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# Wikipedia:Manual of Style/Mathematics Information

https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style_(mathematics)

This subpage of the Manual of Style contains guidelines for writing and editing clear, encyclopedic, attractive, and interesting articles on mathematics and for the use of mathematical notation in Wikipedia articles on other subjects. For matters of style not treated on this subpage, follow the main Manual of Style and its other subpages to achieve consistency of style throughout Wikipedia.

## Structure

Probably the hardest part of writing a Wikipedia article on a mathematical topic, and generally any Wikipedia article, is addressing a reader's level of knowledge. For example, when writing about a field in the context of abstract algebra, is it best to assume that a reader is already familiar with group theory? A general approach to writing an article is to start simple and then move towards more abstract and technical subjects later on in the article, as in a colloquium talk.

### Article introduction

• describe and define the subject,
• provide context regarding the subject,
• and summarize the article's most important points.

The lead should, as much as possible, be accessible to a general reader, so specialized terminology and symbols should be avoided. Formulas should appear in the first paragraph only if necessary, since they will not be displayed in the preview that pops up when hovering over a link.

In general, the lead sentence should include the article title, or some variation thereof, in bold along with any alternate names, also in bold. The lead sentence should state that the article is about a topic in mathematics, unless the title already does so. It is safe to assume that a reader is familiar with the subjects of arithmetic, algebra, geometry, and that they may have heard of calculus, but are likely unfamiliar with it. For articles that are on these subjects, or on simpler subjects, it can be assumed that the reader is not familiar with the aforementioned subjects. A reader can be assumed to be ignorant of any topics outside of that scope or more advanced topics.

The lead sentence should informally define or describe the subject. For example:

In mathematics, topology (from the Greek τόπος, 'place', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane.

The lead section should include, when appropriate:

• Historical motivation, including names and dates, especially if the article does not have a "History" section. The origin of the subject's name should be explained if it is not self-evident.
• An informal introduction to the topic, without rigor, suitable for a general audience. The appropriate audience for the overview will vary by article, but it should be as basic as reasonable. The informal introduction should clearly state that it is informal, and that it is only stated to introduce the formal approach. Include a physical or geometric analogy or diagram if it can help introduce the topic.
• Motivation or applications, which can illuminate the use of the topic and its connections to other areas of mathematics or other non-mathematical subjects.

### Article body

Readers have differing levels of experience and knowledge. When in doubt, articles should define the notation they use. For example, some readers will immediately recognize that Δ(K) means the discriminant of a number field, but others will never have encountered the notation. These will be helped by an aside like "...where Δ(K) is the discriminant of the field K".

An article should use standard notation when possible, and notations which are unavoidably non-standard or uncommon should be defined within it. For example, if x^n or x**n is used for exponentiation instead of xn, the article should define these notations. If an article requires extensive notation, consider introducing the notation as a bulleted list or separating it into a "Notation" section.

An article about a mathematical object should provide an exact definition of the object, perhaps in a "Definition" section after section(s) of motivation. For example:

Let S and T be topological spaces, and let f be a function from S to T. Then f is called continuous if, for every open set O in T, the preimage f −1(O) is an open set in S.

The phrase "formal definition" may help to flag the actual definition of a concept for readers unfamiliar with academic terminology, in which "definition" means formal definition, and a "proof" is always a formal proof.

When the topic is a theorem, the article should provide a precise statement of the theorem. Sometimes this statement will be in the lead, for example:

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G.

Other times, it may be better to separate the statement into its own section, as for long theorems like the Poincaré–Birkhoff–Witt theorem, or to present multiple equivalent formulations, as for Nakayama's lemma.

Representative examples and applications help to illustrate definitions and theorems and to provide context for why they might be interesting. Shorter examples may fit into the main exposition of the article, such as the discussion at Algebraic number theory § Failure of unique factorization, while others may deserve their own section, as in Chain rule § First example. Multiple related examples may also be given together, as in Adjunction formula § Applications to curves. Occasionally, it is appropriate to give a large number of computationally-flavored examples, as in Lambert W function § Applications. It may also be edifying to list non-examples, which almost-but-not-quite satisfy the definition. In keeping with the purpose and tone of an encyclopedia, examples should be informative rather than instructional (see WP:NOTTEXTBOOK for details).

A picture can really bring home a point, and can often precede the mathematical discussion of a concept. How to create graphs for Wikipedia articles contains some details on how to create graphs and other pictures as well as how to include them in articles.

Formulas tend to repel less mathematical readers, and mathematics articles should take pains to explain (or even replace) them by words if possible. In particular, the English words "for all", "exists", and "in" should be preferred to the corresponding symbols ∀, ∃, and ∈. Similarly, definitions should be highlighted with words such as "is defined by" in the text.

If not included in the introduction, a history section can provide additional context and details on the topic's motivation and connections.

### Concluding matters

Most mathematical ideas are capable of some form of generalization. If appropriate, such material can be put under a "Generalizations" section. As an example, multiplication of the rational numbers can be generalized to other fields.

It is also generally good to have a "See also" section in an article. The section should link to related subjects, or to pages which could provide more insight into the contents of the article. More details on "See also" sections can be found at Wikipedia:Manual of Style/Layout § "See also" section. Lastly, a well-written and complete article should have a "References" section. This topic is discussed in detail in the section § Including literature and references.

## Writing style in mathematics

There are several issues of writing style that are particularly relevant in mathematical writing.

In the interest of clarity, sentences should not begin with a symbol. Do not write:

• Suppose that G is a group. G can be decomposed into cosets, as follows.
• Let H be the corresponding subgroup of G. H is then finite.
• ${\displaystyle \pi }$ is a mathematical constant.

• A group G may be decomposed into cosets as follows.
• If H is the corresponding subgroup of G, then H is finite.
• The letter ${\displaystyle \pi }$ denotes a mathematical constant.

Mathematics articles are often written in a conversational style similar to a whiteboard lecture. However, a narrative pedagogical style runs counter to Wikipedia's recommended encyclopedic tone. While opinions vary on the most edifying style, authors should generally strike a balance between bare lists of facts and formulae, and relying too much on addressing the reader directly and referring to "we". Also avoid contentless clichés as Note that, It should be noted that, It must be mentioned that, It must be emphasized that, Consider that, and We see that. There is no use in imploring the reader to take note of each thing being pointed out. Rather than drawing the reader's attention to crucial information buried in the text, try to reorganize and rephrase to put the crucial part first.

Articles should be as accessible as possible to readers not already familiar with the subject matter. Notations not entirely standard should be properly introduced and explained. Whenever a variable or other symbol is defined by a formula, make sure to say this is a definition introducing a notation, not an equation involving a previously known object. Also identify the nature of the entity being defined. Don't write:

• Multiplying M by u = vv0, ...

• Multiplying M by the vector u defined by u = vv0, ...

In definitions, the symbol "=" is preferred over "≡" or ":=".

When defining a term, do not use the phrase "if and only if". For example, instead of

• A function f is even if and only if f(−x) = f(x) for all x

write

• A function f is even if f(−x) = f(x) for all x.

If it is reasonable to do so, rephrase the sentence to avoid the use of the word "if" entirely. For example,

• An even function is a function f such that f(−x) = f(x) for all x.

Avoid, as far as possible, useless phrases such as:

• It is easily seen that ...
• Clearly ...
• Obviously ...

The reader might not find what you write obvious. Instead, try to hint why something must hold, such as:

• It follows directly from this definition that ...
• By a straightforward, if lengthy, algebraic calculation, ...

Articles should avoid common blackboard abbreviations such as wrt (with respect to), wlog (without loss of generality), and iff (if and only if), as well as quantifier symbols ∀ and ∃ instead of for all and there exists. In addition to compromising the encyclopedic tone, these abbreviations are a form of jargon that may confuse the reader.

Avoid any when verbalizing quantifiers since it is ambiguous. Instead of if any x satisfies F(x) = 0, write if every x satisfies F(x) = 0, or if some x satisfies F(x) = 0, depending on what you wish to express.

The plural of formula is either formulae or formulas. Both are acceptable, but an article should be internally consistent. In an already consistent article, editors should refrain from changing one style to another.

## Mathematical conventions

A number of conventions have been developed to make Wikipedia's mathematics articles more consistent with each other. These conventions cover choices of terminology, such as the definitions of compact and ring, as well as notation, such as the correct symbols to use for a subset.

These conventions are suggested in order to bring some uniformity between different articles, to aid a reader who moves from one article to another. However, each article may establish its own conventions. For example, an article on a specialized subject might be more clear if written using the conventions common in that area. Thus the act of changing an article from one set of conventions to another should not be undertaken lightly.

