Symbol
Content
symbol Denotes that certain constants and terms are missing out (e.g. for clarity) and that only the important terms are being listed.
symbol
inference x ⊢ y means y is derivable from x. A → B ⊢ ¬B → ¬A. p ⊢ n means that p is a partition of n.
(4,3,1,1) ⊢ 9, symbol entailment
.
A ⊧ B means the sentence A entails the sentence B, that is in every model in which A is true, B is also true. A ⊧ A ∨ ¬A symbol perpendicular x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.
If l ⊥ m and m ⊥ n in the plane, then l || n. orthogonal complement W⊥ means the orthogonal complement of W (where W is a subspace of theinner product space V), the set of all vectors in V orthogonal to every vector in W. Within coprime x ⊥ y means x has no factor greater than 1 in common with y. 34 ⊥ 55. independent A ⊥ B means A is an event whose probability is independent of event B. If A ⊥ B, then P(A|B) = P(A). bottom element , .
⊥ means the smallest element of a lattice. ∀x : x ∧ ⊥ = ⊥ bottom type ⊥ means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the type system. ∀ types T, ⊥ <: T comparability x ⊥ y means that x is comparable to y. {e, π} ⊥ {1, 2, e, 3, π} under set containment.
symbol
top element ⊤ means the largest element of a lattice. ∀x : x ∨ ⊤ = ⊤ top type of top. ∀ types T, T <: ⊤ ⊤ means the top or universal type; every type in the type system of interest is a subtype
T
T (named tee /ˈtiː/,[1] forms/script: T t T
t) is the 20th letter in the ISO basic Latin alphabet. It is the most
commonly used consonant and the second most common letter in the English language.[2]
History
Taw was the last letter of the Western Semitic and Hebrew alphabets. The sound value of Semitic Taw, Greek alphabet Tαυ (Tau), Old Italic and Latin T has remained fairly constant, representing [t] in each of these; and it has also kept its original basic shape in all of these alphabets.
Usage
In English, 't' often denotes the voiceless alveolar plosive (International Phonetic Alphabet and X-SAMPA: /t/), as in 'tea', 'tee', or 'ties'.
Related letters and other similar characters
Τ η : Greek letter Tau Т т : Cyrillic letter Te
Dagger (typography)
"†" redirects here. For the album by Justice, see † (album). "Double dagger" redirects here. For the punk rock band, see Double Dagger. Not to be confused with the Christian cross. A dagger, or obelisk, U+2020 † DAGGER (HTML: † †), is a typographical symbol or glyph. The term "obelisk" derives from Greek ὀβελίζκος(obeliskos), which means "little obelus"; from Ancient Greek: ὀβελός (obelos) meaning "roasting spit".[1] It was originally represented by the ÷ symbol and was first used by the Ancient Greek scholars as critical marks in manuscripts. A double dagger or diesis, U+2021 ‡ DOUBLE DAGGER (HTML: ‡ ‡), is a variant with two handles. Neither should be confused with theChristian cross symbol.
The dagger symbol originated from a variant of the obelus (plural: obeli), originally depicted by a plain line (-) or a line with one or two dots (÷).[2] It represented an iron roasting spit, a dart, or the sharp end of a javelin,[3] symbolizing the skewering or cutting out of dubious matter.[4][5][6]
Three variants of obelus glyphs.
symbol
cover
x <• y means that x is covered by y.
{1, 8} <• {1, 3, 8} among the subsets of {1, 2, …, 10} ordered by containment. subtype T1 <: T2 means that T1 is a subtype of T2. If S <: T and T <: U then S <: U (transitivity).
Sigma
Sigma (upper case Σ, lower case σ, lower case in word-final position ς; Greek ζίγμα) is the eighteenth letter of the Greek alphabet, it is also very representable and carries the 'S' sound. In the system of Greek numerals it has a value of 200. When used at the end of a word, and the word is not allupper case, the final form (ς) is used, e.g. Ὀδυζζεύς (Odysseus) – note the two sigmas in the center of the name, and the word-final sigma at the end.
Etymology
The name of sigma, according to one hypothesis,[1] may continue that of Phoenician Samekh. According to a different theory,[2] its original name may have been "San" (the name today associated with another, obsolete letter), while "Sigma" was a Greek innovation that simply meant "hissing", based on a nominalization of a verb ζίδω (sízō, from earlier *sig-jō, meaning 'I hiss').
Uppercase of esh
The uppercase form of sigma was re-borrowed into the Latin alphabet to serve as the uppercase of modern esh (lowercase: ʃ).
Lunate sigma
In handwritten Greek during the Hellenistic period (4th and 3rd centuries BC), the epigraphic form of Σ was simplified into a C-like shape.[3] It is also found on coins from the fourth century BC onwards.[4] This became the universal standard form of Sigma during late antiquity and the Middle Ages. It is today known as lunate sigma(upper case Ϲ, lower case ϲ), because of its crescent-like shape. It is still widely used in decorative typefaces in Greece, especially in religious and church contexts, as well as in some modern print editions of classical Greek texts. The forms of the Cyrillic letter С (representing /s/) and Coptic letter Ⲥ sima are derived from lunate sigma. A dotted lunate sigma (sigma periestigmenon, encoded at U+03FE Ͼ) was used by Aristarchus of Samothrace as an editorial sign indicating that the line so marked is at an incorrect position. Similarly, an antisigma or reversed sigma (Ͻ) may mark a line that is out of place. A dotted antisigma or dotted reversed sigma (antisigma periestigmenon: Ͽ) may indicate a line after which rearrangements should be made, or to variant readings of uncertain priority.
Proportionality (mathematics)
In mathematics, two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant. The constant is called the coefficient of proportionality or proportionality constant. Alternatively, we can say that one of the variables is proportional to the other.
If one variable is always the product of the other and a constant, the two are said to be directly proportional. x and y are directly proportional if the ratio is constant.
If the product of the two variables is always equal to a constant, the two are said to be inversely proportional. x and y are inversely proportional if the product is constant.
If a linear function transforms 0, a and b into 0, c and d, and if the product a b c d is not zero, we say a and b are proportional to c and d. An equality of two ratios such as no term is zero, is called a proportion. where
The two rectangles with stripes are similar, the ratios of their dimensions are horizontally written within the image. The duplication scale of a striped triangle is obliquely written, in a proportion obtained by inverting two terms of another proportion horizontally written.
