-lcb- -lcb- pp-move-indef -rcb- -rcb- -lcb- -lcb- redirect -pipe- zero -rcb- -rcb- -lcb- -pipe- class = " infobox nowraplinks " style = " width : 20em ; " -pipe- - ! colspan = " 2 " align = " center " style = " font : 10em times ; background : # ccc ; " -pipe- 0 -pipe- - -pipe- colspan = " 2 " -pipe- -lcb- -lcb- numbers -lrb- digits -rrb- -rcb- -rcb- -pipe- - -pipe- -lsb- -lsb- cardinal number -pipe- cardinal -rsb- -rsb- -pipe- 0 , zero , " oh " -lrb- -lcb- -lcb- ipa-en -pipe- ˈoʊ -pipe- pron -rcb- -rcb- -rrb- , nought , naught , nil , null , zilch , nada -pipe- - -pipe- -lsb- -lsb- ordinal number -lrb- linguistics -rrb- -pipe- ordinal -rsb- -rsb- -pipe- -pipe- 0th , -lsb- -lsb- zeroth -rsb- -rsb- , noughth -pipe- - -pipe- -lsb- -lsb- factorization -rsb- -rsb- -pipe- -pipe- < math > 0 < / math > -pipe- - -pipe- -lsb- -lsb- divisor -rsb- -rsb- s -pipe- -pipe- all numbers -pipe- - -pipe- -lsb- -lsb- roman numerals -pipe- roman numeral -rsb- -rsb- -pipe- -pipe- n / a -pipe- - -pipe- -lsb- -lsb- arabic language -pipe- arabic -rsb- -rsb- -pipe- -pipe- style = " font-size : 150 % " -pipe- ٠ -pipe- - -pipe- -lsb- -lsb- bengali language -pipe- bengali -rsb- -rsb- -pipe- -pipe- style = " font-size : 150 % " -pipe- ০ -pipe- - -pipe- -lsb- -lsb- devanagari -pipe- devanāgarī -rsb- -rsb- -pipe- -pipe- style = " font-size : 150 % " -pipe- ० -pipe- - -pipe- -lsb- -lsb- chinese language -pipe- chinese -rsb- -rsb- -pipe- -pipe- 〇 ， 零 -pipe- - -pipe- -lsb- -lsb- japanese numerals -pipe- japanese numeral -rsb- -rsb- -pipe- -pipe- 〇 ， 零 -pipe- - -pipe- -lsb- -lsb- khmer numerals -pipe- khmer -rsb- -rsb- -pipe- -pipe- ០ -pipe- - -pipe- -lsb- -lsb- thai numerals -pipe- thai -rsb- -rsb- -pipe- -pipe- ๐ -pipe- - -pipe- -lsb- -lsb- binary numeral system -pipe- binary -rsb- -rsb- -pipe- -pipe- 0 -pipe- - -pipe- -lsb- -lsb- octal -rsb- -rsb- -pipe- -pipe- 0 -pipe- - -pipe- -lsb- -lsb- duodecimal -rsb- -rsb- -pipe- -pipe- 0 -pipe- - -pipe- -lsb- -lsb- hexadecimal -rsb- -rsb- -pipe- -pipe- 0 -pipe- -rcb- ' "0 "' -lrb- ' " zero " ' -rrb- is both a -lsb- -lsb- number -rsb- -rsb- and the -lsb- -lsb- numerical digit -rsb- -rsb- used to represent that number in -lsb- -lsb- numeral system -pipe- numerals -rsb- -rsb- .
it plays a central role in -lsb- -lsb- mathematics -rsb- -rsb- as the -lsb- -lsb- additive identity -rsb- -rsb- of the -lsb- -lsb- integer -rsb- -rsb- s , -lsb- -lsb- real number -rsb- -rsb- s , and many other -lsb- -lsb- algebra -rsb- -rsb- ic structures .
as a digit , zero is used as a placeholder in -lsb- -lsb- positional notation -pipe- place value systems -rsb- -rsb- .