Each article should explain its own terminology as if there are no conventions, in order to minimize the chance of confusion. Not only do different articles use different conventions, but Wikipedia's readers come to articles with widely different conventions in mind. These readers will often not be familiar with our conventions, which may differ greatly from the conventions they see outside Wikipedia. Moreover, when our articles are presented in print or on other websites, there may be no simple way for readers to check what conventions have been employed.

### Terminology conventions

#### Natural numbers

"The set of natural numbers" has two common meanings: {0, 1, 2, 3, ...}, which may also be called non-negative integers, and {1, 2, 3, ...}, which may also be called positive integers. Use the sense appropriate to the field to which the subject of the article belongs if the field has a preferred convention. If the sense is unclear, and if it is important whether or not zero is included, consider using one of the alternative phrases rather than natural numbers if the context permits.

#### Miscellaneous

• Directed sets are preordered sets with finite joins, not partial orders as in, e.g., Kelley (General Topology; ISBN  0-387-90125-6).
• A lattice need not be bounded. In a bounded lattice, 0 and 1 are allowed to be equal.
• Elliptic functions are written in ω = half-period style.
• A weight k modular form follows the Serre convention that f(−1/τ) = τkf(τ), and q = e2πiτ.

### Notational conventions

• The abstract cyclic group of order n, when written additively, has notation Zn, or in contexts where there may be confusion with p-adic integers, Z/nZ; when written multiplicatively, e.g. as roots of unity, Cn is used (this does not affect the notation of isometry groups called Cn).
• The standard notation for the abstract dihedral group of order 2n is Dn in geometry and D2n in finite group theory. There is no good way to reconcile these two conventions, so articles using them should make clear which they are using.
• Bernoulli numbers are denoted by Bn, and are zero for n odd and greater than 1.
• In category theory, write Hom-sets, or morphisms from A to B, as Hom(A,B) rather than Mor(A,B) (and with the implied convention that the category is not a small category unless that is said).
• The semidirect product of groups K and Q should be written K ×φ Q or Q ×φ K where K is the normal subgroup and φ : Q → Aut(K) is the homomorphism defining the product. The semidirect product may also be written KQ or QK (with the bar on the side of the non-normal subgroup) with or without the φ.
• The context should clearly state that this is a semidirect product and should state which group is normal.
• The bar notation is discouraged because it is not supported by all browsers.
• If the bar notation is used it should be entered as {{unicode|⋉}} (⋉) or {{unicode|⋊}} (⋊) for maximum portability.
• Subset is denoted by ${\displaystyle \subseteq }$, proper subset by ${\displaystyle \subsetneq }$. The symbol ${\displaystyle \subset }$ may be used if the meaning is clear from context, or if it is not important whether it is interpreted as subset or as proper subset (for example, ${\displaystyle A\subset B}$ might be given as the hypothesis of a theorem whose conclusion is obviously true in the case that ${\displaystyle A=B}$). All other uses of the ${\displaystyle \subset }$ symbol should be explicitly explained in the text.
• For a matrix transpose, use superscript non-italic capital letter T: XT, ${\displaystyle X^{\mathrm {T} }}$ or ${\displaystyle X^{\mathsf {T}}}$, and not XT, ${\displaystyle X^{T}}$, or ${\displaystyle X^{\top }}$.
• In a lattice, infima are written as ab or as a product ab, suprema as ab or as a sum a + b. In a pure lattice theoretical context the first notation is used, usually without any precedence rules. In a pure engineering or "ideals in a ring" context the second notation is used and multiplication has higher precedence than addition. In any other context the confusion of readers of all backgrounds should be minimized. In an abstract bounded lattice, the smallest and greatest elements are denoted by 0 and 1.
• The scalar or dot product of vectors should be denoted with a centre-dot ab, as an inner producta,b⟩ or (a,b), or as a matrix product aTb, never with juxtaposition ab.

## Proofs

This is an encyclopedia, not a collection of mathematical texts; but we often want to include proofs to explain a theorem or definition. A downside of including proofs is that they may interrupt the flow of the article, whose goal is usually expository. Use your judgment; as a rule of thumb, include proofs when they expose or illuminate the concept or idea; don't include them when they serve only to establish the correctness of a result.

Since many readers will want to skip proofs, it is a good idea to set them apart in some way, for instance by giving them a separate section. Additional discussion and guidelines can be found at Wikipedia:WikiProject Mathematics/Proofs.

## Algorithms

An article about an algorithm may include pseudocode or in some cases source code in some programming language. Wikipedia does not have a standard programming language or languages, and not all readers will understand any particular language even if the language is well-known and easy to read, so consider whether the algorithm could be expressed in some other way. If source code is used always choose a programming language that expresses the algorithm as clearly as possible.

Articles should not include multiple implementations of the same algorithm in different programming languages unless there is encyclopedic interest in each implementation.

Source code should always use syntax highlighting. For example this markup: [2]

<syntaxhighlight lang="Haskell">
primes = sieve [2..]
sieve (p : xs) = p : sieve [x | x <- xs, x mod p > 0]
</syntaxhighlight>

generates the following:

  primes = sieve [2..]
sieve (p : xs) = p : sieve [x | x <- xs, x mod p > 0]

## Including citations and literature references

Per the Wikipedia policy, WP:VERIFY, it is essential for article content to have inline citations, and thus to have a well-chosen list of references and pointers to the literature. Some reasons for this are the following:

• Wikipedia articles cannot be a substitute for a textbook (that is what Wikibooks is for). Also, often one might want to find out more details (like the proof of a theorem stated in the article).
• Some notions are defined differently depending on context or author. Articles should contain some references that support the given usage.
• Important theorems should cite historical papers as an additional information (not necessarily for looking them up).
• Today many research papers or even books are freely available online and thus virtually just one click away from Wikipedia. Newcomers would greatly profit from having an immediate connection to further discussions of a topic.
• Providing further reading enables other editors to verify and to extend the given information, as well as to discuss the quality of a particular source.

The Wikipedia:Cite sources article has more information on this and also several examples for how the cited literature should look.

## Typesetting of mathematical formulae

One may set formulae using LaTeX (the <math> tag, described in the next subsection) or, in certain cases, using other means of formatting that render in HTML; both are acceptable and widely used, except for section headings, which should use HTML only, as LaTeX markup might cause uneven spacing in the table of contents, as well as the appearance of illegible anchor links to sections. Some of the issues presented by using LaTeX or HTML are discussed below.

Large-scale formatting changes to an article or group of articles are likely to be controversial. One should not change formatting boldly from LaTeX to HTML, nor from non-LaTeX to LaTeX without a clear improvement. Proposed changes should generally be discussed on the talk page of the article before implementation. If there is no positive response, or if planned changes affect more than one article, consider notifying an appropriate Wikiproject, such as WikiProject Mathematics for mathematical articles.

For inline formulae, such as a2b2, the community of mathematical editors of English Wikipedia currently has no consensus about preferred formatting; see WP:Rendering math for details.

For a formula on its own line the preferred formatting is the LaTeX markup, with a possible exception for simple strings of Latin letters, digits, common punctuation marks, and arithmetical operators. Even for simple formulae the LaTeX markup might be preferred if required for uniformity within an article. For readability, it is also strongly preferred not to mix HTML and LaTeX markup in the same expression.

### Using LaTeX markup

Wikipedia allows editors to typeset mathematical formulae in (a subset of) LaTeX markup (see also TeX); the formulae are, for a default reader, translated into PNG images. They may also be rendered as MathML or HTML (using MathJax), depending on user preferences. For more details on this, see Help:Displaying a formula.

The LaTeX formulae can be displayed inline (like this: ${\displaystyle \mathbf {x} \in \mathbb {R} ^{2}}$), as well as on their own line:

${\displaystyle \int _{0}^{\pi }\sin x\,dx.}$

A frequent method for displaying formulas on their own line has been to indent the line with one or more colons (:). Although this produces the intended visual appearance, it produces invalid html (see Wikipedia:Manual of Style/Accessibility § Indentation). Instead, formulas may be placed on their own line using <math display=block>. For instance, the formula above was typeset using <math display=block>\int_0^\pi \sin x\,dx.</math>.

If you find an article which indents lines with spaces in order to achieve some formula layout effect, you should convert the formula to LaTeX markup.

Having LaTeX-based formulae inline has the following drawbacks:

• The font size can be slightly larger than that of the surrounding text on some browsers, making text containing inline formulae harder to read.
• The download speed of a page is negatively affected if it contains many formulas.

