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Mathematical symbol: ±

The plus–minus sign, ±, is a mathematical symbol with multiple meanings.

A version of the sign, including also the French word ou ("or"), was used in its mathematical meaning by Albert Girard in 1626, and the sign in its modern form was used as early as 1631, in William Oughtred's Clavis Mathematicae.^[6]

In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the plus and minus signs, + or −, allowing the formula to represent two values or two equations.^[7]

For example, given the equation x2 = 9, one may give the solution as x = ±3. This indicates that the equation has two solutions, each of which may be obtained by replacing this equation by one of the two equations x = +3 or x = −3. Only one of these two replaced equations is true for any valid solution. A common use of this notation is found in the quadratic formula

which describes the two solutions to the quadratic equation ax² + bx + c = 0.

Similarly, the trigonometric identity

can be interpreted as a shorthand for two equations: one with + on both sides of the equation, and one with − on both sides. The two copies of the ± sign in this identity must both be replaced in the same way: it is not valid to replace one of them with + and the other of them with −. In contrast to the quadratic formula example, both of the equations described by this identity are simultaneously valid.

The minus–plus sign, ∓, is generally used in conjunction with the ± sign, in such expressions as x ± y ∓ z, which can be interpreted as meaning x + y − z or x − y + z, but not x + y + z nor x − y − z. The upper − in ∓ is considered to be associated to the + of ± (and similarly for the two lower symbols), even though there is no visual indication of the dependency.

However, the ± sign is generally preferred over the ∓ sign, so if both of them appear in an equation, it is safe to assume that they are linked. On the other hand, if there are two instances of the ± sign in an expression, without a ∓, it is impossible to tell from notation alone whether the intended interpretation is as two or four distinct expressions.

The original expression can be rewritten as x ± (y − z) to avoid confusion, but cases such as the trigonometric identity are most neatly written using the "∓" sign:

which represents the two equations:

Another example where the minus–plus sign appears is

A third related usage is found in this presentation of the formula for the Taylor series of the sine function:

Here, the plus-or-minus sign indicates that the term may be added or subtracted, in this case depending on whether n is odd or even, the rule can be deduced from the first few terms. A more rigorous presentation of the same formula would multiply each term by a factor of (−1), which gives +1 when n is even, and −1 when n is odd. In older texts one occasionally finds (−), which means the same.

When the standard presumption that the plus-or-minus signs all take on the same value of +1 or all −1  is not true, then the line of text that immediately follows the equation must contain a brief description of the actual connection, if any, most often of the form “where the ‘±’ signs are independent” or similar. If a brief, simple description is not possible, the equation must be re-written to provide clarity; e.g. by introducing variables such as s₁, s₂, ... and specifying a value of +1 or −1 separately for each, or some appropriate relation, like s 3 = s 1 ⋅ ( s 2 ) n , {\displaystyle s_{3}=s_{1}\cdot (s_{2})^{n}\,,} or similar.

The use of ± for an approximation is most commonly encountered in presenting the numerical value of a quantity, together with its tolerance or its statistical margin of error.^[2] For example, 5.7 ± 0.2 may be anywhere in the range from 5.5 to 5.9 inclusive. In scientific usage, it sometimes refers to a probability of being within the stated interval, usually corresponding to either 1 or 2 standard deviations (a probability of 68.3% or 95.4% in a normal distribution).

Operations involving uncertain values should always try to preserve the uncertainty—in order to avoid propagation of error. If   n = a ± b , {\displaystyle ~n=a\pm b\;,} any operation of the form   m = f ( n )   {\displaystyle ~m=f(n)~} must return a value of the form   m = c ± d   {\displaystyle ~m=c\pm d~} , where c is f ( n ) {\displaystyle \,f(n)\,} and d is range updated using interval arithmetic.

A percentage may also be used to indicate the error margin. For example, 230 ±10% V refers to a voltage within 10% of either side of 230 V (from 207 V to 253 V inclusive).^{[citation needed]} Separate values for the upper and lower bounds may also be used. For example, to indicate that a value is most likely 5.7, but may be as high as 5.9 or as low as 5.6, one may write 5.7+0.2−0.1.

