] and [[Technology|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.
<math>\mathbb{\emptyset}</math> denotes the [[empty set]] (which has no elements at all),
 
<math>\mathbb{N}</math> denotes the set of [[integer number]]s,
== Overview ==
<math>\mathbb{R}</math> denotes the set of [[real number]]s,
<math>\mathbb{C}</math> denotes the set of [[complex number]]s, and so on.
==Overview==
The basic concepts of this slang include [[grouping]], that allows to combine several objects in one.
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 incifate the argument of operations; especially, if some operations <math>A,B,C</math> from some set (called [[group]]) can be applied sequentially, one by one, in raw, for example,  <math>A\Big(B\big(C(z)\big)\Big)</math>.  
Usually, the grouping is denoted with [[parenthesis]]. Also, the parenthesis are used to indicate the argument of operations; especially, if some operations <math>A,B,C</math> from some set (called [[group]]) can be applied sequentially, one by one, in raw, for example,  <math>A\Big(B\big(C(z)\big)\Big)</math>.  


All things mathematicians deal with are called [[object]]s, and each object is supposed to belong to some set of objects, which is either already defined,  or allows some independent definition.  
All things mathematicians deal with are called [[object]]s, 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 [[paradox]]es, at very beginning.
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 [[paradox]]es, 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 <math>\pi</math>, etvetera. Such names form the basics of the '''mathematical notations'''. Variety of such mathematical notations is numerous, but mathematicians (and especially, [[editor]]s) apply certain efforts in order to keep some standard of mathematical notations.
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 <math>\pi</math>, etvetera. Such names form the basics of the '''mathematical notations'''.  
For ecample, if some author needs to use characters e or <math>\pi</math> or <math>\sum</math> or <math>\prod</math> in some non-usual way, say, as variables, then this author has to present serious reasons for such a strange choice of names of  variables.
<!--
Variety of such mathematical notations is numerous, but mathematicians (and especially, [[editor]]s) apply certain efforts in order to keep some standard of mathematical notations.
For example, if some author needs to use characters e or <math>\pi</math> or <math>\sum</math> or <math>\prod</math> in some non-usual way, say, as variables, then this author has to present serious reasons for such a strange choice of names of  variables.
!-->


==Equvalence <math>=</math> ==
==Equvalence <math>=</math> ==
Equivalence is renoted with symbol <math>=</math>. 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 hight hand side.
Equivalence is renoted with symbol <math>=</math>. 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 equvalence is used to compare [[numbers]].
In the most of cases, the equivalence is used to compare [[numbers]].


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


==True and False ==
==True and False ==
In the simplest case, the mathematical exression may have only one of two values, '''true''' or '''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.
These are basic mathematical expressions and basic mathematical notations.
By default, are expressions in a scientific text are supposed to have value '''true'''.
By default, are expressions in a scientific text are supposed to have value '''true'''.
In some programming lamguages, at initialization, all the logical variables are set to '''false'''.
In some [[programming language]]s, 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:
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==
==Sets==
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One of ways to define the set is following.
One of ways to define the set is following.


One begins with curly parenthesis <math>\{ </math>, which is past of the [[group operation]].
One begins with a curly parenthesis <math>\{ </math>, which is part of the [[group operation]].


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


Then one writes character <math>\in </math> and specify some known (standard) set, indicating, that kind of elements can be considered, for example, will it be from set of words, od from set of animals, of from set of numbers, etc.
Then one writes the character <math>\in </math> 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 character : , and list, separating with comma, all the properties which are specific for all elements of namely this set.
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, closing curly parenthesis <math>\} </math>.
Such a definition should finish with the closing of the grouping, id est, a closing curly parenthesis <math>\} </math>.


===Examples===
===Examples===
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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.
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.


==Quantors==
==Quantifiers==
*<math>\forall</math> [[forall|for all]]
*<math>\forall</math> [[forall|for all]]
*<math>\exist</math>
*<math>\exist</math> there exists
==Nubers==
 
==Numbers==
Basic numbers are 0 and 1.
Basic numbers are 0 and 1.
With respect to these numbers, the arithmetic operations are defined. They are denoted with
With respect to these numbers, the arithmetic operations are defined. They are denoted with
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: <math>\exp</math>
: <math>\exp</math>
and so on.
and so on.
At least for integer, all next operation in this raw  can be constructed as recurrence of operations from previous rows.
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 1-3 above.
Most of conventional calculus is based on the operations summation, multiplication, exponentiation and the inverse functions.
Some of numbers have special single-character notations: 0,1,2,3,4,5,6,7,8,9. For example,  
Some of numbers have own single-character names: 0,1,2,3,4,5,6,7,8,9. For example,  
: <math> 1=0++</math>
: <math> 1=0++</math>
: <math> 2=1++</math>
: <math> 2=1++</math>
and so on. Most of larger numbers have no special manes; the integer numbers are denoted using the [[positional numeral system]].
and so on. Most of larger numbers have no single-character mames; the integer numbers are denoted using the [[positional numeral system]].
Inverse operations (if exist) are denoted, correspondently, with symbols
The inverse operations (if exist) of basic arithmetic operations are denoted, correspondingly, with symbols
: <math>--</math>
: <math>--</math>
: <math>-</math>
: <math>-</math>
: <math>/</math>
: <math>/</math>
: <math>\log</math> and <math>~^*\sqrt{~}</math>
: <math>\log</math> and <math>~^*\sqrt{~}</math>
and so on.
and so on.[[Category:Suggestion Bot Tag]]

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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 from some set (called group) can be applied sequentially, one by one, in raw, for example, .

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 , 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
  • 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 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:

.

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

.

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

  • for all
  • 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)

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,

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

and

and so on.