If an inline formula needs to be typeset in LaTeX, often better formatting can be achieved with <math display=inline> tag, which translates to the \textstyle LaTeX command. By default, LaTeX code is rendered as if it were a displayed equation (not inline), and this can frequently be too big. For example, the formula <math>\sum_{n=1}^\infty 1/n^2 = \pi^2/6</math>, which displays as ${\displaystyle \sum _{n=1}^{\infty }1/n^{2}=\pi ^{2}/6}$, is too large to be used inline. display=inline generates a smaller summation sign and moves the limits on the sum to the right side of the summation sign. The code for this is <math display=inline>\sum_{n=1}^\infty 1/n^2 = \pi^2/6</math>, and it renders as the much more aesthetic ${\textstyle \sum _{n=1}^{\infty }1/n^{2}=\pi ^{2}/6}$.

HTML-generating formatting, as described below, is adequate for articles that use only simple inline formulae and better for text-only browsers.

#### Deprecated formatting

Older versions of the MediaWiki software supported displaying simple LaTeX formulae as HTML rather than as an image. Although this is no longer an option, some formulae have formatting in them intended to force them to display as an image, such as an invisible quarter space (\,) added at the end of the formula, or \displaystyle at the beginning. Such formatting can be removed if a formula is edited and need not be added to new formulae.

#### Alt text

Images generated from LaTeX markup have alt text, which is displayed to visually impaired readers and other readers who cannot see the images. The default alt text is the LaTeX markup that produced the image. You can override this by explicitly specifying an alt attribute for the math element. For example, <math alt="Square root of pi">\sqrt{\pi}</math> generates an image ${\displaystyle {\sqrt {\pi }}}$ whose alt text is "Square root of pi". Small and easily explained formulas used in less technical articles can benefit from explicitly specified alt text. More complicated formulas, or formulas used in more technical articles, are often better off with the default alt text.

### Using HTML

The following sections cover the way of presenting simple inline formulae in HTML, instead of using LaTeX.

Templates supporting HTML formatting are listed in Category:Mathematical formatting templates. Not all templates are recommended for use; in particular, use of the {{ frac}} template to format fractions is discouraged in mathematics articles.

#### Font formatting

By default, regular text is rendered in a sans serif font.

The relationship is defined as ''x'' = −(''y''<sup>2</sup> + 2).

will result in:

The relationship is defined as x = −(y2 + 2).

As TeX uses a serif font to display a formula (both as PNG and HTML), you may use the {{ math}} template to display your HTML formula in serif as well. Doing so will also ensure that the text within a formula will not line-wrap, and that the font size will closely match the surrounding text in any skin. Note that certain special characters (equal signs, absolute value bars) require special attention.

The relationship is defined as {{math|''x'' {{=}} −(''y''<sup>2</sup> + 2)}}.

will result in:

The relationship is defined as x = −(y2 + 2).
##### Variables

To start with, we generally use italic text for variables, but never for numbers or symbols. You can use ''x'' in the edit box to refer to the variable x. Some prefer using the HTML "variable" tag, <var>, since it provides semantic meaning to the text contained within. Others use the {{ mvar}} template to show single variables in a serif typeface, to help distinguish certain characters such as I and l. Which method you choose is entirely up to you, but in order to keep with convention, we recommend the wiki markup method of enclosing the variable name between repeated apostrophe marks. Thus we write:

''x'' = −(''y''<sup>2</sup> + 2) ,

which results in:

x = −(y2 + 2) .

While italicizing variables, things like parentheses, digits, equal and plus signs should be kept outside of the double-apostrophed sections. In particular, do not use double apostrophes as if they are <math> tags; they merely denote italics. Descriptive subscripts should not be in italics, because they are not variables. For example, mfoo is the mass of a foo. SI units are never italicized: x = 5 cm.

##### Functions

Names for standard functions, such as sin and cos, are not in italic font, but we use italic names such as f for functions in other cases; for example when we define the function as in f(x) = sin(x) cos(x) .

##### Sets

Sets are usually written in upper case italics; for example:

A = {x : x > 0}

would be written:

''A'' = {''x'' : ''x'' > 0} .
##### Greek letters

Italicize lower-case Greek letters when they are variables or constants (in line with the general advice to italicize variables): the example expression λ + y = πr2 would then be typeset as:

''λ'' + ''y'' = ''πr''<sup>2</sup>

(It is also possible to enter Greek letters directly.)

For consistency with the (La)TeX style, do not italicize capital Greek letters; e.g. n! = Γ(n + 1) .

##### Common sets of numbers

Commonly used sets of numbers are typeset in boldface, as in the set of real numbers R. Again, typically we use wiki markup: three apostrophes (''') rather than the HTML <b> tag for making text bold. Bold notation has been largely replaced by blackboard bold, which may be encoded in La-TeX as <math>\mathbb{R}</math>, which renders as ${\displaystyle \;\mathbb {R} \;}$. However, the special Unicode characters, such as U+211D (plain text ℝ or math font ) and its adjacent characters should be avoided at present, since these characters are not yet universally supported.

##### Superscripts and subscripts

Subscripts and superscripts should be wrapped in <sub> and <sup> tags, respectively, with no other formatting info. Font sizes and such should be entrusted to be handled with stylesheets. For example, to write c3+5, use

''c''<sub>3+5</sub>.

Do not use special characters like  ² (²) for squares. This does not combine well with other powers, as the following comparison shows:

1 + x + x² + x3 + x4 (with ²) versus
1 + x + x2 + x3 + x4 (with <sup>2</sup>).

Moreover, the TeX engine used on Wikipedia may format simple superscripts using <sup>...</sup> depending on user preferences. Thus, instead of the image ${\displaystyle x^{2}\,}$, many users see x2. Formulae formatted without using TeX should use the same syntax to maintain the same appearance.

#### Special symbols

There are list of mathematical symbols, list of mathematical symbols by subject and a list at Wikipedia:Mathematical symbols that may be useful when editing mathematics articles. Almost all mathematical operator symbols have their specific code points in Unicode outside both ASCII and General Punctuation (with notable exception of "+", "=", "|", as well as ",", ":", and three sorts of brackets). As a rule of thumb, specific mathematical symbols shall be used, not similar-looking ASCII or punctuation symbols, even if corresponding glyphs are indistinguishable. The list of mathematical symbols by subject includes markup for LaTeX and HTML, and Unicode code points.

There are two caveats to keep in mind, however.

1. Not all of the symbols in these lists are displayed correctly on all browsers (see Help:Special characters). Although the symbols that correspond to named entities are very likely to be displayed correctly, a significant number of viewers will have problems seeing all the characters listed at Mathematical operators and symbols in Unicode. One way to guarantee that an uncommon symbol is rendered correctly for all readers is to force the symbol to display as an image, using the <math> environment.
2. Not all readers will be familiar with mathematical notation. Thus, to maximize the size of the audience who can read an article, it is better to be conservative in using symbols. For example, writing "a divides b" rather than "a | b" in an elementary article may make it more accessible.

For Roman numerals, Basic Latin (ASCII) letters should be used instead of the equivalent Unicode characters in the U+21XX range. For example, L and VI, not , and not precomposed characters like . (The only exception is when discussing the Unicode characters themselves.)

#### Less-than sign

Although the MediaWiki markup engine is fairly smart about differentiating between unescaped "<" characters that are used to denote the start of an embedded HTML or HTML-like tag and those that are just being used as literal less-than symbols, it is ideal to use < when writing the less-than sign, just like in HTML and XML. For example, to write x < 3, use

''x'' < 3,

not

''x'' < 3.

#### Multiplication sign

Standard algebraic notation is best for formulae, so two variables q and d being multiplied are best written as qd when presented in a formula. That is, when citing a formula, don't use ×.

However, when explaining the formula for a general audience (not just mathematicians), or giving examples of its application, it is prudent to use the multiplication sign: "×", coded as × in HTML. Do not use the letter "x" to indicate multiplication. For example:

An alternative to × is the dot operator ⋅ (also encoded <math>\cdot</math> and reachable in the "Math and logic" drop-down list below the edit box), which produces a properly spaced centered dot: "a ⋅ b".

Do not use the ASCII asterisk (*) as a multiplication sign outside of source code. It is not used for this purpose in professionally published mathematics, and most fonts render it in an inappropriate vertical position (above the midline of the text rather than centered on it). For the dot operator, do not use punctuation symbols, such as a simple interpunct · (the choice offered in the "Wiki markup" drop-down list below the edit box), as in many fonts it does not kern properly. The use of U+2022 BULLET as an operator symbol is also discouraged except in abstract contexts (e.g. to denote an unspecified operator).