When the duplication of a given rectangle preserves its shape, the ratio of the large dimension to the small dimension is a constant number in all the copies, and in the original rectangle. The largest rectangle of the drawing is similar to one or the other rectangle with stripes. From their width to their height, the coefficient is A ratio of their dimensions
horizontally written within the image, at the top or the bottom, determines the common shape of the three similar rectangles. The common diagonal of the similar rectangles divides each rectangle into two
superposable triangles, with two different kinds of stripes. The four striped triangles and the two striped rectangles have a common vertex: the center of an homothetic transformation with a negative ratio – k or , that transforms one triangle and its stripes into another triangle
with the same stripes, enlarged or reduced. The duplication scale of a striped triangle is the proportionality constant between the corresponding sides lengths of the triangles, equal to a positive or In the proportion , the terms a and d are called the extremes, while b and c are ratio obliquely written within the image:
the means, because aand d are the extreme terms of the list (a, b, c, d), while b and c are in the middle of the list. From any proportion, we get another proportion by inverting the extremes or the means. And the product of the extremes equals the product of the means. Within the image, a double arrow indicates two inverted terms of the first proportion. Consider dividing the largest rectangle in two triangles, cutting along the diagonal. If we remove two triangles from either half rectangle, we get one of the plain gray rectangles. Above and below this diagonal, the areas of the two biggest triangles of the drawing are equal, because these triangles are superposable. Above and below the subtracted areas are equal for the same reason. Therefore, the two plain gray rectangles have the same area: a d = b c. Symbol ∝ The mathematical symbol ∝ is used to indicate that two values are proportional. For example, A ∝ B means the variable A is directly proportional to the variable B.
In Unicode this is symbol U+221D. Direct proportionality Given two variables x and y, y is directly proportional to x (x and y vary directly, or x and y are in direct variation) if there is a non-zero constant k such that
The relation is often denoted, using the ∝ symbol, as
and the constant ratio
is called the proportionality constant or constant of proportionality. Examples
If an object travels at a constant speed, then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality.
The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to π.
On a map drawn to scale, the distance between any two points on the map is directly proportional to the distance between the two locations that the points represent, with the constant of proportionality being the scale of the map.
The force acting on a certain object due to gravity is directly proportional to the object's mass; the constant of proportionality between the mass and the force is known as gravitational acceleration.
Properties Since
is equivalent to
it follows that if y is directly proportional to x, with (nonzero) proportionality constant k, then x is also directly proportional to y with proportionality constant 1/k. If y is directly proportional to x, then the graph of y as a function of x will be a straight line passing through the origin with the slope of the line equal to the constant of proportionality: it corresponds to linear growth.
Inverse proportionality The concept of inverse proportionality can be contrasted against direct proportionality. Consider two variables said to be "inversely proportional" to each other. If all other variables are held constant, the magnitude or absolute value of one inversely proportional variable will decrease if the other variable increases, while their product (the constant of proportionality k) is always the same. Formally, two variables are inversely proportional (or varying inversely, or in inverse variation, or in inverse proportion or in reciprocal proportion) if one of the variables is directly proportional with the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant. It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that
The constant can be found by multiplying the original x variable and the original y variable.
As an example, the time taken for a journey is inversely proportional to the speed of travel; the time needed to dig a hole is (approximately) inversely proportional to the number of people digging. The graph of two variables varying inversely on the Cartesian coordinate plane is a hyperbola. The product of the X and Y values of each point on the curve will equal the constant of proportionality (k). Since neither x nor y can equal zero (if k is non-zero), the graph will never cross either axis. Hyperbolic coordinates Main article: Hyperbolic coordinates The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that locates a point on a ray and the constant of inverse proportionality that locates a point on a hyperbola. Exponential and logarithmic proportionality A variable y is exponentially proportional to a variable x, if y is directly proportional to the exponential function of x, that is if there exist nonzero constants k and a
Likewise,
a
variable y is logarithmically
proportional to
a
variable x, if y is directly proportional to the logarithm of x, that is if there exist non-zero constants k and a
Experimental determination To determine experimentally whether two physical quantities are directly proportional, one performs several measurements and plots the resulting data points in a Cartesian coordinate system. If
the points lie on or close to a straight line that passes through the origin (0, 0), then the two variables are probably proportional, with the proportionality constant given by the line's slope.
Plus and minus signs The plus and minus signs (+ and −) are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has been extended to many other meanings, more or less analogous. Plus and minus are Latin terms meaning "more" and "less", respectively. Plus sign The plus sign is a binary operator that indicates addition, as in 2 + 3 = 5. It can also serve as a unary operator that leaves its operand unchanged (+x means the same as x). This notation may be used when it is desired to emphasise the positiveness of a number, especially when contrasting with the negative (+5 versus −5). The plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structures have some operation which is called, or equivalent to, addition. Moreover, the symbolism has been extended to very different operations. Plus can mean:
exclusive or (usually written ⊕): 1 + 1 = 0, 1 + 0 = 1 logical disjunction (usually written ∨): 1 + 1 = 1, 1 + 0 = 1
Minus sign The minus sign has three main uses in mathematics:[5] 1. The subtraction operator: A binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition. 2. Directly in front of a number and when it is not a subtraction operator it means a negative number. For instance −5 is negative 5.
3. A unary operator that acts as an instruction to replace the operand by its opposite. For example, if x is 3, then −x is −3, but if x is −3, then −x is 3. Similarly, −(−2) is equal to 2. All three uses can be referred to as "minus" in everyday speech. In modern US usage, −5 (for example) is normally pronounced "negative five" rather than "minus five". "Minus" may be used by speakers born before 1950, and is still popular in some contexts, but "negative" is usually taught as the only correct reading.[6] In most other parts of the English-speaking world, "minus five" is more common. Textbooks in America encourage −x to be read as "the opposite of x" or even "the additive inverse of x" to avoid giving the impression that −x is necessarily negative.[7] In some contexts, different glyphs are used for these meanings; for instance in the computer language APL a raised minus sign is used in negative numbers (as in 2 − 5 gives −3), but such usage is rare. In mathematics and most programming languages, the rules for the order of operations mean that −52 is equal to −25. Powers bind more strongly than multiplication or division which binds more strongly than addition or subtraction. While strictly speaking, the unary minus is not subtraction, it is given the same place as subtraction. However in some programming languages and Excel in particular, unary operators bind strongest, so in these −5^2 is 25 but 0−5^2 is −25.[8] Use in elementary education Some elementary teachers use raised plus and minus signs before numbers to show they are positive or negative numbers.[9] For example subtracting −5 from 3 might be read as positive three take away negative 5 and be shown as 3 − −5 becomes 3 + 5 = 8, or even as
+
3 − −5 becomes +3 + +5 which is +8
Use as a qualifier In grading systems (such as examination marks), the plus sign indicates a grade one level higher and the minus sign a grade lower. For example, B− ("B minus") is one grade lower than B. Sometimes this is extended to two plus or minus signs; for example A++ is two grades higher than A.
