table of mathematical symbols

In mathematics, there are numerous methods for calculating the average or central tendency of a list of n numbers. The most common method, and the one generally referred to simply as the average, is the arithmetic mean. Please see the table of mathematical symbols for explanations of the symbols used. Average - Arithmetic mean. The arithmetic mean is the standard "average", often simply called the "mean". It is used for many purposes and may be abused by using it to describe skewed dist ...

Including:

• Average - Arithmetic mean
• Average - Median
• Average - Mode
• Average - Other averages
• Average - Etymology

Read more here: » Average: Encyclopedia - Average

In statistics, mean has two related meanings: the average in ordinary English, which is more correctly called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. The average is also called sample mean. the expected value of a random variable, which is also called the population mean. As well as statistics, means are often used in geometry and analysis; a wide range of means have been developed for these purposes, which are not much used in statistics. See the Other m ...

Including:

• Mean - Arithmetic mean
• Mean - An example
• Mean - Geometric mean
• Mean - An example
• Mean - Harmonic mean
• Mean - An example
• Mean - Generalized mean
• Mean - Weighted mean
• Mean - Truncated mean
• Mean - Interquartile mean
• Mean - Mean of a function
• Mean - Other means

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Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...

Including:

• Mathematics - History
• Mathematics - Inspiration pure and applied mathematics and aesthetics
• Mathematics - Notation language and rigor
• Mathematics - Is mathematics a science?
• Mathematics - Overview of fields of mathematics
• Mathematics - Major themes in mathematics
• Mathematics - Quantity
• Mathematics - Structure
• Mathematics - Space
• Mathematics - Change
• Mathematics - Foundations and methods
• Mathematics - Discrete mathematics
• Mathematics - Applied mathematics
• Mathematics - Important theorems
• Mathematics - Important conjectures
• Mathematics - History and the world of mathematicians
• Mathematics - Mathematics and other fields
• Mathematics - Common misconceptions

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Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is also called the diagonalization argument or the diagonal slash argument or the diagonal method.) The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, but was published three years after his first proof. His original argument did not mention decimal expansions, nor any other numeral system. Since this technique was first used, si ...

Including:

• Cantor's diagonal argument - Real numbers
• Cantor's diagonal argument - Why this does not work on integers
• Cantor's diagonal argument - General sets

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In Zermelo-Fränkel set theory, Cantor's theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantor's theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. In particular, the power set of a countably infinite set is un-countably infinite. To illustrate the validity of Cantor's theorem for infinite sets, just test an infinite set in the proof below. Cantor's theorem - The proof. Including:

• Cantor's theorem - The proof
• Cantor's theorem - A detailed explanation of the proof when X is countably infinite
• Cantor's theorem - History

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To get a handle on the proof, let's examine it for the specific case when X is countably infinite. Without loss of generality, we may take X = N = {1, 2, 3,...}, the set of natural numbers. Suppose that N is bijective with its power set P(N). Let us see a sample of what P(N) looks like: P(N) contains in ...

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Cantor's theorem, Cantor's theorem - The proof, Cantor's theorem - A detailed explanation of the proof when X is countably infinite, Cantor's theorem - History

Read more here: » Cantor's theorem: Encyclopedia II - Cantor's theorem - A detailed explanation of the proof when X is countably infinite

Cantor's original proof shows that the interval [0,1] is not countably infinite. The proof by contradiction proceeds as follows: Assume (for the sake of argument) that the interval [0,1] is countably infinite. We may then enumerate all numbers in this interval as a sequence, ( r1, r2, r3, ... ) We already know that each of these numbers may be represented as a decimal expansion. We arrange the numbers in a list (they do not need to be in orde ...

See also:

Cantor's diagonal argument, Cantor's diagonal argument - Real numbers, Cantor's diagonal argument - Why this does not work on integers, Cantor's diagonal argument - General sets

Read more here: » Cantor's diagonal argument: Encyclopedia II - Cantor's diagonal argument - Real numbers

A contravariant tensor of order 1(Ti) is defined as: A covariant tensor of order 1(Ti) is defined as: ...

See also:

Classical treatment of tensors, Classical treatment of tensors - Contravariant and covariant tensors, Classical treatment of tensors - General tensors

Read more here: » Classical treatment of tensors: Encyclopedia II - Classical treatment of tensors - Contravariant and covariant tensors

Mathematical notation - Counting. It is believed that a mathematical notation was first developed at least 50,000 years ago in order to assist with counting. Early mathematical ideas for counting were represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The tally stick is a timeless way of counting. Perhaps the oldest known mathematical texts are those of ancient Sumer. The Census Quipu of the Andes and the Ishango Bone from Africa both used the tally mark method of accounting for numerical concepts. Mathemat ...

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Mathematical notation, Mathematical notation - Definition, Mathematical notation - Expressions, Mathematical notation - Precise semantic meaning, Mathematical notation - History, Mathematical notation - Counting, Mathematical notation - Geometry becomes analytic, Mathematical notation - Counting is mechanized, Mathematical notation - Computerized notation, Mathematical notation - Ideographic notation, Mathematical notation - Notes

Read more here: » Mathematical notation: Encyclopedia II - Mathematical notation - History

When predicate logic was introduced to the mathematical community by Frege (and independently — and more influentially — by Peirce, who coined the term Second-order logic), he did use different variables to distinguish quantification over objects from quantification over properties and sets; but he did not see himself as doing two different kinds of logic. After the discovery of Russell's paradox it was realized that something was wrong with his system. Eventually logicians found that restricting Frege's logic in various ways—to ...