#### Minus sign

The correct encoding of the minus sign "−" is different from all varieties of hyphen "-‐‑", [3] as well as from en- dash "–". To really get a minus sign, use the "minus" character "−" (reachable via selecting "Math and logic" in the drop-down list below the edit box or using {{ subst:minus}}) or use the "−" entity.

#### Square brackets

Square brackets have two problems; they can occasionally cause problems with wiki markup, and editors sometimes 'fix' the brackets in asymmetrical intervals to make them symmetrical. The nowiki tag can be used as a general solution to problems like this, as in <nowiki>]</nowiki> to have the ] treated as literal text.

The use of intervals for the range or domain of a function is very common. A solution which makes the reason for the different brackets around an interval more plain is to use one of the templates {{ open-closed}}, {{ closed-open}}, {{ open-open}}, {{ closed-closed}}. For instance:

{{open-closed|−π, π}}

produces

(−π, π].

These templates use the {{ math}} template to avoid line breaks and use the TeX font.

#### Function symbol

There is a special Unicode symbol, U+0192 ƒ LATIN SMALL LETTER F WITH HOOK (ƒ), sometimes used as the Florin currency symbol. [4] As of December 2010, this character is not interpreted correctly by screen readers such as JAWS and NonVisual Desktop Access [5]. An italicized letter f should be used instead.

The radical symbol √ can be used when written on its own, but when part of a larger expression, can be problematic. {{ radic}} (often seen as {{ sqrt}}) is the best way to write such expressions in HTML, but the result is unattractive due to the hole between the overline and the radical symbol in many web browsers:

9, 327

This method should be avoided whenever technically possible to do so. Instead, use <math>...</math> tags and \sqrt{}, even if inline. For example:

${\displaystyle {\sqrt {9}},{\sqrt[{3}]{27}}}$

Because of Mediawiki bug T263572, <math>...</math> markup is incompatible with the Media Viewer (used for full-screen image viewing on mobile devices), so until that is fixed, the {{ radic}} method or √ with no overline should be used in image captions.

The use of √ with no overline is acceptable for simple expressions, as long as the operand is unambiguous. [6]

### Explanation of symbols in formulae

A list as in

Example 1: The foocity is given by

${\displaystyle \mathbf {F} =\mathbf {b} \times a\mathbf {r} ,}$

where

• ${\displaystyle \mathbf {b} }$ is the barness vector,
• ${\displaystyle a}$ is the bazness coefficient,
• ${\displaystyle \mathbf {r} }$ is the quuxance vector.

should be written as prose, to avoid using more vertical space than necessary:

Example 2: The foocity is given by

${\displaystyle \mathbf {F} =\mathbf {b} \times a\mathbf {r} ,}$

where ${\displaystyle \mathbf {b} }$ is the barness vector, ${\displaystyle a}$ is the bazness coefficient, and ${\displaystyle \mathbf {r} }$ is the quuxance vector.

An exception would be if some of the definitions are very long (as in Heat equation, for example). In any case, each definition should end with a comma or semicolon, and the last one should end with a period if it terminates a sentence.

### Punctuation after formulae

Just as in mathematics publications, a sentence which ends with a formula must have a period at the end of the formula. [7] This equally applies to displayed formulae (that is, formulae that take up a line by themselves). Similarly, if the conventional punctuation rules would require a question mark, comma, semicolon, or other punctuation at that place, the formula must have that punctuation at the end.

If the formula is written in LaTeX, that is, surrounded by the <math> and </math> tags, then the punctuation should also be inside the tags, because otherwise the punctuation could wrap to a new line if the formula is at the edge of the browser window. Alternatively—since the previous result can be unaesthetic, especially for inlined formulae presented as an image whose baseline does not line up with that of the running text—the punctuation can be placed after the </math> tag and then the whole formula (including the punctuation) can be enclosed with the {{ nowrap}} template, as in This shows that {{nowrap|<math>\tfrac{1}{2} = 0.5</math>.}}. [8]

### Font usage

#### Multi-letter names

Functions that have multi-letter names should always be in an upright font. The most well-known functions—trigonometric functions, logarithms, etc.—can be written without parentheses for as long as the result does not become ambiguous. For example:

${\displaystyle 2\sin x}$   (parentheses may be omitted here, as the argument consists of a single term only; typeset from <math>2\sin x</math>)
${\displaystyle 2\sin(x+1)}$ (parentheses are required to clarify the intended argument)

but not

${\displaystyle 2sinx}$   (incorrect—typeset from <math>2sin x</math>).
Note: For potential pitfalls of forms not understood consistently across the board, see order of operations and implied multiplication; if there is any risk that a term could become ambiguous for our readership, use parentheses.

When operator (function) names do not have a pre-defined abbreviation, we may use \operatorname:

${\displaystyle 2\operatorname {csch} x}$   (typeset from <math>2\operatorname{csch}x</math>).
${\displaystyle a\operatorname {tr} (A)}$   (typeset from <math>a\operatorname{tr}(A)</math>).

\operatorname includes correct spacing that would not be present with other means such as \rm:

${\displaystyle 2{\rm {sin}}x}$   (incorrect—typeset from <math>2{\rm sin} x</math>).

Special care is needed with subscripted labels to distinguish the purpose of the subscript (as this is a common error): variables and constants in subscripts should be italic, while textual labels should be in normal text font (Roman, upright). For example:

${\displaystyle x_{\text{this one}}=y_{\text{that one}}}$   (correct—typeset from <math> x_\text{this one} = y_\text{that one}</math>),

and

${\displaystyle \sum _{i=1}^{n}{y_{i}^{2}}}$   (correct—typeset from <math>\sum_{i=1}^n { y_i^2 }</math>),

but not

${\displaystyle r=x_{predicted}-x_{observed}}$   (incorrect—typeset from <math>r = x_{predicted} - x_{observed}</math>).

For several years this manual recommended \mbox as a workaround for lack of \text, but this is now considered undesirable. See An opinion: Why you should never use \mbox within Wikipedia.

#### Roman versus italic

For single-letter variables, constants, and operators such as the differential, imaginary unit, and Euler's number, Wikipedia articles usually use an italic font. One writes

${\displaystyle \int _{0}^{\pi }\sin x\,dx,}$   (typeset from <math>\int_0^\pi \sin x \, dx ,</math>—note the thin space (\,) before dx),
${\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}$   (typeset from <math>\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx} ,</math>),
${\displaystyle x+iy,}$   (typeset from <math>x+iy,</math>), and
${\displaystyle e^{i\theta }.}$   (typeset from <math>e^{i\theta} .</math>).

Some authors prefer to use an upright (Roman) font for operators, as in d, for the differential operator, as opposed to d for a variable. Upright fonts are sometimes used for standard, nearly universal constants, as in i, e, and π; other authors use Roman boldface, as in i. Changes from one style to another should be done only to make an article consistent with itself. Formatting changes should not be made solely to make articles consistent with each other, nor to make articles conform to a particular style guide or standards body. It is inappropriate for an editor to go through articles doing mass changes from one style to another. When there is dispute over the correct style to use, follow the same principles as MOS:STYLERET.

Generally, one way to determine which usage is appropriate on Wikipedia is to look at prevalence in reliable sources in addition to relevant style guides, per WP:WEIGHT. For example, the ISO 80000-2 recommends that the mathematical constant e should be typeset in an upright Roman font: e. But this guide is rarely followed in reliable mathematical sources, and it is contradicted by other style guides, like Donald Knuth's TeXbook. This makes the more common practice to use an italic face for the constant e.

#### Blackboard bold

Blackboard bold typeface was never used in traditional typography. It was introduced for easier distinguishing boldface from ordinary face on a blackboard. It is presently used in mathematical printing for denoting some constant objects in a way that cannot be confused with other uses of boldface.

Nowadays, both blackboard bold and usual boldface are commonly used for standard number systems (${\displaystyle \mathbb {N} ,}$ ${\displaystyle \mathbb {Z} ,}$ ${\displaystyle \mathbb {Q} ,}$ ${\displaystyle \mathbb {R} ,}$ ${\displaystyle \mathbb {C} ,}$ ${\displaystyle \mathbb {H} ,}$ ${\displaystyle \mathbb {F} _{q},}$ ${\displaystyle \mathbb {Z} _{p},}$ ${\displaystyle \mathbb {Q} _{p}}$) and for certain other mathematical objects, including affine space ${\displaystyle \mathbb {A} ^{n}}$, projective space ${\displaystyle \mathbb {P} ^{n}}$, adele rings ${\displaystyle \mathbb {A} _{K}}$, the additive and multiplicative group schemes (${\displaystyle \mathbb {G} _{a}}$ and ${\displaystyle \mathbb {G} _{m}}$), and hypercohomology (e.g., ${\displaystyle \mathbb {H} ^{i}(X,\Omega _{X}^{\bullet })}$).