in the -lsb- -lsb- english language -rsb- -rsb- , zero may also be called ' " oh "' , ' " null "' , ' " nil "' , or ' " nought "' . < ref > -lcb- -lcb- cite book -pipe- editor = catherine soanes -pipe- others = maurice waite , sara hawker -pipe- title = the oxford dictionary , thesaurus and wordpower guide -pipe- format = hardback -pipe- accessdate = 2007-12-21 -pipe- edition = 2nd -pipe- year = 2001 -pipe- publisher = -lsb- -lsb- oxford university press -rsb- -rsb- -pipe- location = -lsb- -lsb- new york -pipe- new york , united states -rsb- -rsb- -pipe- isbn = 978-0-19-860393-3 -rcb- -rcb- < / ref >
<p>
= = as number = = ' "0 "' is the -lsb- -lsb- integer -rsb- -rsb- preceding -lsb- -lsb- 1 -lrb- number -rrb- -pipe- 1 -rsb- -rsb- .
in most systems , 0 was identified before the idea of negative things that go lower than zero was accepted .
-lsb- -lsb- evenness of zero -pipe- zero is an even number -rsb- -rsb- . < ref > -lsb- -lsb- lemma -lrb- mathematics -rrb- -pipe- lemma -rsb- -rsb- b.2.2 , ' ' the integer 0 is even and is not odd " , in -lcb- -lcb- cite book -pipe- first = robert c. -pipe- last = penner -pipe- year = 1999 -pipe- title = discrete mathematics : proof techniques and mathematical structures -pipe- publisher = world scientific -pipe- isbn = 9810240880 -pipe- pages = 34 -rcb- -rcb- < / ref > 0 is neither positive nor negative .
by some definitions 0 is also a -lsb- -lsb- natural number -rsb- -rsb- , and then the only natural number not to be positive .
<p>
zero is a number which quantifies a count or an amount of -lsb- -lsb- empty set -pipe- null -rsb- -rsb- size .
almost all -lsb- -lsb- historian -rsb- -rsb- s omit the -lsb- -lsb- year zero -rsb- -rsb- from the -lsb- -lsb- proleptic gregorian calendar -pipe- proleptic gregorian -rsb- -rsb- and -lsb- -lsb- proleptic julian calendar -pipe- julian calendar -rsb- -rsb- s , but -lsb- -lsb- astronomer -rsb- -rsb- s include it in these same calendars .
however , the phrase -lsb- -lsb- year zero -lrb- political notion -rrb- -pipe- year zero -rsb- -rsb- may be used to describe any event considered so significant that it serves as a new base point in time .
<p>
= = as digit = = -lsb- -lsb- image : text figures 036.svg -pipe- 71px -pipe- left -rsb- -rsb- the modern numerical digit 0 is usually written as a circle , an ellipse , or a rounded rectangle .
in most modern -lsb- -lsb- typeface -rsb- -rsb- s , the height of the 0 character is the same as the other digits .
however , in typefaces with -lsb- -lsb- text figures -rsb- -rsb- , the character is often shorter -lrb- -lsb- -lsb- x-height -rsb- -rsb- -rrb- .
<p>
-lcb- -pipe- align = " right " -pipe- -lsb- -lsb- image : 7-segment cdeg.svg -pipe- 80px -pipe- unusual smaller appearance of zero on seven-segment displays -rsb- -rsb- -pipe- -lsb- -lsb- image : 7-segment abcdef.svg -pipe- 80px -pipe- usual appearance of zero on seven-segment displays -rsb- -rsb- -pipe- -rcb-
<p>
on the -lsb- -lsb- seven-segment display -rsb- -rsb- s of calculators , watches , and household appliances , 0 is usually written with six line segments , though on some historical calculator models it was written with four line segments .
<p>
the value , or ' ' number " , zero is not the same as the ' ' digit " zero , used in -lsb- -lsb- numeral system -rsb- -rsb- s using -lsb- -lsb- positional notation -rsb- -rsb- .
successive positions of digits have higher weights , so inside a numeral the digit zero is used to skip a position and give appropriate weights to the preceding and following digits .
a zero digit is not always necessary in a positional number system , for example , in the number 02 .