Positive and negative are sometimes abbreviated as +ve and −ve. In mathematics the one-sided limit x→a+ means x approaches a from the right,
and x→a− means x approaches a from the left. For example, when calculating what x−1 is when x approaches 0, because x−1→+∞ when x→0+ but x−1→ −∞ when x→0−. Uses in computing As well as the normal mathematical usage plus and minus may be used for a number of other purposes in computing. Plus and minus signs are often used in tree view on a computer screen to show if a folder is collapsed or not. In some programming languages concatenation of strings is written: "a" + "b" = "ab", although this usage is questioned by some for violating commutativity, a property addition is expected to have. In most programming languages, subtraction and negation are indicated with the ASCII hyphen-minus character, -. In APL a raised minus sign (Unicode U+00AF) is used to denote a negative number, as in ¯3) and in J a negative number is denoted by an underscore, as in _5. In C and some other computer programming languages, two plus signs indicate the increment operator and two minus signs a decrement. For example, x++ means "increment the value of x by one" and x-- means "decrement the value of x by one". By extension, "++" is sometimes used in computing terminology to signify an improvement, as in the name of the language C++. There is no concept of negative zero in mathematics, but in computing −0 may have a separate representation from zero. In the IEEE floating-point standard 1/−0 is negative infinity whereas 1/0 is positive infinity. Other uses In chemistry, the minus sign (rather than an en dash) is used for a single covalent bond between two atoms, in skeletal formula.
Subscripted plus and minus signs are used as diacritics in the International Phonetic Alphabet to indicate advanced or retracted articulations of speech sounds. The minus sign is also used as tone letter in the orthographies
of Dan, Krumen, Karaboro, Mwan, Wan, Yaouré, Wè, Nyabwa and Godié.[10] The Unicode character used for the tone letter (U+02D7) is different from the mathematical minus sign. Character codes
Plus, minus, and hyphen-minus. The Unicode minus sign is designed to be the same length and height as the plus and equals signs. In most fonts these are the same width as digits in order to facilitate the alignment of numbers in tables. The hyphen-minus sign (-) is the ASCII version of the minus sign, and doubles as a hyphen. It is usually shorter in length than the plus sign and sometimes at a different height. It can be used as a substitute for the true minus sign when the character set is limited to ASCII. There is a commercial minus sign (⁒), which looks somewhat like an obelus, at U+2052 (HTML &x2052;). Alternative plus sign
A Jewish tradition that dates from at least the 19th century is to write plus using a symbol like an inverted T. This practice was adopted into Israeli schools (this practice goes back to at least the 1940s)[11] and is still commonplace today in elementary schools (including secular schools) but in fewer secondary schools.[12] It is also used occasionally in books by religious authors, but most books for adults use the
international symbol "+". The usual explanation for this practice is that it avoids the writing of a symbol "+" that looks like a Christian cross.[12] Unicode has this symbol at position U+FB29 "Hebrew letter alternative plus sign" (﬩).[13] See also: up tack.
Character codes
Plus, minus, and hyphen-minus. The Unicode minus sign is designed to be the same length and height as the plus and equals signs. In most fonts these are the same width as digits in order to facilitate the alignment of numbers in tables. The hyphen-minus sign (-) is the ASCII version of the minus sign, and doubles as a hyphen. It is usually shorter in length than the plus sign and sometimes at a different height. It can be used as a substitute for the true minus sign when the character set is limited to ASCII. There is a commercial minus sign (⁒), which looks somewhat like an obelus, at U+2052 (HTML &x2052;). Alternative plus sign
A Jewish tradition that dates from at least the 19th century is to write plus using a symbol like an inverted T. This practice was adopted into Israeli schools (this practice goes back to at least the 1940s)[11] and is still commonplace today in elementary schools (including secular schools) but in fewer secondary schools.[12] It is also used occasionally in books by religious authors, but most books for adults use the international symbol "+". The usual explanation for this practice is that it avoids the
writing of a symbol "+" that looks like a Christian cross.[12] Unicode has this symbol at position U+FB29 "Hebrew letter alternative plus sign" (﬩).[13] Equals sign . Due to technical restrictions, ":=" redirects here. For the computer programming assignment operator, see Assignment (computer programming). For the definition symbol, see List of mathematical symbols#Symbols. For other uses, see Equals (disambiguation).
A well-known equality featuring the equals sign The equals sign, equality sign, or "=" is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde. The equals sign is placed between the things stated to have the same value, as in an equation. It is assigned to the Unicode and ASCII character 003D in hexadecimal, 0061 in decimal. History The etymology of the word equal is from the Latin word aequalis, meaning "uniform, identical, or equal," from aequus "level, even, just."