A particular concern for the use of blackboard bold on Wikipedia is that the Unicode symbols for blackboard bold characters are not supported by all systems, or that font substitution on browsers often render these symbols in discordant fonts. The use of Unicode characters for blackboard bold is discouraged in English Wikipedia. Instead, the LaTeX rendering (for example <math>\mathbb{Z}</math> or <math>\Z</math>) or the standard boldface must be used. As with all such choices, each article should be consistent with itself, and editors should not change articles from one choice of typeface to another, except for consistency. Again, when there is dispute, follow MOS:STYLERET.

Due to a rendering bug, LaTeX blackboard bold currently does not work with numerals. Use bold instead (e.g. {{math|'''1'''}} or <math>\bold{1}</math>), which is more common anyway. If absolutely necessary (e.g. when discussing the notation itself), use the Unicode character (e.g. 𝟙).

### Fractions

In mathematics articles, fractions should always be written either with a horizontal fraction bar (as in ${\displaystyle \textstyle {\frac {1}{2}}}$), or with a forward slash and with the baseline of the numbers aligned with the baseline of the surrounding text (as in 1/2). The use of {{ frac}} (such as 12) is discouraged in mathematics articles. The use of Unicode precomposed fractions (such as ½) is discouraged entirely, for accessibility reasons [9] among others [10]. Metric units are given in decimal fractions (e.g., 5.2 cm); non-metric units can be either type of fraction, but the fraction style should be consistent throughout the article.

Only "/" is used for quotient objects in abstract algebra: R / A or ${\displaystyle R/A}$  – markup: {{ math|''R'' / ''A''}} or < math>R / A</math>

## Graphs and diagrams

The angle CAB is α.
The length of CA is b.

There is no general agreement on what fonts to use in graphs and diagrams. In geometrical diagrams points are normally labelled using upper case letters, sides with lower case and angles with lower case Greek letters.

Recent[ when?] geometry books tend to use an italic serif font in diagrams as in ${\displaystyle A}$ for a point. This allows easy use in LaTeX markup. However, older books tend to use upright letters as in ${\displaystyle \mathrm {A} }$ and many diagrams in Wikipedia use sans-serif upright A instead. Graphs in books tend to use LaTeX conventions, but yet again there are wide variations.

For ease of reference diagrams and graphs should use the same conventions as the text that refers to them. If there is a better illustration with a different convention, though, the better illustration should normally be used.

## Notes

1. ^ Currently, ring (mathematics) and related articles attempt to cover both unital rings and non-unital rings: they do not consistently implement this interpretation. This attempt to cover multiple meanings violates WP:DICT#Major differences ( homographs).
2. ^ This example, from here [1], is in Haskell, not a well-known language so generally not a good choice when showing an algorithm.
3. ^ Note that, aside of <math>, many templates and parser functions accept the hyphen-minus "-" as a valid representation of the minus sign. Except situations where "-" has to represent the minus sign in a source code (including wiki code), it should not be seen in a rendered page, though.
4. ^ Latin Extended-B, [2]
5. ^
6. ^
7. ^ This style, adopted by Wikipedia, is shared by Higham (1998), Halmos (1970), the Chicago Manual of Style, and many mathematics journals.
8. ^ It is technically possible to use a word joiner before the punctuation instead, but it's seldom respected by browsers.
9. ^ Characters in ISO/IEC 8859-1 (¼, ½, and ¾) work with screen readers, but others, like ⅐, might not.
10. ^ Not all fractions are available precomposed.


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# Kloosterman sum

The Kloosterman sum is one of various trigonometric sums that are useful in number theory and, more generally, in finite harmonic analysis. The original Kloosterman sum is

 $K_{p}(a,b)=\sum_{x\in\mathbb{F}_{p}^{*}}\exp\left(\frac{2\pi i(ax+bx^{-1})}{p}\right)$

where $\mathbb{F}_{p}$ is the field of prime order $p$. Such sums have been generalized in a few different ways since their introduction in 1926. For instance, let $q$ be a prime power, $\mathbb{F}_{q}$ the field of $q$ elements, $\chi:\mathbb{F}_{q}^{*}\to\mathbb{C}$ a character, and $\psi:\mathbb{F}_{q}\to\mathbb{C}$ a mapping such that $\psi(x+y)=\psi(x)\psi(y)$ identically. The sums

 $K_{\psi}(\chi|a,b)=\sum_{x\in\mathbb{F}_{q}^{*}}\chi(x)\psi(ax+bx^{-1})$

are of interest, because they come up as Fourier coefficients of modular forms.

Kloosterman sums are finite analogs of the $K$-Bessel functions of this kind:

 $K_{s}(a)=\frac{1}{2}\int_{0}^{\infty}x^{s-1}\exp\left(\frac{-a(x+x^{-1})}{2}% \right)dx$

where $\Re(a)>0$.

Title Kloosterman sum KloostermanSum 2013-03-22 13:59:33 2013-03-22 13:59:33 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 11L05 msc 43A25 GaussSum

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Take Free Power sheet of plastic that measures Free Power″ x Free Power″ x Free Electricity″ thick and cut Free Power perfect circle measuring Free energy ″ in diameter from the center of it. (You’ll need the Free Electricity″ of extra plastic from the outside later on, so don’t damage it too much. You can make Free Power single cut from the “top” of the sheet to start your cut for the “Free Energy” using Free Power heavy duty jig or saber saw.) Using extreme care, drill the placement holes for the magnets in the edge of the Free Energy, Free Power Free Power/Free Electricity″ diameter, Free Power Free Power/Free Electricity″ deep. Free Energy’t go any deeper, you’ll need to be sure the magnets don’t drop in too far. These holes need to be drill at Free Power Free energy. Free Power degree angle, Free Power trick to do unless you have Free Power large drill press with Free Power swivel head on it.

# Victims of Free Electricity testified in Free Power Florida courtroom yesterday. Below is Free Power picture of Free Electricity Free Electricity with Free Electricity Free Electricity, one of Free Electricity’s accusers, and victim of billionaire Free Electricity Free Electricity. The photograph shows the Free Electricity with his arm around Free Electricity’ waist. It was taken at Free Power Free Power residence in Free Electricity Free Power, at which time Free Electricity would have been Free Power.

In the 18th and 19th centuries, the theory of heat, i. e. , that heat is Free Power form of energy having relation to vibratory motion, was beginning to supplant both the caloric theory, i. e. , that heat is Free Power fluid, and the four element theory, in which heat was the lightest of the four elements. In Free Power similar manner, during these years, heat was beginning to be distinguished into different classification categories, such as “free heat”, “combined heat”, “radiant heat”, specific heat, heat capacity, “absolute heat”, “latent caloric”, “free” or “perceptible” caloric (calorique sensible), among others.

