|
|
| Line 1: |
Line 1: |
| {{Other uses|plus-minus (disambiguation)}}
| | Luke is often a superstar during the building plus the profession growth first second to his 3rd stadium record, & , is the evidence. He broken to the picture in 2015 together with his funny mix of straight down-home ease of access, motion picture superstar wonderful appears and lyrics, is scheduled t within a major way. The brand new re about the nation graph and [http://lukebryantickets.sgs-suparco.org concert tickets luke bryan] #2 in the [http://www.ladyhawkshockey.org discount luke bryan tickets] pop [http://okkyunglee.com zac brown band tour dates] maps, building it the next greatest very first during those times of 2014 for a region artist. <br><br>The kid of a , is aware of persistence and perseverance are important elements in relation to an effective occupation- . His to start with recording, Remain Me, generated the Top reaches “All My Buddies Say” and “Country Person,” whilst his energy, Doin’ Factor, found the [http://lukebryantickets.asiapak.net luke bryan sold out] vocalist-3 straight No. 2 single men and women: Else Calling Is usually a Good Thing.”<br><br>During the tumble of 2015, Concert tours: Luke & which in fact had an outstanding selection of , which include Metropolitan. “It’s almost like you are receiving a acceptance to travel to a higher level, affirms these performers that were an element of the Concertsover into a bigger degree of designers.” It twisted among the most successful organized tours in its ten-calendar year background.<br><br>my webpage; [http://www.netpaw.org luke bryan my kinda night tour] |
| <div style="float:right; margin: 0 0 10px 10px; padding:40px; font-size:400%; font-family: Times; background-color: #ddddff; border: 1px solid #aaaaff;">'''{{unicode|±}}'''</div>
| |
| The '''plus-minus sign''' ({{unicode|±}}) is a mathematical symbol with multiple meanings.
| |
| *In [[mathematics]], it generally indicates a choice of exactly two possible values, one of which is the negation of the other.
| |
| *In [[experimental science]]s, the sign commonly indicates the [[confidence interval]] or [[Errors and residuals in statistics|error]] in a measurement, often the [[standard deviation]] or [[standard error]].<ref name="stderror"/> The sign may also represent an inclusive range of values that a reading might have.
| |
| *In [[chess]], the sign indicates a clear advantage for the white player; the complementary sign ∓ indicates the same advantage for the black player.<ref name="chess">{{citation|title=Chess For Dummies|first=James|last=Eade|edition=2nd|publisher=John Wiley & Sons|year=2005|isbn=9780471774334|url=http://books.google.com/books?id=7eZxKNQu-JoC&pg=PA272|page=272}}.</ref>
| |
| The sign is normally pronounced "plus or minus".
| |
| | |
| ==History==
| |
| A version of the sign, including also the French word "ou" (meaning "or") was used in its mathematical meaning by [[Albert Girard]] in 1626, and the sign in its modern form was used as early as [[William Oughtred]]'s ''Clavis Mathematicae'' (1631).<ref>{{citation|title=A History of Mathematical Notations, Volumes 1-2|first=Florian|last=Cajori|authorlink=Florian Cajori|publisher=Dover|year=1928|isbn=9780486677668|url=http://books.google.com/books?id=7juWmvQSTvwC&pg=PA245|page=245}}.</ref>
| |
| | |
| ==Usage==
| |
| ===In mathematics===
| |
| <!-- [[± shorthand]] redirects here -->
| |
| In [[mathematical formula]]s, the {{unicode|±}} symbol may be used to indicate a symbol that may be replaced by either the [[Plus and minus signs|+ or −]] symbols, allowing the formula to represent two values or two equations.
| |
| For example, given the equation ''x''<sup>2</sup> = 1, one may give the solution as ''x'' = ±1. 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'' = +1 or ''x'' = −1. Only one of these two replaced equations is true for any valid solution. This notation is commonly used in the [[quadratic formula]]
| |
| :<math>\displaystyle x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}.</math> | |
| describing the two solutions to the [[quadratic equation]] ''ax''<sup>2</sup> + ''bx'' + ''c'' = 0.