<p>
in rare instances , a leading 0 may distinguish a number . this appears in -lsb- -lsb- roulette -rsb- -rsb- in the united states , where ' 00 ' is distinct from ' 0 ' -lrb- a wager on ' 0 ' will not win if the ball lands in ' 00 ' , and vice versa -rrb- . sports where competitors are numbered follow this as well ; a -lsb- -lsb- stock car -rsb- -rsb- numbered ' 07 ' would be considered distinct from one numbered ' 7 ' . this is most common with single-digit numbers .
<p>
= = = distinguishing the digit 0 from the letter o = = = -lsb- -lsb- image : zero o comparison.svg -pipe- a comparison of the letter o and the number 0 . -rsb- -rsb-
<p>
traditionally , many print typefaces made the capital letter -lsb- -lsb- o -rsb- -rsb- more rounded than narrower , elliptical digit 0 . < ref name = " bemer " / > -lsb- -lsb- typewriter -rsb- -rsb- s originally made no distinction in shape between o and 0 ; some models did not even have a separate key for the digit 0 .
the distinction came into prominence on modern character -lsb- -lsb- visual display unit -pipe- displays -rsb- -rsb- . < ref name = " bemer " > r. w. bemer .
" towards standards for handwritten zero and oh : much ado about nothing -lrb- and a letter -rrb- , or a partial dossier on distinguishing between handwritten zero and oh " .
' ' communications of the acm " , volume 10 , issue 8 -lrb- august 1967 -rrb- , pp . & nbsp ; 513 – 518 . < / ref >
<p>
the digit 0 with a dot in the centre seems to have originated as an option on -lsb- -lsb- ibm 3270 -rsb- -rsb- displays .
its appearance has continued with the -lsb- -lsb- microsoft windows -rsb- -rsb- typeface -lsb- -lsb- andalé mono -rsb- -rsb- .
one variation used a short vertical bar instead of the dot .
this could be confused with the -lsb- -lsb- greek alphabet -pipe- greek letter -rsb- -rsb- -lsb- -lsb- theta -rsb- -rsb- on a badly focused display , but in practice there was no confusion because theta was not -lrb- then -rrb- a displayable character and very little used anyway .
<p>
an alternative , the -lsb- -lsb- slashed zero -rsb- -rsb- -lrb- looking similar to the letter o except for the slash -rrb- , was primarily used in hand-written coding sheets before transcription to punched cards or tape , and is also used in old-style -lsb- -lsb- ascii -rsb- -rsb- graphic sets descended from the default typewheel on the -lsb- -lsb- asr-33 teletype -rsb- -rsb- .
this form is similar to the symbol < math > \ emptyset < / math > , or " ∅ " -lrb- -lsb- -lsb- unicode -rsb- -rsb- character u + 2205 -rrb- , representing the -lsb- -lsb- empty set -rsb- -rsb- , as well as to the letter -lsb- -lsb- ø -rsb- -rsb- used in several -lsb- -lsb- north germanic languages -pipe- scandinavian languages -rsb- -rsb- .
some -lsb- -lsb- burroughs corporation -pipe- burroughs -rsb- -rsb- / -lsb- -lsb- unisys -rsb- -rsb- equipment displays a digit 0 with a ' 'reversed " slash .
<p>
the opposing convention that has the letter o ' ' with " a slash and the digit 0 ' ' without " was advocated by share , a prominent -lsb- -lsb- ibm -rsb- -rsb- user group , < ref name = " bemer " / > and recommended by ibm for writing -lsb- -lsb- fortran -pipe- fortran -rsb- -rsb- programs , < ref name = " einarsson " > bo einarsson and yurij shokin .