The symbols ± and ∓ are used in chess notation to denote an advantage for white and black, respectively. However, the more common chess notation would be to only use + and –.^[5] If several different symbols are used together, then the symbols + and − denote a clearer advantage than ± and ∓. When finer evaluation is desired, three pairs of symbols are used: ⩲ and ⩱ for only a slight advantage; ± and ∓ for a significant advantage; and +– and –+ for a potentially winning advantage, in each case for white or black respectively.^[8]

The plus–minus sign resembles the Chinese characters 土 (Radical 32) and 士 (Radical 33), whereas the minus–plus sign resembles 干 (Radical 51).

Letter A with overring

The letter A (a in lower case) represents various (although often very similar) sounds in several languages. It is a separate letter in Danish, Swedish, Norwegian, Finnish, North Frisian, Low Saxon, Walloon, Chamorro, Lule Sami, Pite Sami, Skolt Sami, Southern Sami, Ume Sami, and Greenlandic alphabets. Additionally, it is part of the alphabets used for some Alemannic and Austro-Bavarian dialects of German.^{[citation needed]}

Though A is derived from A by adding an overring, it is considered a separate letter. It developed as a form of semi-ligature of an A with a smaller o above it to denote a long and darker A, a process similar to how the umlaut mark developed from a small e written above certain letters.

The A-sound originally had the same origin as the long /aː/ sound in German Aal and Haar (Scandinavian al, har, English eel, hair).

Historically, the a derives from the Old Norse long /aː/ vowel (spelled with the letter a), but over time, it developed to an [ɔː] sound in most Scandinavian language varieties (in Swedish and Norwegian, it has eventually reached the pronunciation [oː]). Medieval writing often used doubled letters for long vowels, and the vowel continued to be written Aa.

In Old Swedish the use of the ligature AE and of O (originally also a variant of the ligature OE) that represented the sounds [ae] and [o] respectively were gradually replaced by new letters. Instead of using ligatures, a minuscule (that is, lower-case) E was placed above the letters A and O to create new graphemes. They later evolved into the modern letters A and O, where the E was simplified into the two dots now referred to as umlaut. A similar process was used to construct a new grapheme where an "aa" had previously been used. A minuscule O was placed on top of an A to create a new letter. It was first used in print in the Gustav Vasa Bible that was published in 1541 and replaced Aa in the 16th century.^[1]

In an attempt to modernize the orthography, linguists tried to introduce the A to Danish and Norwegian writing in the 19th century. Most people felt no need for the new letter, although the letter group Aa had already been pronounced like A for centuries in Denmark and Norway. Aa was usually treated as a single letter, spoken like the present A when spelling out names or words. Orthography reforms making A official were carried out in Norway in 1917 and in Denmark in 1948. According to Jorgen Norby Jensen, senior consultant at Dansk Sprognaevn, the cause for the change in Denmark was a combination of anti-German and pro-Nordic sentiment.^[2] Danish had been the only language apart from German and Luxembourgish to use capitalized nouns in the last decades, but abolished them at the same occasion.

In a few names of Danish cities or towns, the old spelling has been retained as an option due to local resistance, e.g. Aalborg and Aabenraa; however, Alborg and Abenra are the spellings recommended by the Danish Language Board.^[3] Between 1948 and 2010, the city of Aarhus was officially spelled Arhus. However, the city has changed to the Aa spelling starting 2011, in a controversial decision citing internationalization and web compatibility advantages.

Icelandic and Faroese are the only North Germanic languages not to use the a. The Old Norse letter a is retained, but the sound it now expresses is a diphthong, pronounced [au] in Icelandic and [ɔa] in Faroese. The short variation of Faroese a is pronounced [ɔ], though.