The first use of an equals sign, equivalent to 14x+15=71 in modern notation. From The
Recorde's introduction of "=" The "=" symbol that is now universally accepted by mathematics for equality was first recorded by Welsh mathematician Robert Recorde in The Whetstone of Witte (1557). The original form of the symbol was much wider than the present form. In his book Recorde explains his design of the "Gemowe lines" (meaning twin lines, from the Latin gemellus):
...to auoide the tediouſe repetition of theſe woordes : is equalle to : I will ſette as I doe often in woorke vſe, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicauſe noe .2. thynges, can be moare equalle. ...to avoid the tedious repetition of these words: "is equal to", I will set (as I do often in work use) a pair of parallels (or Gemowe lines) of one length (thus =), because no two things can be more equal. According to Scotland's University of St Andrews History of Mathematics website:[1] The symbol '=' was not immediately popular. The symbol || was used by some and æ (or œ), from the Latin word aequalis meaning equal, was widely used into the 1700s. Not equal The symbol used to denote inequation (when items are not equal) is a slashed equals sign "≠" (Unicode 2260). In LaTeX, this is done with the "\neq" command. Most programming languages, limiting themselves to the ASCII character set,
use ~=, !=, /=, =/=, or <> to represent their boolean inequality operator. Identity The triple bar symbol "≡" (U+2261) is often used to indicate an identity, a definition (which can also be represented by "≝", U+225D or ":="), or a congruence relation in modular arithmetic. The symbol "≘" can be used to express that an item corresponds to another. Isomorphism The symbol "≅" is often used to indicate isomorphic algebraic structures or congruent geometric figures.]In logic Equality of truth values, i.e. bi-implication or logical equivalence, may be denoted by various symbols including =, ~, and ⇔. In a double-barreled name A possibly unique case of the equals sign in a person's name, specifically in a double-barreled name, was by pioneer aviator Alberto Santos=Dumont, as he is also known to not only have often used an equals sign (=) between his two surnames in place of a hyphen, but also seems to
have personally preferred that practice, to display equal respect for his father's French ethnicity and the Brazilian ethnicity of his mother.[9] Incorrect usage The equals sign can be used incorrectly within a mathematical argument, if used in a manner that connects steps of math in a non-standard way, rather than to show equality. For example, if one were finding the sum, step by step, of the numbers 1, 2, 3, 4, and 5, one might write: 1 + 2 = 3 + 3 = 6 + 4 = 10 + 5 = 15 Structurally, this is shorthand for ([(1 + 2 = 3) + 3 = 6] + 4 = 10) + 5 = 15 but the notation is incorrect, because each part of the equality has a different value. If interpreted strictly as it says, it implies 3 = 6 = 10 = 15 = 15 A correct version of the argument would be 1 + 2 = 3; 3 + 3 = 6; 6 + 4 = 10; 10 + 5 = 15[10]
Inequation In mathematics, an inequation is a statement that an inequality holds between two values.[1] It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific inequality relation. Some examples of inequations are:
Some authors apply the term only to inequations in which the inequality relation is, specifically, not-equal-to (≠).[2] Chains of inequations
A shorthand notation is used for the conjunction of several inequations involving common expressions, by chaining them together. For example, the chain
is shorthand for
nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals x
where n is the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred to using ordinal numbers, as in fourth root,twentieth root, etc. For example:
2 is a square root of 4, since 22 = 4. −2 is also a square root of 4, since (−2)2 = 4.
A real number or complex number has n roots of degree n. While the roots of 0 are not distinct (all equaling 0), the n nth roots of any other real or complex number are all distinct. If n is even and the number is real and positive, one of its nth roots is positive, one is negative, and the rest are complex but not real; if n is even and the number is real and negative, none of the nth roots are real. If n is odd and the number is real, one nth root is real and has the same sign as the number, while the other roots are not real. Roots are usually written using the radical symbol or radix square root, denoting the cube root, or , with or denoting the
denoting the fourth root, and so on. In the expression
, n is called the index,
is the radical sign or radix, and x is called the radicand. When a number is
presented under the radical symbol, it must return only one result like afunction, so a non-negative real root, called the principal nth root, is preferred rather than others. An unresolved root, especially one using the radical symbol, is often referred to as a surd[1] or a radical.[2] Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a radical expression. In calculus, roots are treated as special cases of exponentiation, where the exponent is a fraction:
Turned A
Turned A (capital: Ɐ or Ɐ, lowercase: ɐ or ɒ, math symbol ∀) is a symbol based upon the letter A.
Lowercase ɐ (in two story form) is used in the International Phonetic Alphabet to identify the near-open central vowel. A variant, turned alpha, ɒ, is also used in the IPA as the open back rounded vowel. It was used in the 18th century by Edward Lhuyd and William Pryce as phonetic character for the Cornish language. In their books, both Ɐ and ɐ have been used. [1] It was used in the 19th century by Charles Sanders Peirce as a logical symbol for 'un-American' ("unamerican").[2] The symbol ∀ has the same shape of a capital turned A, sans serif. It is used to represent universal quantification in predicate logic. When it appears in a formula together with a predicate variable, they are referred to as a universal quantifier. In traffic engineering it is used to represent flow, the number of units (vehicles) passing a point in a unit of time.
Bracket
Brackets are tall punctuation marks used in matched pairs within text, to set apart or interject other text. Used unqualified, brackets refer to different types of brackets in different parts of the world and in different contexts.
List of types
( ) — parentheses, round brackets or soft brackets [ ] — square brackets, closed brackets, hard brackets, or brackets (US) { } — braces (UK and US), French brackets, curly brackets, definite brackets, swirly brackets, curly braces, birdie brackets, Scottish brackets, squirrelly brackets, gullwings, seagull, squiggly brackets or fancy brackets ⟨ ⟩ — pointy brackets, angle brackets, triangular brackets, diamond brackets, tuples, or chevrons < > — inequality signs, pointy brackets, or brackets. Sometimes referred to as angle brackets, in [1] such cases as HTML markup. Occasionally known as broken brackets or brokets. ⁒ ⁒; ｢ ｣ — corner brackets
Characters ‹ › and « », known as guillemets or angular quote brackets, are actually quotation mark glyphs used in several European languages.
Ø (disambiguation)
Ø, a Scandinavian letter and vowel [ø], the IPA symbol for a close-mid front rounded vowel Ø, a piece of land in Denmark ∅, a zero in linguistics Slashed zero, a representation of the number 0 (zero) to distinguish it from the letter O , an empty set in mathematical set theory
⌀, the symbol for a diameter
Not to be confused with:
Ф / ф, the Cyrillic letter corresponding to "f" or "ph" Φ / θ, the Greek letter phi, corresponding to "f" or "ph" Θ / θ, the Greek letter theta, corresponding to "th" o, the Plimsoll line used to represent standard states in chemistry.
Summation
Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group (or even monoid). For finite sequences of such elements, summation always produces a welldefined sum (possibly by virtue of the convention for empty sums).
The value of this summation is 5050. It can be found without performing 99 additions, since it can be shown (for instance by mathematical induction) that
for all natural numbers n. More generally, formulas exist for many summations of terms following a regular pattern.
symbol
inference x ⊢ y means y is derivable from x. A → B ⊢ ¬B → ¬A. p ⊢ n means that p is a partition of n.