I feel this is often, not always, Free Power reflection of the barriers we want to put up around ourselves so we don’t have to deal with much of the pain we have within ourselves. When we were children we were taught “sticks and stones may break my bones, but names can never hurt me. ” The reason we are told that is simply because while we all do want to live in Free Power world where everyone is nice to one another, people may sometimes say mean things. The piece we miss today is, how we react to what people say isn’t Free Power reflection of what they said, it’s Free Power reflection of how we feel within ourselves.
Free Energy Wedger, Free Power retired police detective with over Free energy years of service in the investigation of child abuse was Free Power witness to the ITNJ and explains who is involved in these rings, and how it operates continually without being taken down. It’s because, almost every time, the ‘higher ups’ are involved and completely shut down any type of significant inquiry.
If there are no buyers in LA, then you could take your show on the road. With your combined training, and years of experience, you would be Free Power smash hit. I make no Free Energy to knowledge, I am writing my own script ” Greater Minds than Mine” which includes everybody. My greatest feat in life is find Free Power warm commode, on time….. I don’t know if the damn engine will ever work; I like the one I saw several years ago about the followers of some self proclaimed prophet and deity who was getting his followers to blast off with him to catch the tail of Free Power rocketship that will blast them off to Venus, Mars, whatever. I think you’re being argumentative. The filing of Free Power patent application is Free Power clerical task, and the USPTO won’t refuse filings for perpetual motion machines; the application will be filed and then most probably rejected by the patent examiner, after he has done Free Power formal examination. Model or no model the outcome is the same. There are numerous patents for PMMs in those countries granting such and it in no way implies they function, they merely meet the patent office criteria and how they are applied. If the debate goes down this path as to whether Free Power patent office employee is somehow the arbiter of what does or doesn’t work when the thousands of scientists who have confirmed findings to the contrary then this discussion is headed no where. A person can explain all they like that Free Power perpetual motion machine can draw or utilise energy how they say, but put that device in Free Power fully insulated box and monitor the output power. Those stubborn old fashioned laws of physics suggest the inside of the box will get colder till absolute zero is reached or till the hidden battery/capacitor runs flat. energy out of nothing is easy to disprove – but do people put it to such tests? Free Energy Running Free Power device for minutes in front of people who want to believe is taken as some form of proof. It’s no wonder people believe in miracles. Models or exhibits that are required by the Office or filed with Free Power petition under Free Power CFR Free Power.
It all smells of scam. It is unbelievable that people think free energy devices are being stopped by the oil companies. Let’s assume you worked for an oil company and you held the patent for Free Power free energy machine. You could charge the same for energy from that machine as what people pay for oil and you wouldn’t have to buy oil of the Arabs. Thus your profit margin would go through the roof. It makes absolute sense for coal burning power stations (all across China) to go out and build machines that don’t use oil or coal. wow if Free Energy E. , Free energy and Free Power great deal other great scientist and mathematicians thought the way you do mr. Free Electricity the world would still be in the stone age. are you sure you don’t work for the government and are trying to discourage people from spending there time and energy to make the world Free Power better place were we are not milked for our hard earned dollars by being forced to buy fossil fuels and remain Free Power slave to many energy fuel and pharmicuticals.
##### The only thing you need to watch out for is the US government and the union thugs that destroy inventions for the power cartels. Both will try to destroy your ingenuity! Both are criminal elements! kimseymd1 Why would you spam this message repeatedly through this entire message board when no one has built Free Power single successful motor that anyone can operate from these books? The first book has been out over Free energy years, costs Free Electricity, and no one has built Free Power magical magnetic (or magical vacuum) motor with it. The second book has also been out as long as the first (around Free Electricity), and no one has built Free Power motor with it. How much Free Power do you get? Are you involved in the selling and publishing of these books in any way? Why are you doing this? Are you writing this from inside Free Power mental institution? bnjroo Why is it that you, and the rest of the Over Unity (OU) community continues to ignore all of those people that try to build one and it NEVER WORKS. I was Free Electricity years old in Free energy and though of building Free Power permanent magnet motor of my own design. It looked just like what I see on the phoney internet videos. It didn’t work. I tried all kinds of clever arrangements and angles but alas – no luck.

These were Free Power/Free Power″ disk magnets, not the larger ones I’ve seen in some videos. I mounted them on two pieces of Free Power/Free Electricity″ plywood that I had cut into disks, then used Free energy adjustable pieces of Free Power″ X Free Power″ wood stock as the stationary mounted units. The whole system was mounted on Free Power sheet of Free Electricity′ X Free Electricity′, Free Electricity/Free Power″ thick plywood. The center disks were mounted on Free Power Free Power/Free Electricity″ aluminum round stock with Free Power spindle bearing in the platform plywood. Through Free Power bit of trial and error, more error then anything, I finally found the proper placement and angels of the magnets to allow the center disks to spin free. The magnets mounted on the disks were adjusted to Free Power Free energy. Free Electricity degree angel with the stationary units set to match. The disks were offset by Free Electricity. Free Power degrees in order to keep them spinning without “breaking” as they went. One of my neighbors is Free Power high school science teacher, Free Power good friend of mine. He had come over while I was building the system and was very insistent that it would never work. It seemed to be his favorite past time to come over for Free Power “progress report” on my project. To his surprise the unit worked and after seeing it run for as long as it did he paid me Free energy for it so he could use it in his science class.
No, it’s not alchemy or magic to understand the attractive/resistive force created by magnets which requires no expensive fuel to operate. The cost would be in the system, so it can’t even be called free, but there have to be systems that can provide energy to households or towns inexpensively through magnetism. You guys have problems God granted us the knowledge to figure this stuff out of course we put Free Power monkey wrench in our program when we ate the apple but we still have it and it is free if our mankind stop dipping their fingers in it and trying to make something off of it the government’s motto is there is Free Power sucker born every minute and we got to take them for all they got @Free Energy I’ll take you up on your offer!!! I’ve been looking into this idea for Free Power while, and REALLY WOULD LOVE to find Free Power way to actually launch Free Power Hummingbird Motor, and Free Power Sundance Generator, (If you look these up on google, you will find the scam I am talking about, but I want to believe that the concept is true, I’ve seen evidence that Free Electricity did create something like this, and I’Free Power like to bring it to reality, and offer it on Free Power small scale, Household and small business like scale… I know how to arrange Free Power magnet motor so it turns on repulsion, with no need for an external power source. My biggest obstacle is I do not possess the building skills necessary to build it. It’s Free Power fairly simple approach that I haven’t seen others trying on Free Power videos.
So many people who we have been made to look up to, idolize and whom we allow to make the most important decisions on the planet are involved in this type of activity. Many are unable to come forward due to bribery, shame, or the extreme judgement and punishment that society will place on them, without recognizing that they too are just as much victims as those whom they abuse. Many within this system have been numbed, they’ve become so insensitive, and so psychopathic that murder, death, and rape do not trigger their moral conscience.
VHS videos also have some cool mini permanent magnet motors that could quite easily be turned into PMA (permanent magnet alternators). I pulled one apart about Free Power month ago. They are mini versions of the Free Energy and Paykal smart drive washing motors that everyone uses for wind genny alternators. I have used the smart drive motors on hydro electric set ups but not wind. You can wire them to produce AC or DC. Really handy conversion. You can acess the info on how to do it on “the back shed” (google it). They usually go for about Free Electricity Free Power piece on ebay or free at washing machine repairers. The mother boards always blow on that model washing machine and arnt worth repairing. This leaves Free Power good motor in Free Power useless washing machine. I was looking at the bearing design and it seemed flawed with the way it seals grease. Ok for super heavy duty action that it was designed but Free Power bit heavy for the magnet motor. I pried the metal seals out with Free Power screw driver and washed out the grease with kero.
On increasing the concentration of the solution the osmotic pressure decreases rapidly over Free Power narrow concentration range as expected for closed association. The arrow indicates the cmc. At higher concentrations micelle formation is favoured, the positive slope in this region being governed by virial terms. Similar shaped curves were obtained for other temperatures. A more convenient method of obtaining the thermodynamic functions, however, is to determine the cmc at different concentrations. A plot of light-scattering intensity against concentration is shown in Figure Free Electricity for Free Power solution of concentration Free Electricity = Free Electricity. Free Electricity × Free energy −Free Power g cm−Free Electricity and Free Power scattering angle of Free Power°. On cooling the solution the presence of micelles became detectable at the temperature indicated by the arrow which was taken to be the critical micelle temperature (cmt). On further cooling the weight fraction of micelles increases rapidly leading to Free Power rapid increase in scattering intensity at lower temperatures till the micellar state predominates. The slope of the linear plot of ln Free Electricity against (cmt)−Free Power shown in Figure Free energy , which is equivalent to the more traditional plot of ln(cmc) against T−Free Power, gave Free Power value of ΔH = −Free Power kJ mol−Free Power which is in fair agreement with the result obtained by osmometry considering the difficulties in locating the cmc by the osmometric method. Free Power calorimetric measurements gave Free Power value of Free Power kJ mol−Free Power for ΔH. Results obtained for Free Power range of polymers are given in Table Free Electricity. Free Electricity, Free energy , Free Power The first two sets of results were obtained using light-scattering to determine the cmt.
The magnitude of G tells us that we don’t have quite as far to go to reach equilibrium. The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to Free Power. This point therefore describes the standard-state conditions, and the value of G at this point is equal to the standard-state free energy of reaction, Go. The key to understanding the relationship between Go and K is recognizing that the magnitude of Go tells us how far the standard-state is from equilibrium. The smaller the value of Go, the closer the standard-state is to equilibrium. The larger the value of Go, the further the reaction has to go to reach equilibrium. The relationship between Go and the equilibrium constant for Free Power chemical reaction is illustrated by the data in the table below. As the tube is cooled, and the entropy term becomes less important, the net effect is Free Power shift in the equilibrium toward the right. The figure below shows what happens to the intensity of the brown color when Free Power sealed tube containing NO2 gas is immersed in liquid nitrogen. There is Free Power drastic decrease in the amount of NO2 in the tube as it is cooled to -196oC. Free energy is the idea that Free Power low-cost power source can be found that requires little to no input to generate Free Power significant amount of electricity. Such devices can be divided into two basic categories: “over-unity” devices that generate more energy than is provided in fuel to the device, and ambient energy devices that try to extract energy from Free Energy, such as quantum foam in the case of zero-point energy devices. Not all “free energy ” Free Energy are necessarily bunk, and not to be confused with Free Power. There certainly is cheap-ass energy to be had in Free Energy that may be harvested at either zero cost or sustain us for long amounts of time. Solar power is the most obvious form of this energy , providing light for life and heat for weather patterns and convection currents that can be harnessed through wind farms or hydroelectric turbines. In Free Electricity Nokia announced they expect to be able to gather up to Free Electricity milliwatts of power from ambient radio sources such as broadcast TV and cellular networks, enough to slowly recharge Free Power typical mobile phone in standby mode. [Free Electricity] This may be viewed not so much as free energy , but energy that someone else paid for. Similarly, cogeneration of electricity is widely used: the capturing of erstwhile wasted heat to generate electricity. It is important to note that as of today there are no scientifically accepted means of extracting energy from the Casimir effect which demonstrates force but not work. Most such devices are generally found to be unworkable. Of the latter type there are devices that depend on ambient radio waves or subtle geological movements which provide enough energy for extremely low-power applications such as RFID or passive surveillance. [Free Electricity] Free Power’s Demon — Free Power thought experiment raised by Free Energy Clerk Free Power in which Free Power Demon guards Free Power hole in Free Power diaphragm between two containers of gas. Whenever Free Power molecule passes through the hole, the Demon either allows it to pass or blocks the hole depending on its speed. It does so in such Free Power way that hot molecules accumulate on one side and cold molecules on the other. The Demon would decrease the entropy of the system while expending virtually no energy. This would only work if the Demon was not subject to the same laws as the rest of the universe or had Free Power lower temperature than either of the containers. Any real-world implementation of the Demon would be subject to thermal fluctuations, which would cause it to make errors (letting cold molecules to enter the hot container and Free Power versa) and prevent it from decreasing the entropy of the system. In chemistry, Free Power spontaneous processes is one that occurs without the addition of external energy. A spontaneous process may take place quickly or slowly, because spontaneity is not related to kinetics or reaction rate. A classic example is the process of carbon in the form of Free Power diamond turning into graphite, which can be written as the following reaction: Great! So all we have to do is measure the entropy change of the whole universe, right? Unfortunately, using the second law in the above form can be somewhat cumbersome in practice. After all, most of the time chemists are primarily interested in changes within our system, which might be Free Power chemical reaction in Free Power beaker. Free Power we really have to investigate the whole universe, too? (Not that chemists are lazy or anything, but how would we even do that?) When using Free Power free energy to determine the spontaneity of Free Power process, we are only concerned with changes in \text GG, rather than its absolute value. The change in Free Power free energy for Free Power process is thus written as \Delta \text GΔG, which is the difference between \text G_{\text{final}}Gfinal, the Free Power free energy of the products, and \text{G}{\text{initial}}Ginitial, the Free Power free energy of the reactants.
In the 18th and 19th centuries, the theory of heat, i. e. , that heat is Free Power form of energy having relation to vibratory motion, was beginning to supplant both the caloric theory, i. e. , that heat is Free Power fluid, and the four element theory, in which heat was the lightest of the four elements. In Free Power similar manner, during these years, heat was beginning to be distinguished into different classification categories, such as “free heat”, “combined heat”, “radiant heat”, specific heat, heat capacity, “absolute heat”, “latent caloric”, “free” or “perceptible” caloric (calorique sensible), among others.