| |
| | |
| Similarly, the [[trigonometric identity]]
| |
| :<math>\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B).\,</math>
| |
| 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.
| |
| | |
| A third related usage is found in this presentation of the formula for the [[Taylor series]] of the sine function:
| |
| :<math>\sin\left( x \right) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \pm \frac{1}{(2n+1)!} x^{2n+1} + \cdots. </math>
| |
| Here, the plus-or-minus sign indicates that the signs of the terms alternate, where (starting the count at 0) the terms with an even index ''n'' are added while those with an odd index are subtracted. A more rigorous presentation of the same formula would multiply each term by a factor of (−1)<sup>''n''</sup>, which gives +1 when ''n'' is even and −1 when ''n'' is odd.
| |
| | |
| ===In statistics===
| |
| The use of {{unicode|±}} for an approximation is most commonly encountered in presenting the numerical value of a quantity together with its [[tolerance (engineering)|tolerance]] or its statistical [[margin of error]].<ref name="stderror">{{citation|title=Standard Deviation, Standard Error: Which 'Standard' Should We Use?|first=George W.|last=Brown|journal=American Journal of Diseases of Children|year=1982|volume=136|issue=10|pages=937–941|doi=10.1001/archpedi.1982.03970460067015}}.</ref> | |
| For example, "{{Val|5.7|0.2}}" denotes a quantity that is specified or estimated to be within 0.2 units of 5.7; it may be anywhere in the range from {{Nowrap|5.7 − 0.2}} (i.e., 5.5) to {{Nowrap|5.7 + 0.2}} (5.9). In scientific usage it sometimes refers to a probability of being within the stated interval, usually corresponding to either 1 or 2 [[standard deviation]]s (a probability of 68.3% or 95.4% in a Normal distribution).
| |
| | |
| A [[percentage]] may also be used to indicate the error margin. For example, {{Nowrap|230 {{unicode|±}} 10% V}} refers to a voltage within 10% of either side of 230 V (207 V to 253 V). 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 could write {{Val|5.7|+0.2|-0.1}}.
| |
| | |
| ===In chess===
| |
| The symbols {{unicode|±}} and {{unicode|∓}} are used in [[Punctuation (chess)#Position evaluation symbols|chess notation]] to denote an advantage for white and black respectively.<ref name="chess"/>
| |
| | |
| ==Minus-plus sign==
| |
| There is another symbol, the '''minus-plus sign''' ({{unicode|∓}}). It is generally used in conjunction with the "{{unicode|±}}" sign, in such expressions as "x {{unicode|±}} y {{unicode|∓}} z", which can be interpreted as meaning "''x'' + ''y'' − ''z''" or/and "''x'' − ''y'' + ''z''", but ''not'' "''x'' + ''y'' + ''z''" nor "''x'' − ''y'' − ''z''". The upper "−" in "{{unicode|∓}}" is considered to be associated to the "+" of "{{unicode|±}}" (and similarly for the two lower symbols) even though there is no visual indication of the dependency. (However, the "{{unicode|±}}" sign is generally preferred over the "{{unicode|∓}}" sign, so if they both appear in an equation it is safe to assume that they are linked. On the other hand, if there are two instances of the "{{unicode|±}}" sign in an expression, 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'' {{unicode|±}} (''y'' − ''z'')" to avoid confusion, but cases such as the trigonometric identity
| |
| | |
| :<math>\cos(A \pm B) = \cos(A) \cos(B) \mp \sin(A) \sin(B) </math> | |
| | |
| are most neatly written using the "{{unicode|∓}}" sign. The trigonometric equation above thus represents the two equations:
| |
| :<math>\cos(A + B) = \cos(A)\cos(B) - \sin(A) \sin(B)\,</math>
| |
| :<math>\cos(A - B) = \cos(A)\cos(B) + \sin(A) \sin(B)\,</math>
| |
| but ''never''
| |
| :<math>\cos(A + B) = \cos(A)\cos(B) + \sin(A) \sin(B)\,</math>
| |
| :<math>\cos(A - B) = \cos(A)\cos(B) - \sin(A) \sin(B)\,</math>
| |
| | |
| because the signs are exclusively alternating.