(4,3,1,1) ⊢ 9, symbol entailment
.
A ⊧ B means the sentence A entails the sentence B, that is in every model in which A is true, B is also true. A ⊧ A ∨ ¬A symbol perpendicular x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.
If l ⊥ m and m ⊥ n in the plane, then l || n. orthogonal complement W⊥ means the orthogonal complement of W (where W is a subspace of theinner product space V), the set of all vectors in V orthogonal to every vector in W. Within coprime x ⊥ y means x has no factor greater than 1 in common with y. 34 ⊥ 55. independent A ⊥ B means A is an event whose probability is independent of event B. If A ⊥ B, then P(A|B) = P(A). bottom element , .
⊥ means the smallest element of a lattice. ∀x : x ∧ ⊥ = ⊥ bottom type ⊥ means the bottom type (a.k.a. the zero type or empty type); bottom is the subtype of every type in the type system. ∀ types T, ⊥ <: T comparability x ⊥ y means that x is comparable to y. {e, π} ⊥ {1, 2, e, 3, π} under set containment.
symbol
top element ⊤ means the largest element of a lattice. ∀x : x ∨ ⊤ = ⊤ top type of top. ∀ types T, T <: ⊤ ⊤ means the top or universal type; every type in the type system of interest is a subtype
T
T (named tee /ˈtiː/,[1] forms/script: T t T
t) is the 20th letter in the ISO basic Latin alphabet. It is the most
commonly used consonant and the second most common letter in the English language.[2]
History
Taw was the last letter of the Western Semitic and Hebrew alphabets. The sound value of Semitic Taw, Greek alphabet Tαυ (Tau), Old Italic and Latin T has remained fairly constant, representing [t] in each of these; and it has also kept its original basic shape in all of these alphabets.
Usage
In English, 't' often denotes the voiceless alveolar plosive (International Phonetic Alphabet and X-SAMPA: /t/), as in 'tea', 'tee', or 'ties'.
Related letters and other similar characters
Τ η : Greek letter Tau Т т : Cyrillic letter Te
Dagger (typography)
"†" redirects here. For the album by Justice, see † (album). "Double dagger" redirects here. For the punk rock band, see Double Dagger. Not to be confused with the Christian cross. A dagger, or obelisk, U+2020 † DAGGER (HTML: † †), is a typographical symbol or glyph. The term "obelisk" derives from Greek ὀβελίζκος(obeliskos), which means "little obelus"; from Ancient Greek: ὀβελός (obelos) meaning "roasting spit".[1] It was originally represented by the ÷ symbol and was first used by the Ancient Greek scholars as critical marks in manuscripts. A double dagger or diesis, U+2021 ‡ DOUBLE DAGGER (HTML: ‡ ‡), is a variant with two handles. Neither should be confused with theChristian cross symbol.
The dagger symbol originated from a variant of the obelus (plural: obeli), originally depicted by a plain line (-) or a line with one or two dots (÷).[2] It represented an iron roasting spit, a dart, or the sharp end of a javelin,[3] symbolizing the skewering or cutting out of dubious matter.[4][5][6]
Three variants of obelus glyphs.
symbol
cover
x <• y means that x is covered by y.
{1, 8} <• {1, 3, 8} among the subsets of {1, 2, …, 10} ordered by containment. subtype T1 <: T2 means that T1 is a subtype of T2. If S <: T and T <: U then S <: U (transitivity).
Sigma
Sigma (upper case Σ, lower case σ, lower case in word-final position ς; Greek ζίγμα) is the eighteenth letter of the Greek alphabet, it is also very representable and carries the 'S' sound. In the system of Greek numerals it has a value of 200. When used at the end of a word, and the word is not allupper case, the final form (ς) is used, e.g. Ὀδυζζεύς (Odysseus) – note the two sigmas in the center of the name, and the word-final sigma at the end.
Etymology
The name of sigma, according to one hypothesis,[1] may continue that of Phoenician Samekh. According to a different theory,[2] its original name may have been "San" (the name today associated with another, obsolete letter), while "Sigma" was a Greek innovation that simply meant "hissing", based on a nominalization of a verb ζίδω (sízō, from earlier *sig-jō, meaning 'I hiss').
Uppercase of esh
The uppercase form of sigma was re-borrowed into the Latin alphabet to serve as the uppercase of modern esh (lowercase: ʃ).
Lunate sigma
In handwritten Greek during the Hellenistic period (4th and 3rd centuries BC), the epigraphic form of Σ was simplified into a C-like shape.[3] It is also found on coins from the fourth century BC onwards.[4] This became the universal standard form of Sigma during late antiquity and the Middle Ages. It is today known as lunate sigma(upper case Ϲ, lower case ϲ), because of its crescent-like shape. It is still widely used in decorative typefaces in Greece, especially in religious and church contexts, as well as in some modern print editions of classical Greek texts. The forms of the Cyrillic letter С (representing /s/) and Coptic letter Ⲥ sima are derived from lunate sigma. A dotted lunate sigma (sigma periestigmenon, encoded at U+03FE Ͼ) was used by Aristarchus of Samothrace as an editorial sign indicating that the line so marked is at an incorrect position. Similarly, an antisigma or reversed sigma (Ͻ) may mark a line that is out of place. A dotted antisigma or dotted reversed sigma (antisigma periestigmenon: Ͽ) may indicate a line after which rearrangements should be made, or to variant readings of uncertain priority.
Proportionality (mathematics)
In mathematics, two variables are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant. The constant is called the coefficient of proportionality or proportionality constant. Alternatively, we can say that one of the variables is proportional to the other.
If one variable is always the product of the other and a constant, the two are said to be directly proportional. x and y are directly proportional if the ratio is constant.
If the product of the two variables is always equal to a constant, the two are said to be inversely proportional. x and y are inversely proportional if the product is constant.
If a linear function transforms 0, a and b into 0, c and d, and if the product a b c d is not zero, we say a and b are proportional to c and d. An equality of two ratios such as no term is zero, is called a proportion. where
The two rectangles with stripes are similar, the ratios of their dimensions are horizontally written within the image. The duplication scale of a striped triangle is obliquely written, in a proportion obtained by inverting two terms of another proportion horizontally written.