See also:

Second-order logic, Second-order logic - Why second-order logic is not reducible to first-order logic, Second-order logic - Second-order logic and metalogical results, Second-order logic - The history and disputed value of second-order logic, Second-order logic - Power of the existential fragment on finite structures, Second-order logic - Applications to complexity

Read more here: » Second-order logic: Encyclopedia II - Second-order logic - The history and disputed value of second-order logic

The arithmetic mean is the standard "average", often simply called the "mean". It is used for many purposes and may be abused by using it to describe skewed distributions, with highly misleading results. A classic example is average income. The arithmetic mean may be used to imply that most people's incomes are higher than is in fact the case. When presented with an "average" one may be led to believe that most people's incomes are near this number. This "average" (arithmetic mean) income is higher than most peopl ...

See also:

Average, Average - Arithmetic mean, Average - Median, Average - Mode, Average - Other averages, Average - Etymology

Read more here: » Average: Encyclopedia II - Average - Arithmetic mean

An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possible view. For a fuller treatment, see areas of mathematics or the list of mathematics lists. Mathematics - Quantity. This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. See also:

Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Common misconceptions

Read more here: » Mathematics: Encyclopedia II - Mathematics - Major themes in mathematics

The most general notion is the intersection of an arbitrary nonempty collection of sets. If M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if and only if for every element A of M, x is an element of A. In symbols: This idea subsumes the above paragraphs, in that for example, A ∩B ∩C ...

See also:

Intersection set theory, Intersection set theory - Basic definition, Intersection set theory - Arbitrary intersections, Intersection set theory - Nullary intersection

Read more here: » Intersection set theory: Encyclopedia II - Intersection set theory - Arbitrary intersections

The arithmetic mean is the "standard" average, often simply called the "mean". The mean may often be confused with the median or mode. The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not the same as the middle value (median), or most likely (mode). For example, mean income is skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the l ...

See also:

Mean, Mean - Arithmetic mean, Mean - An example, Mean - Geometric mean, Mean - An example, Mean - Harmonic mean, Mean - An example, Mean - Generalized mean, Mean - Weighted mean, Mean - Truncated mean, Mean - Interquartile mean, Mean - Mean of a function, Mean - Other means

Read more here: » Mean: Encyclopedia II - Mean - Arithmetic mean

Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence o ...

See also:

Mathematics, Mathematics - History, Mathematics - Inspiration, pure and applied mathematics, and aesthetics, Mathematics - Notation, language, and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Common misconceptions

Read more here: » Mathematics: Encyclopedia II - Mathematics - Notation, language, and rigor

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that ins ...

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Mathematics, Mathematics - History, Mathematics - Inspiration, pure and applied mathematics, and aesthetics, Mathematics - Notation, language, and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Common misconceptions

Read more here: » Mathematics: Encyclopedia II - Mathematics - Inspiration, pure and applied mathematics, and aesthetics

The geometric mean is an average that is useful for sets of numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean). For example rates of growth. Mean - An example. An experiment yields the following data: 34,27,45,55,22,34 To get the geometric mean How many items? There are 6. Therefore n=6 What is the product of all items? It is 1699493400. To get the geometric mean take the nth (the 6th) root of that pro ...

See also:

Mean, Mean - Arithmetic mean, Mean - An example, Mean - Geometric mean, Mean - An example, Mean - Harmonic mean, Mean - An example, Mean - Generalized mean, Mean - Weighted mean, Mean - Truncated mean, Mean - Interquartile mean, Mean - Mean of a function, Mean - Other means

Read more here: » Mean: Encyclopedia II - Mean - Geometric mean

Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence of ...

See also:

Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Common misconceptions

Read more here: » Mathematics: Encyclopedia II - Mathematics - Notation language and rigor

A multi-order (general) tensor is simply the tensor product of single order tensors: such that: This is sometimes termed the tensor transformation law. ...

See also:

Classical treatment of tensors, Classical treatment of tensors - Contravariant and covariant tensors, Classical treatment of tensors - General tensors

Read more here: » Classical treatment of tensors: Encyclopedia II - Classical treatment of tensors - General tensors

An optimist might attempt to reduce second-order logic to first-order logic in the following way. Expand the domain from the set of all real numbers to the union of that set with the set of all sets of real numbers. Add a new binary predicate to the language: the membership relation. Then sentences that were second-order become first-order. But notice that the domain was asserted to include all sets of real numbers. That requirement has not been reduced to a first-order sentence! But might there be some way to acc ...

See also:

Second-order logic, Second-order logic - Why second-order logic is not reducible to first-order logic, Second-order logic - Second-order logic and metalogical results, Second-order logic - The history and disputed value of second-order logic, Second-order logic - Power of the existential fragment on finite structures, Second-order logic - Applications to complexity

Read more here: » Second-order logic: Encyclopedia II - Second-order logic - Why second-order logic is not reducible to first-order logic