| |
| | |
| ==Encodings==
| |
| * In Unicode: {{unichar|00B1|PLUS-MINUS SIGN|html=}}
| |
| *In [[ISO 8859-1]], [[ISO 8859-7|-7]], [[ISO 8859-8|-8]], [[ISO 8859-9|-9]], [[ISO 8859-13|-13]], [[ISO 8859-15|-15]], and [[ISO 8859-16|-16]], the plus-minus symbol is given by the code 0xB1<sub>[[hexadecimal|hex]]</sub> Since the first 256 code points of Unicode are identical to the contents of ISO-8859-1 this symbol is also at [[Unicode]] code point U+00B1.
| |
| *The symbol also has a [[Character entity reference|HTML entity]] representation of <code>&plusmn;</code>.
| |
| *On [[Microsoft Windows|Windows]] systems, it may be entered by means of [[Alt code]]s, by holding the ALT key while typing the numbers 0177 or 241 on the [[numeric keypad]].
| |
| *On Unix-like systems, it can be entered by typing the sequence [[Compose key|compose]] + -.
| |
| *On Macintosh systems, it may be entered by pressing option shift = (on the non-numeric keypad).
| |
| *The rarer minus-plus sign ({{Unicode|∓}}) is not generally found in legacy encodings and does not have a named HTML entity but is available in Unicode with codepoint U+2213 and so can be used in HTML using <code>&#x2213;</code> or <code>&#8723;</code>.
| |
| *In [[TeX]] 'plus-or-minus' and 'minus-or-plus' symbols are denoted <code>\pm</code> and <code>\mp</code>, respectively.
| |
| *These characters may also be produced as an underlined or overlined + symbol ( <u>+</u> or {{Overline|+}} ).
| |
| | |
| ==Similar characters==
| |
| The plus-minus sign resembles the [[Chinese character]]s [[wikt:士|士]] and [[wikt:土|土]], whereas the minus-plus sign resembles [[wikt:干|干]].
| |
| | |
| ==See also==
| |
| *[[Plus and minus signs]]
| |
| *[[Table of mathematical symbols]]
| |
| *[[≈]] (approximately equal to)
| |
| *[[Engineering tolerance]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| {{DEFAULTSORT:Plus-Minus Sign}}
| |
| [[Category:Elementary arithmetic]]
| |
| [[Category:Mathematical symbols]]
| |
Luke is often a superstar during the building plus the profession growth first second to his 3rd stadium record, & , is the evidence. He broken to the picture in 2015 together with his funny mix of straight down-home ease of access, motion picture superstar wonderful appears and lyrics, is scheduled t within a major way. The brand new re about the nation graph and concert tickets luke bryan #2 in the discount luke bryan tickets pop zac brown band tour dates maps, building it the next greatest very first during those times of 2014 for a region artist.
The kid of a , is aware of persistence and perseverance are important elements in relation to an effective occupation- . His to start with recording, Remain Me, generated the Top reaches “All My Buddies Say” and “Country Person,” whilst his energy, Doin’ Factor, found the luke bryan sold out vocalist-3 straight No. 2 single men and women: Else Calling Is usually a Good Thing.”
During the tumble of 2015, Concert tours: Luke & which in fact had an outstanding selection of , which include Metropolitan. “It’s almost like you are receiving a acceptance to travel to a higher level, affirms these performers that were an element of the Concertsover into a bigger degree of designers.” It twisted among the most successful organized tours in its ten-calendar year background.
my webpage; luke bryan my kinda night tour