When the duplication of a given rectangle preserves its shape, the ratio of the large dimension to the small dimension is a constant number in all the copies, and in the original rectangle. The largest rectangle of the drawing is similar to one or the other rectangle with stripes. From their width to their height, the coefficient is A ratio of their dimensions
horizontally written within the image, at the top or the bottom, determines the common shape of the three similar rectangles. The common diagonal of the similar rectangles divides each rectangle into two
superposable triangles, with two different kinds of stripes. The four striped triangles and the two striped rectangles have a common vertex: the center of an homothetic transformation with a negative ratio – k or , that transforms one triangle and its stripes into another triangle
with the same stripes, enlarged or reduced. The duplication scale of a striped triangle is the proportionality constant between the corresponding sides lengths of the triangles, equal to a positive or In the proportion , the terms a and d are called the extremes, while b and c are ratio obliquely written within the image:
the means, because aand d are the extreme terms of the list (a, b, c, d), while b and c are in the middle of the list. From any proportion, we get another proportion by inverting the extremes or the means. And the product of the extremes equals the product of the means. Within the image, a double arrow indicates two inverted terms of the first proportion. Consider dividing the largest rectangle in two triangles, cutting along the diagonal. If we remove two triangles from either half rectangle, we get one of the plain gray rectangles. Above and below this diagonal, the areas of the two biggest triangles of the drawing are equal, because these triangles are superposable. Above and below the subtracted areas are equal for the same reason. Therefore, the two plain gray rectangles have the same area: a d = b c. Symbol ∝ The mathematical symbol ∝ is used to indicate that two values are proportional. For example, A ∝ B means the variable A is directly proportional to the variable B.
In Unicode this is symbol U+221D. Direct proportionality Given two variables x and y, y is directly proportional to x (x and y vary directly, or x and y are in direct variation) if there is a non-zero constant k such that
The relation is often denoted, using the ∝ symbol, as
and the constant ratio
is called the proportionality constant or constant of proportionality. Examples
If an object travels at a constant speed, then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality.
The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to π.
On a map drawn to scale, the distance between any two points on the map is directly proportional to the distance between the two locations that the points represent, with the constant of proportionality being the scale of the map.
The force acting on a certain object due to gravity is directly proportional to the object's mass; the constant of proportionality between the mass and the force is known as gravitational acceleration.
Properties Since
is equivalent to
it follows that if y is directly proportional to x, with (nonzero) proportionality constant k, then x is also directly proportional to y with proportionality constant 1/k. If y is directly proportional to x, then the graph of y as a function of x will be a straight line passing through the origin with the slope of the line equal to the constant of proportionality: it corresponds to linear growth.
Inverse proportionality The concept of inverse proportionality can be contrasted against direct proportionality. Consider two variables said to be "inversely proportional" to each other. If all other variables are held constant, the magnitude or absolute value of one inversely proportional variable will decrease if the other variable increases, while their product (the constant of proportionality k) is always the same. Formally, two variables are inversely proportional (or varying inversely, or in inverse variation, or in inverse proportion or in reciprocal proportion) if one of the variables is directly proportional with the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant. It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that
The constant can be found by multiplying the original x variable and the original y variable.
As an example, the time taken for a journey is inversely proportional to the speed of travel; the time needed to dig a hole is (approximately) inversely proportional to the number of people digging. The graph of two variables varying inversely on the Cartesian coordinate plane is a hyperbola. The product of the X and Y values of each point on the curve will equal the constant of proportionality (k). Since neither x nor y can equal zero (if k is non-zero), the graph will never cross either axis. Hyperbolic coordinates Main article: Hyperbolic coordinates The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that locates a point on a ray and the constant of inverse proportionality that locates a point on a hyperbola. Exponential and logarithmic proportionality A variable y is exponentially proportional to a variable x, if y is directly proportional to the exponential function of x, that is if there exist nonzero constants k and a
Likewise,
a
variable y is logarithmically
proportional to
a
variable x, if y is directly proportional to the logarithm of x, that is if there exist non-zero constants k and a
Experimental determination To determine experimentally whether two physical quantities are directly proportional, one performs several measurements and plots the resulting data points in a Cartesian coordinate system. If
the points lie on or close to a straight line that passes through the origin (0, 0), then the two variables are probably proportional, with the proportionality constant given by the line's slope.
Plus and minus signs The plus and minus signs (+ and −) are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has been extended to many other meanings, more or less analogous. Plus and minus are Latin terms meaning "more" and "less", respectively. Plus sign The plus sign is a binary operator that indicates addition, as in 2 + 3 = 5. It can also serve as a unary operator that leaves its operand unchanged (+x means the same as x). This notation may be used when it is desired to emphasise the positiveness of a number, especially when contrasting with the negative (+5 versus −5). The plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structures have some operation which is called, or equivalent to, addition. Moreover, the symbolism has been extended to very different operations. Plus can mean:
exclusive or (usually written ⊕): 1 + 1 = 0, 1 + 0 = 1 logical disjunction (usually written ∨): 1 + 1 = 1, 1 + 0 = 1
Minus sign The minus sign has three main uses in mathematics:[5] 1. The subtraction operator: A binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition. 2. Directly in front of a number and when it is not a subtraction operator it means a negative number. For instance −5 is negative 5.
3. A unary operator that acts as an instruction to replace the operand by its opposite. For example, if x is 3, then −x is −3, but if x is −3, then −x is 3. Similarly, −(−2) is equal to 2. All three uses can be referred to as "minus" in everyday speech. In modern US usage, −5 (for example) is normally pronounced "negative five" rather than "minus five". "Minus" may be used by speakers born before 1950, and is still popular in some contexts, but "negative" is usually taught as the only correct reading.[6] In most other parts of the English-speaking world, "minus five" is more common. Textbooks in America encourage −x to be read as "the opposite of x" or even "the additive inverse of x" to avoid giving the impression that −x is necessarily negative.[7] In some contexts, different glyphs are used for these meanings; for instance in the computer language APL a raised minus sign is used in negative numbers (as in 2 − 5 gives −3), but such usage is rare. In mathematics and most programming languages, the rules for the order of operations mean that −52 is equal to −25. Powers bind more strongly than multiplication or division which binds more strongly than addition or subtraction. While strictly speaking, the unary minus is not subtraction, it is given the same place as subtraction. However in some programming languages and Excel in particular, unary operators bind strongest, so in these −5^2 is 25 but 0−5^2 is −25.[8] Use in elementary education Some elementary teachers use raised plus and minus signs before numbers to show they are positive or negative numbers.[9] For example subtracting −5 from 3 might be read as positive three take away negative 5 and be shown as 3 − −5 becomes 3 + 5 = 8, or even as
+
3 − −5 becomes +3 + +5 which is +8
Use as a qualifier In grading systems (such as examination marks), the plus sign indicates a grade one level higher and the minus sign a grade lower. For example, B− ("B minus") is one grade lower than B. Sometimes this is extended to two plus or minus signs; for example A++ is two grades higher than A.