In some place names, the old Aa spelling dominates, more often in Denmark than in Norway (where it has been abolished in official use since 1917). Locals of Aalborg and Aabenraa resist the A, whereas Alesund is rarely seen with Aa spelling. Official rules allow both forms in the most common cases, but A is always correct. A as a word means "small river" in Danish, Swedish, and Norwegian and can be found in place names.

Before 1917, when spelling with the double A was common, some Norwegian place names contained three or four consecutive A letters: for instance Haaa (now Haa, a river) and Blaaaasen (Blaasen, 'the blue ("bla") ridge ("as")').

In family names, the bearer of the name uses Aa or A according to their choice, but since family names are inherited they are resistant to change and the traditional Aa style is often kept. For instance, the last name Aagaard is much more common than Agard. The surname Aa is always spelled with double A, never with the single a. However, given names - which are less commonly inherited - have largely changed to the use of the A. For instance, in Norway more than 12,000 male citizens spell their name Hakon, while only around 2,500 are named Haakon.

Company names are sometimes spelled with the double A by choice, usually in order to convey an impression of old-fashionedness or traditionality. The double A, representing a single sound, is usually kept in initials e.g. for people whose first, middle, and/or last name begins with the double A. Accordingly, a man named "Hans Aagard Hauge" would spell his initials "H. Aa. H." (not "H. A. H." nor "H. A. H."), while a woman named Aase Vestergaard would spell her initials "Aa. V." (not "A. V." nor "A. V.").

Correct alphabetization in Danish and Norwegian places A as the last letter in the alphabet, the sequence being AE, O, A. This is also true for the alternative spelling "Aa". Unless manually corrected, sorting algorithms of programs localised for Danish or Norwegian will place e.g., Aaron after Zorro.

In Danish the correct sorting of aa depends on pronunciation: If the sound is pronounced as one sound it is sorted as A regardless of the sound is 'a' or 'a'; thus, for example, the German city Aachen is listed under A, as well as the Danish city Aabenraa. (This is §3 in the Danish Retskrivningsreglerne.)

In the Swedish and Finnish alphabets, A is sorted after Z, as the third letter from the end, the sequence being A, A, O. This is easiest to remember across the Nordic languages, that Danish and Norwegian follow Z first with E-mutated letters AE and O and then the symbol with a one-stroke diacritic A. Swedish and Finnish follow Z with a one-stroke diacritic A and then a two-stroke (or two-dot) diacritic A, O. A combined Nordic sorting mnemonic is AE, O, A, A, O.

Alternative spellings of the Scandinavian A have become a concern because of globalization, and particularly because of the popularization of the World Wide Web. This is to a large extent due to the fact that prior to the creation of IDNA system around 2005, internet domains containing Scandinavian letters were not recognized by the DNS system, and anyway do not feature on keyboards adapted for other languages. While it is recommended to keep the A intact wherever possible, the next best thing is to use the older, double A spelling (e.g. "www.raade.com" instead of "www.rade.com"). This is because, as previously discussed, the A/Aa indicates a separate sound. If the A is represented as a common A without the overring (e.g. "www.rade.com") there is no indication that the A is supposed to represent another sound entirely. Even so, representing the A as just an A is particularly common in Sweden, as compared to Norway and Denmark, because the spelling Aa has no traditional use there.

Because the Finnish alphabet is derived from the Swedish alphabet, A is carried over, but it has no native Finnish use and is treated as in Swedish. Its usage is limited to loanwords and names of Swedish, Danish or Norwegian origin. In Finland there are many Swedish-speaking as well as many Finnish-speaking people with Swedish surnames, and many Swedish surnames include A. In addition, there are many geographical places in the Finnish coastal areas that have a in their Swedish names, such as Krako and Langnas. The Finnish name for A is ruotsalainen O ("Swedish O"), and is pronounced identically to O, which has the value [o̞].

It is not advised to substitute aa for a in Finnish, as aa is already a common letter combination with the value [aː].