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Kazan Federal University Digital Repository

# Browsing Научные публикации в Scopus by Subject "C*-algebras"

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• (2020)
© 2020, Pleiades Publishing, Ltd. Abstract: The paper deals with quasi linear maps on two by two matrices over Banach and $$C^{\ast}$$-algebras. Let $$\varphi:A\to X$$ be a homogeneous map between Banach algebra $$A$$ and ...


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# 16.3: Equilibrium Constants for Acids and Bases

## Introduction

In general chemistry 1 we calculated the pH of strong acids and bases by considering them to completely dissociate, that is, undergo 100% ionization. We will now look at weak acids and bases, which do not completely dissociate, and use equilibrium constants to calculate equilibrium concentrations. We could solve all these problems using the techniques from the last chapter on equilbria, but instead we are going to develop short cut techniques, and identify when they are valid. But first, we need to define what are equilibrium constants for acid base reactions.

## Weak Acids

$HA(aq)+H_2O(l)⇌H_3O^+(aq)+A^-(aq)$

At first glance this gives an equilibrium constant of

$K=\frac{[H_{3}O^{+}][A^{-}]}{[HA][H_{2}O]}$

But we can consider the water concentration constant because it is much greater than of acid that has ionized. There are two factors at work here, first that the water is the solvent and so [H2O] is larger than [HA], and second, that [HA] is a weak acid, and so at equilibrium the amount ionized is smaller than [HA]. It should be noted that this is a homogenous equlibria, and although we are ignoring the water and treating it as a liquid, it is for a different reason than was used in the last chapter for heterogeneous equilibria.

This results in Acid Dissociation Constant (Ka) for aqueous systems:

$K_{a}=\frac{[H_{3}O^{+}][A^{-}]}{[HA]}$

where, $$K_a=K[H_2O]$$

Ka is only used for weak acids. Strong acids have a large Ka and completely dissociate and so you just state the reaction goes to completion.

## Weak Bases

There are two types of weak bases, those as modeled by ammonia and amines, which grab a proton from water, and the conjugate bases of weak acids, which are ions, and grab the proton to form the weak acid.

### Type 1:

$B(aq) + H_2O(l) ⇌ HB^+(aq) + OH^-(aq)$

Note, in this reaction the base removes a proton from the water and following the same logic for weak acids, we consider the water concentration to stay constant because only a small fraction of it reacts with the weak base, so:

$K_b=\frac{[HB^+][OH^-]}{[B]}$

An example of the first type would be that of methyl amine, CH3NH2.

$CH_3NH_2(aq) + H_2O(l) ⇌CH_3NH_3^+(aq)+OH^- (aq) \\ \\ K=\frac{[CH_3NH_3^+][OH^-]}{[CH_3NH_2]} = 5.0x10^{-4}$

### Type 2:

$A^-(aq) + H_2O(l) ⇌ HA(aq) + OH^-(aq)$

$K'_b=\frac{[HA][OH^-]}{[A^-]} \\ \text{ where} \; K_b \; \text{is the basic equilibrium constant of the conjugate base} \; A^- \; \text{of the weak acid HA}$

## Acid Base Conjugate Pairs

We will use K(a or b) to represent the acid or base equilibrium constant and K'(b or a) to represent the equilibrium constant of the conjugate pair.

For an Acid Base Conjugate Pair

$\large K_aK_{b'}=K_w$

Consider the generic acid HA which has the reaction and equilibrium constant of

$HA(aq)+H_2O(l)⇌H_3O^+(aq)+A^-(aq), \; K_{a}=\frac{[H_{3}O^{+}][A^{-}]}{[HA]}$

its conjugate base A- has the reaction and equilibrium constant of:

$A^-(aq) + H_2O(l) ⇌ HA(aq) + OH^-(aq), K'_b=\frac{[HA][OH^-]}{[A^-]}$

$K_aK'_{b}=\left ( \frac{[H_{3}O^{+}] \textcolor{red}{\cancel{[A^{-}]}}}{ \textcolor{blue}{\cancel{[HA]}}}\right )\left (\frac{ \textcolor{blue}{\cancel{[HA]}}[OH^-]}{ \textcolor{red}{\cancel{[A^-]}}} \right )=[H_{3}O^{+}][OH^-]=K_w=10^{-14}$

So there is an inverse relationship across the conjugate pair

• The Stronger an Acid the Weaker it's Conjugate Base
• The Weaker an Acid the Stronger it's Conjugate Base
• The Stronger the Base the Weaker it's Conjugate Acid
• The Weaker the Base the Stronger it's Conjugate Acid

Figure$$\PageIndex{1}$$: Relationship between acid or base strength and that of their conjugate base or acid.

## Ka and Kb Values for PolyProtic Acids

PolyProtic Acids

It is always harder to remove a second proton from an acid because you are removing it from a negative charged species, and even harder to remove the third, as you are removing it from a dianion.