Positive and negative are sometimes abbreviated as +ve and −ve. In mathematics the one-sided limit x→a+ means x approaches a from the right,
and x→a− means x approaches a from the left. For example, when calculating what x−1 is when x approaches 0, because x−1→+∞ when x→0+ but x−1→ −∞ when x→0−. Uses in computing As well as the normal mathematical usage plus and minus may be used for a number of other purposes in computing. Plus and minus signs are often used in tree view on a computer screen to show if a folder is collapsed or not. In some programming languages concatenation of strings is written: "a" + "b" = "ab", although this usage is questioned by some for violating commutativity, a property addition is expected to have. In most programming languages, subtraction and negation are indicated with the ASCII hyphen-minus character, -. In APL a raised minus sign (Unicode U+00AF) is used to denote a negative number, as in ¯3) and in J a negative number is denoted by an underscore, as in _5. In C and some other computer programming languages, two plus signs indicate the increment operator and two minus signs a decrement. For example, x++ means "increment the value of x by one" and x-- means "decrement the value of x by one". By extension, "++" is sometimes used in computing terminology to signify an improvement, as in the name of the language C++. There is no concept of negative zero in mathematics, but in computing −0 may have a separate representation from zero. In the IEEE floating-point standard 1/−0 is negative infinity whereas 1/0 is positive infinity. Other uses In chemistry, the minus sign (rather than an en dash) is used for a single covalent bond between two atoms, in skeletal formula.
Subscripted plus and minus signs are used as diacritics in the International Phonetic Alphabet to indicate advanced or retracted articulations of speech sounds. The minus sign is also used as tone letter in the orthographies
of Dan, Krumen, Karaboro, Mwan, Wan, Yaouré, Wè, Nyabwa and Godié.[10] The Unicode character used for the tone letter (U+02D7) is different from the mathematical minus sign. Character codes
Plus, minus, and hyphen-minus. The Unicode minus sign is designed to be the same length and height as the plus and equals signs. In most fonts these are the same width as digits in order to facilitate the alignment of numbers in tables. The hyphen-minus sign (-) is the ASCII version of the minus sign, and doubles as a hyphen. It is usually shorter in length than the plus sign and sometimes at a different height. It can be used as a substitute for the true minus sign when the character set is limited to ASCII. There is a commercial minus sign (⁒), which looks somewhat like an obelus, at U+2052 (HTML &x2052;). Alternative plus sign
A Jewish tradition that dates from at least the 19th century is to write plus using a symbol like an inverted T. This practice was adopted into Israeli schools (this practice goes back to at least the 1940s)[11] and is still commonplace today in elementary schools (including secular schools) but in fewer secondary schools.[12] It is also used occasionally in books by religious authors, but most books for adults use the
international symbol "+". The usual explanation for this practice is that it avoids the writing of a symbol "+" that looks like a Christian cross.[12] Unicode has this symbol at position U+FB29 "Hebrew letter alternative plus sign" (﬩).[13] See also: up tack.
Character codes
Plus, minus, and hyphen-minus. The Unicode minus sign is designed to be the same length and height as the plus and equals signs. In most fonts these are the same width as digits in order to facilitate the alignment of numbers in tables. The hyphen-minus sign (-) is the ASCII version of the minus sign, and doubles as a hyphen. It is usually shorter in length than the plus sign and sometimes at a different height. It can be used as a substitute for the true minus sign when the character set is limited to ASCII. There is a commercial minus sign (⁒), which looks somewhat like an obelus, at U+2052 (HTML &x2052;). Alternative plus sign
A Jewish tradition that dates from at least the 19th century is to write plus using a symbol like an inverted T. This practice was adopted into Israeli schools (this practice goes back to at least the 1940s)[11] and is still commonplace today in elementary schools (including secular schools) but in fewer secondary schools.[12] It is also used occasionally in books by religious authors, but most books for adults use the international symbol "+". The usual explanation for this practice is that it avoids the
writing of a symbol "+" that looks like a Christian cross.[12] Unicode has this symbol at position U+FB29 "Hebrew letter alternative plus sign" (﬩).[13] Equals sign . Due to technical restrictions, ":=" redirects here. For the computer programming assignment operator, see Assignment (computer programming). For the definition symbol, see List of mathematical symbols#Symbols. For other uses, see Equals (disambiguation).
A well-known equality featuring the equals sign The equals sign, equality sign, or "=" is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde. The equals sign is placed between the things stated to have the same value, as in an equation. It is assigned to the Unicode and ASCII character 003D in hexadecimal, 0061 in decimal. History The etymology of the word equal is from the Latin word aequalis, meaning "uniform, identical, or equal," from aequus "level, even, just."