In Emilian-Romagnol, a is used to represent the open-mid back unrounded vowel [ʌ], e.g. Modenese dialect amm, danna [ˈʌmː], [ˈdʌnːa] "man, woman";

e.g. Bolognese dialect Bulaggna, dapp [buˈlʌɲːa] [ˈdʌpː] "Bologna, later".

A was introduced to some eastern local variants of Walloon at the beginning of the 16th century and initially noted the same sound as in Danish. Its use quickly spread to all eastern dialects, but the cultural influence Liege and covered three sounds, a long open o, a long close o or a long a, depending on the local varieties. The use of a single a letter to cover such pronunciations has been embraced by the new pan-Walloon orthography, with one orthography for words regardless of the local phonetic variations. The Walloon use of A became the most popular use outside a Scandinavian language, even being used in the International Phonetic Alphabet drafted by Otto Jespersen.

In standardized writings outside the Liege area, words containing a are written with uh, a or o. For example, the word majhon (house), in the standardized orthography is spelled mojo, mahon, mohone, maujon in dialectal writings.

The Istro-Romanian alphabet is based on the standard Romanian alphabet with three additional letters used to mark sounds specific only to this language: a, l and n.

A and a are also used in the practical orthography of Chamorro, a language indigenous to the people of Northern Mariana Islands and Guam. The Chamorro name for Guam is Guahan, and its capital is called Hagatna.

In Greenlandic, a is not used in native words, but is used in several loanwords from Danish, such as bandoptageri (Danish bandoptager) 'tape recorder'. Like in Danish, a is sorted last in the alphabet.

The letter "A" (U+00C5) is also used throughout the world as the international symbol for the non-SI unit angstrom, a physical unit of length named after the Swedish physicist Anders Jonas Angstrom. It is always upper case in this context (symbols for units named after persons are generally upper-case). The angstrom is a unit of length equal to 10−10 m (one ten-billionth of a meter) or 0.1 nm.

Unicode also has encoded U+212B Å ANGSTROM SIGN. However, that is canonically equivalent to the ordinary letter A. The duplicate encoding at U+212B is due to round-trip mapping compatibility with an East-Asian character encoding, but is otherwise not to be used.^[4]

The logo of the Major League Baseball team known as the Los Angeles Angels is a capital "A" with a halo. Due to the resemblance, some Angels fans stylize the name as "Angels".

The logo of the Stargate series similarly features a stylized A with a circle above it, making it resemble an A as in Stargate; in Norwegian, gate means "riddle".

Cirque du Soleil's Kooza production uses this character in its logo, although it is pronounced by the main singer as a regular "a".

British producer and singer Lapsley uses it in her stage name.

Vuelta al cole… A¡pero con ganas!

Los dA­as van acortando yA parece que el verano ha acabado con la vuelta al cole (y eso que aun le quedan unos dA­as pobre…). Aunque yo confA­o en que quede algAon mes mA¡s de calor y temperaturas agradables :)

DespuA©s de andar un tiempo a medio gas, puesto que los meses de verano son complicados y todos tenemos/queremos hacer muchas cosas en un corto espacio de tiempo… A¡A¡he vuelto!!.

Este verano no he hecho nada en especial. He trabajado (A¡pecado mortal!), he ido unos dA­as a ver a mis padres a Soria, y excepto algAon par de salidas por temas laborales, y alguna tarde dominguera, no me he movido de Donosti.

Aunque eso no significa que no haya estado aprovechando el tiempo al mA¡ximo. Disfrutando de esta ciudad, que tiene mucho que ofrecer, yA preparando diferentes entradas para el blog. Entre ellas, alguna que otra sorpresita… ;)

AquA­ os dejo unas migajas de lo que tengo en borradores.

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Pues a partir de ahora hasta no sA© cuando me voy a encargar de actualizar la pA¡gina.

La mera neta no hallo ni quA© poner. El seA±or Woms, uno de los blogueros mA¡s respetados de la blogosfera recomienda no postear cosas como aEoeAvisar cA³mo me siento, hacer una lista, escribir por escribir (tambiA©n conocido como "escribir a lo pendejo"), hablar de mis inconscienciasaE� y ese tipo de cosa que a nadie le interesan.