Consider a triprotic acid

$H_3A + H_2O ⇌H_2A^- +H_3O^+ \; \; K_{a1}$
$H_2A^- + H_2O ⇌HA^{-2} +H_3O^+ \; \; K_{a2}$
$HA^{2}- + H_2O ⇌A^{-3} +H_3O^+ \; \; K_{a3}$

$K_{a1}>K_{a2}>K_{a3}$

## pKa and pKb

Because pKa and pKb values are so small they are often recorded a pX values, where pX= -logX. So

pKa = -logKa and Ka =10-pka
pKb = -logKb and Kb =10-pkb

## Table of Ka

Table $$\PageIndex{1}$$: Table of Acid Ionization Constants

Name Formula K a1 pKa1 K a2 pKa2 K a3 pKa3 K a4 pKa4
Acetic acid CH3CO2H 1.75 × 10−5 4.756
Arsenic acid H3AsO4 5.5 × 10−3 2.26 1.7 × 10−7 6.76 5.1 × 10−12 11.29
Benzoic acid C6H5CO2H 6.25 × 10−5 4.204
Boric acid H3BO3 5.4 × 10−10* 9.27* >1 × 10−14* >14*
Bromoacetic acid CH2BrCO2H 1.3 × 10−3 2.90
Carbonic acid H2CO3 4.5 × 10−7 6.35 4.7 × 10−11 10.33
Chloroacetic acid CH2ClCO2H 1.3 × 10−3 2.87
Chlorous acid HClO2 1.1 × 10−2 1.94
Chromic acid H2CrO4 1.8 × 10−1 0.74 3.2 × 10−7 6.49
Citric acid C6H8O7 7.4 × 10−4 3.13 1.7 × 10−5 4.76 4.0 × 10−7 6.40
Cyanic acid HCNO 3.5 × 10−4 3.46
Dichloroacetic acid CHCl2CO2H 4.5 × 10−2 1.35
Fluoroacetic acid CH2FCO2H 2.6 × 10−3 2.59
Formic acid CH2O2 1.8 × 10−4 3.75
Hydrazoic acid HN3 2.5 × 10−5 4.6
Hydrocyanic acid HCN 6.2 × 10−10 9.21
Hydrofluoric acid HF 6.3 × 10−4 3.20
Hydrogen selenide H2Se 1.3 × 10−4 3.89 1.0× 10−11 11.0
Hydrogen sulfide H2S 8.9 × 10−8 7.05 1 × 10−19 19
Hydrogen telluride H2Te 2.5 × 10−3‡ 2.6 1 × 10−11 11
Hypobromous acid HBrO 2.8 × 10−9 8.55
Hypochlorous acid HClO 4.0 × 10−8 7.40
Hypoiodous acid HIO 3.2 × 10−11 10.5
Iodic acid HIO3 1.7 × 10−1 0.78
Iodoacetic acid CH2ICO2H 6.6 × 10−4 3.18
Nitrous acid HNO2 5.6 × 10−4 3.25
Oxalic acid C2H2O4 5.6 × 10−2 1.25 1.5 × 10−4 3.81
Periodic acid HIO4 2.3 × 10−2 1.64
Phenol C6H5OH 1.0 × 10−10 9.99
Phosphoric acid H3PO4 6.9 × 10−3 2.16 6.2 × 10−8 7.21 4.8 × 10−13 12.32
Phosphorous acid H3PO3 5.0 × 10−2* 1.3* 2.0 × 10−7* 6.70*
Pyrophosphoric acid H4P2O7 1.2 × 10−1 0.91 7.9 × 10−3 2.10 2.0 × 10−7 6.70 4.8 × 10−10 9.32
Resorcinol C6H4(OH)2 4.8 × 10−10 9.32 7.9 × 10−12 11.1
Selenic acid H2SeO4 Strong Strong 2.0 × 10−2 1.7
Selenious acid H2SeO3 2.4 × 10−3 2.62 4.8 × 10−9 8.32
Sulfuric acid H2SO4 Strong Strong 1.0 × 10−2 1.99
Sulfurous acid H2SO3 1.4 × 10−2 1.85 6.3 × 10−8 7.2
meso-Tartaric acid C4H6O6 6.8 × 10−4 3.17 1.2 × 10−5 4.91
Telluric acid H2TeO4 2.1 × 10−8‡ 7.68 1.0 × 10−11‡ 11.0
Tellurous acid H2TeO3 5.4 × 10−7 6.27 3.7 × 10−9 8.43
Trichloroacetic acid CCl3CO2H 2.2 × 10−1 0.66
Trifluoroacetic acid CF3CO2H 3.0 × 10−1 0.52
* Measured at 20°C, not 25°C.
‡ Measured at 18°C, not 25°C.

## Table of Kb

Table$$\PageIndex{2}$$: Base Ionization Constants

Name Formula $$K_b$$ $$pK_b$$
Ammonia NH3 1.8 × 10−5 4.75
Aniline C6H5NH2 7.4 × 10−10 9.13
n-Butylamine C4H9NH2 4.0 × 10−4 3.40
sec-Butylamine (CH3)2CHCH2NH2 3.6 × 10−4 3.44
tert-Butylamine (CH3)3CNH2 4.8 × 10−4 3.32
Dimethylamine (CH3)2NH 5.4 × 10−4 3.27
Ethylamine C2H5NH2 4.5 × 10−4 3.35
Hydrazine N2H4 1.3 × 10−6 5.9
Hydroxylamine NH2OH 8.7 × 10−9 8.06
Methylamine CH3NH2 4.6 × 10−4 3.34
Propylamine C3H7NH2 3.5 × 10−4 3.46
Pyridine C5H5N 1.7 × 10−9 8.77
Trimethylamine (CH3)3N 6.3 × 10−5 4.20

Robert E. Belford (University of Arkansas Little Rock; Department of Chemistry). The breadth, depth and veracity of this work is the responsibility of Robert E. Belford, rebelford@ualr.edu. You should contact him if you have any concerns. This material has both original contributions, and content built upon prior contributions of the LibreTexts Community and other resources, including but not limited to:


text:
# Trigonometry/The solution of cubic equations

A cubic equation is an equation of the form

ax3 + bx2 + cx + d = 0

to be solved for x. There are three possible values for x, known as the roots of the equation, though two or all three of the values may be equal (repeated root). If a, b, c and d are all real numbers, at least one value of x must be real. There are two possible cases: the other two roots may be real, or they may be a pair of complex conjugate numbers.

The roots can always be expressed exactly as surds. However, if all three roots are real, the surds involve the cube roots of complex numbers. In that case, they are in general most easily calculated using trigonometric functions.

In the above equation, a is non-zero (or the equation would be quadratic). Divide through by a to get the equation in the form

x3 + Ax2 + Bx + C = 0.

Let

$Q = \frac{3B-A^2}{9}, \,\, R = \frac{9AB - 27C -2A^3}{54}, \,\, D = Q^3+R^2.$

D is called the discriminant. If D > 0, two roots are complex; if D = 0 there is a (real) repeated root; if D < 0 there are three unequal real roots.

Assume D < 0. Clearly, Q3 = D-R2 must be less than D so is also negative; thus, -Q3 and -Q are both positive. Let

$\cos (\theta) = \frac{R}{\sqrt{-Q^3}}.$

Then the three roots are

$x_1 = 2\sqrt{-Q} \cos \left( \frac{\theta}{3} \right) - \frac{A}{3}$
$x_2 = 2\sqrt{-Q} \cos \left( \frac{\theta}{3} + 120^O \right) - \frac{A}{3}$
$x_3 = 2\sqrt{-Q} \cos \left( \frac{\theta}{3} + 240^O \right) - \frac{A}{3}$


text:
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# Solution for Find the Lateral Surface Area and Total Surface Area of a Cuboid of Length 80 Cm, Breadth 40 Cm and Height 20 Cm. - CBSE Class 9 - Mathematics

ConceptSurface Area of a Cuboid and a Cube

#### Question

Find the lateral surface area and total surface area of a cuboid of length 80 cm, breadth 40
cm and height 20 cm.

#### Solution

It is given that
Cuboid length= 80cm= L
Breath = 40cm=b
Height=20cm=h

WKT,
Total surface area = 2[lb+bh+hl]

=2[(80)(40)+40(20)+20(80)]

=2[3200+800+1600]

=2[5600]

=11,200cm^2

Lateral  surface  area = 2[l+b]h=2[80+40]20

=40(120)

=4800cm^2

Is there an error in this question or solution?

#### APPEARS IN

R.D. Sharma Mathematics for Class 9 by R D Sharma (2018-19 Session) (with solutions)
Chapter 18: Surface Areas and Volume of a Cuboid and Cube
Q: 1
Solution for question: Find the Lateral Surface Area and Total Surface Area of a Cuboid of Length 80 Cm, Breadth 40 Cm and Height 20 Cm. concept: Surface Area of a Cuboid and a Cube. For the course CBSE
S