The first use of an equals sign, equivalent to 14x+15=71 in modern notation. From The
Recorde's introduction of "=" The "=" symbol that is now universally accepted by mathematics for equality was first recorded by Welsh mathematician Robert Recorde in The Whetstone of Witte (1557). The original form of the symbol was much wider than the present form. In his book Recorde explains his design of the "Gemowe lines" (meaning twin lines, from the Latin gemellus):
...to auoide the tediouſe repetition of theſe woordes : is equalle to : I will ſette as I doe often in woorke vſe, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicauſe noe .2. thynges, can be moare equalle. ...to avoid the tedious repetition of these words: "is equal to", I will set (as I do often in work use) a pair of parallels (or Gemowe lines) of one length (thus =), because no two things can be more equal. According to Scotland's University of St Andrews History of Mathematics website:[1] The symbol '=' was not immediately popular. The symbol || was used by some and æ (or œ), from the Latin word aequalis meaning equal, was widely used into the 1700s. Not equal The symbol used to denote inequation (when items are not equal) is a slashed equals sign "≠" (Unicode 2260). In LaTeX, this is done with the "\neq" command. Most programming languages, limiting themselves to the ASCII character set,
use ~=, !=, /=, =/=, or <> to represent their boolean inequality operator. Identity The triple bar symbol "≡" (U+2261) is often used to indicate an identity, a definition (which can also be represented by "≝", U+225D or ":="), or a congruence relation in modular arithmetic. The symbol "≘" can be used to express that an item corresponds to another. Isomorphism The symbol "≅" is often used to indicate isomorphic algebraic structures or congruent geometric figures.]In logic Equality of truth values, i.e. bi-implication or logical equivalence, may be denoted by various symbols including =, ~, and ⇔. In a double-barreled name A possibly unique case of the equals sign in a person's name, specifically in a double-barreled name, was by pioneer aviator Alberto Santos=Dumont, as he is also known to not only have often used an equals sign (=) between his two surnames in place of a hyphen, but also seems to
have personally preferred that practice, to display equal respect for his father's French ethnicity and the Brazilian ethnicity of his mother.[9] Incorrect usage The equals sign can be used incorrectly within a mathematical argument, if used in a manner that connects steps of math in a non-standard way, rather than to show equality. For example, if one were finding the sum, step by step, of the numbers 1, 2, 3, 4, and 5, one might write: 1 + 2 = 3 + 3 = 6 + 4 = 10 + 5 = 15 Structurally, this is shorthand for ([(1 + 2 = 3) + 3 = 6] + 4 = 10) + 5 = 15 but the notation is incorrect, because each part of the equality has a different value. If interpreted strictly as it says, it implies 3 = 6 = 10 = 15 = 15 A correct version of the argument would be 1 + 2 = 3; 3 + 3 = 6; 6 + 4 = 10; 10 + 5 = 15[10]
Inequation In mathematics, an inequation is a statement that an inequality holds between two values.[1] It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific inequality relation. Some examples of inequations are:
Some authors apply the term only to inequations in which the inequality relation is, specifically, not-equal-to (≠).[2] Chains of inequations
A shorthand notation is used for the conjunction of several inequations involving common expressions, by chaining them together. For example, the chain
is shorthand for
nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals x
where n is the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred to using ordinal numbers, as in fourth root,twentieth root, etc. For example:
2 is a square root of 4, since 22 = 4. −2 is also a square root of 4, since (−2)2 = 4.
A real number or complex number has n roots of degree n. While the roots of 0 are not distinct (all equaling 0), the n nth roots of any other real or complex number are all distinct. If n is even and the number is real and positive, one of its nth roots is positive, one is negative, and the rest are complex but not real; if n is even and the number is real and negative, none of the nth roots are real. If n is odd and the number is real, one nth root is real and has the same sign as the number, while the other roots are not real. Roots are usually written using the radical symbol or radix square root, denoting the cube root, or , with or denoting the
denoting the fourth root, and so on. In the expression
, n is called the index,
is the radical sign or radix, and x is called the radicand. When a number is
presented under the radical symbol, it must return only one result like afunction, so a non-negative real root, called the principal nth root, is preferred rather than others. An unresolved root, especially one using the radical symbol, is often referred to as a surd[1] or a radical.[2] Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a radical expression. In calculus, roots are treated as special cases of exponentiation, where the exponent is a fraction:
Turned A
Turned A (capital: Ɐ or Ɐ, lowercase: ɐ or ɒ, math symbol ∀) is a symbol based upon the letter A.
Lowercase ɐ (in two story form) is used in the International Phonetic Alphabet to identify the near-open central vowel. A variant, turned alpha, ɒ, is also used in the IPA as the open back rounded vowel. It was used in the 18th century by Edward Lhuyd and William Pryce as phonetic character for the Cornish language. In their books, both Ɐ and ɐ have been used. [1] It was used in the 19th century by Charles Sanders Peirce as a logical symbol for 'un-American' ("unamerican").[2] The symbol ∀ has the same shape of a capital turned A, sans serif. It is used to represent universal quantification in predicate logic. When it appears in a formula together with a predicate variable, they are referred to as a universal quantifier. In traffic engineering it is used to represent flow, the number of units (vehicles) passing a point in a unit of time.
Bracket
Brackets are tall punctuation marks used in matched pairs within text, to set apart or interject other text. Used unqualified, brackets refer to different types of brackets in different parts of the world and in different contexts.
List of types
( ) — parentheses, round brackets or soft brackets [ ] — square brackets, closed brackets, hard brackets, or brackets (US) { } — braces (UK and US), French brackets, curly brackets, definite brackets, swirly brackets, curly braces, birdie brackets, Scottish brackets, squirrelly brackets, gullwings, seagull, squiggly brackets or fancy brackets ⟨ ⟩ — pointy brackets, angle brackets, triangular brackets, diamond brackets, tuples, or chevrons < > — inequality signs, pointy brackets, or brackets. Sometimes referred to as angle brackets, in [1] such cases as HTML markup. Occasionally known as broken brackets or brokets. ⁒ ⁒; ｢ ｣ — corner brackets
Characters ‹ › and « », known as guillemets or angular quote brackets, are actually quotation mark glyphs used in several European languages.
Ø (disambiguation)
Ø, a Scandinavian letter and vowel [ø], the IPA symbol for a close-mid front rounded vowel Ø, a piece of land in Denmark ∅, a zero in linguistics Slashed zero, a representation of the number 0 (zero) to distinguish it from the letter O , an empty set in mathematical set theory
⌀, the symbol for a diameter
Not to be confused with:
Ф / ф, the Cyrillic letter corresponding to "f" or "ph" Φ / θ, the Greek letter phi, corresponding to "f" or "ph" Θ / θ, the Greek letter theta, corresponding to "th" o, the Plimsoll line used to represent standard states in chemistry.
Summation
Summation is the operation of adding a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group (or even monoid). For finite sequences of such elements, summation always produces a welldefined sum (possibly by virtue of the convention for empty sums).
The value of this summation is 5050. It can be found without performing 99 additions, since it can be shown (for instance by mathematical induction) that
for all natural numbers n. More generally, formulas exist for many summations of terms following a regular pattern.