Aunque bueno, el RaztambiA©n publicA³ un post en el que hablaba sobre su blog y dijo que A©l lo escribA­a para A©l y le valA­a madre quiA©n lo leyera.

Pero pues a este blog no le valen madre sus lectores. Cuando el ZA³nico me invitA³ a chambear con A©l me dijo que nuestro Aonico objetivo era la conquista de la blogosfera y creo que eso significa en convertirnos en la pagina mA¡s popular del intenet y pues la cosa es encontrar un equilibrio entre escribir las cosas que nos pasar y utilizar la pagina como un diario personal y tambiA©n que sea entretenido y interesante para nuestros lectores.

Bueno, pues tenA­a pensado platicarles que hoy lunes, en menos de una hora, me encontrA© 100 pesos en la bolsa de un pantalA³n (mA­o pues) y luego me encontrA© un monedero de plA¡stico con la cara de la pantera rosa que tenA­a 50 pesos en puras moneditas. Iba en el camiA³n cuando me lo encontrA© y me parA© en medio de todos enseA±ando el monedero y les dije aEoeEy A¿Es de alguien esta fregadera?aE� pero nadie dijo nada y entonces fui con el chofer, pensA© dejA¡rselo por si alguien lo iba a reclamar, pero en cuanto le dije que era dinero lo que tenA­a, el gA¼ey me lo quiso arrebatar y mejor no se o di y ahorita traigo 150 baros que no pensA© traer.

TambiA©n tenA­a pensado platicarles sobre mi fin de semana que casi no estuve con Mar. Quien sabe quA© pasa que la morra anda bien quiA©n sabe quA© modo conmigo, no sA©, anda medio distante y indiferente, como si anduviera en sus dA­as. A¿Se acuerdan del poema que el ZA³nico escribiA³ que se llama La Mujer es el Mar? Pos se lo leA­, segAon yo para ser romA¡ntico y tierno y esas cosas que les gustan a las mujeres. Pero pues la morra se puso bien rara y dijo que se sentA­a mal y se metiA³ a su casa. Mujeres A¿QuiA©n las entiende? Si hay alguien que las entienda le agradecerA­a mucho que me lo explicara porque yo si que no entiendo ni maA­z.

Otra de las cosas que les querA­a platicar es acerca de la selecciA³n y el partido contra Brasil.

No habA­a dicho nada porque la neta no le voy mucho a MA©xico (Al equipo de fAotbol), A³sea, sA­ le voy porque es mi paA­s y porque siempre me gusta ver ganar a cualquier mexicano. Pero hablando en tA©rminos estrictamente futboleros, pues no siento que uno porque nomA¡s naciA³ en algAon lugar tenga que irle a tal equipo, imagA­nense a alguien que sea de Venezuela o algAon otro paA­s que nomA¡s no la arman pal fucho pero que le gusto un resto el fut, pues se la pasarA­a toda la vida frustrado. Yo sA­ tengo esperanzas que algAon dA­a MA©xico haga algo importante dentro del fAotbol internacional, pero de eso a que le vaya, pues es muy diferente (bueno, no tanto).

El chiste es que de los torneos internacionales, la copa Amierdica se encuentra dentro de los mA¡s chafas y mA¡s aburridos. DespuA©s de ver la Eurocopa, ver la copa Amierdica es como ver juegos en cA¡mara lenta, no mamen. Partidazos como Bolivia contra Peru, o Venezuela vs Ecuador son para jetearse instantA¡neamente.

Pues MA©xico no jugA³ chido, se veA­an como lentos, como cansados y se equivocaban bien mucho, y pues Brasil no jugA³ bien tampoco, pero Brasil es Brasil y no por nada han ganado 5 campeonatos mundiales y dos de ellos los jugaron en Guadalajara, asA­ que como buen tapatA­o. Arriba Brasil.