' ' fortran 90 for the fortran 77 programmer " . -lsb- http : / / www.nsc.liu.se / ~ boein / f77to90 / a7.html appendix 7 : " the historical development of fortran " -rsb- < / ref > and by a few other early mainframe makers ; this is even more problematic for -lsb- -lsb- scandinavia -rsb- -rsb- ns because it means two of their letters collide .
others advocated the opposite convention , < ref name = " bemer " / > including ibm for writing -lsb- -lsb- algol -rsb- -rsb- programs . < ref name = " einarsson " / > another convention used on some early -lsb- -lsb- line printer -rsb- -rsb- s left digit 0 unornamented but added a tail or hook to the capital o so that it resembled an inverted -lsb- -lsb- q -rsb- -rsb- or cursive capital letter-o -lrb- < math > \ mathcal o < / math > -rrb- . < ref name = " bemer " / >
<p>
some fonts designed for use with computers made one of the capital-o – digit-0 pair more rounded and the other more angular -lrb- closer to a rectangle -rrb- .
the -lsb- -lsb- texas instruments ti-99/4a -rsb- -rsb- computer featured a more angular capital o and a more rounded digit 0 , whereas others made the choice the other way around .
-lsb- -lsb- image : deutsches kfz-kennzeichen für behördenfahrzeuge -lrb- nummernbereich 3 -rrb- .jpg -pipe- thumb -pipe- german license plate with slit zeros -rsb- -rsb-
<p>
the typeface used on most -lsb- -lsb- europe -rsb- -rsb- an -lsb- -lsb- vehicle registration plate -rsb- -rsb- s distinguishes the two symbols partially in this manner -lrb- having a more rectangular or wider shape for the capital o than the digit 0 -rrb- , but in several countries a further distinction is made by slitting open the digit 0 on the upper right side -lrb- as in -lsb- -lsb- vehicle registration plates of germany -pipe- german plates -rsb- -rsb- using the " -lsb- -lsb- fe-schrift -pipe- fälschungserschwerende schrift -rsb- -rsb- ' ' , " harder-to-falsify script " -rrb- .
<p>
sometimes the digit 0 is used either exclusively , or not at all , to avoid confusion altogether .
for example , -lsb- http : / / southwest.com / content / travel _ center / retrievecheckindoc.html confirmation numbers -rsb- used by -lsb- -lsb- southwest airlines -rsb- -rsb- use only the capital letters o and i instead of the digits 0 and 1 , while -lsb- -lsb- canadian postal code -rsb- -rsb- s use only the digits 1 and 0 and never the capital letters o and i , although letters and numbers always alternate .
<p>
= = names = = -lcb- -lcb- main -pipe- names for the number 0 -rcb- -rcb- in 976 -lsb- -lsb- muhammad ibn ahmad al-khwarizmi -rsb- -rsb- , in his keys of the sciences , remarked that if , in a calculation , no number appears in the place of tens , a little circle should be used " to keep the rows . "
this circle the arabs called sifr . < ref > -lsb- http : / / www.archive.org / details / ageoffaithahisto012288mbp will durant , < i > ' the story of civilization ' < / i > , volume 4 , < i > ' the age of faith ' < / i > , pp. 241 . -rsb- < / ref >
<p>
the word " ' " zero " " ' came via -lsb- -lsb- french language -pipe- french -rsb- -rsb- ' ' zéro " from -lsb- -lsb- venetian language -pipe- venetian -rsb- -rsb- ' ' zero " , which -lrb- together with " -lsb- -lsb- wikt : cipher -pipe- cipher -rsb- -rsb- " -rrb- came via -lsb- -lsb- italian language -pipe- italian -rsb- -rsb- ' ' zefiro " from arabic صفر , ' ' ṣafira " = " it was empty " , ' ' ṣifr " = " zero " , " -lsb- -lsb- nothing -rsb- -rsb- " . < ref > -lsb- http : / / www.merriam-webster.com / dictionary / zero merriam webster online dictionary -rsb- < / ref >
<p>
italian ' ' zefiro " already meant " west wind " from latin and greek " -lsb- -lsb- anemoi -pipe- zephyrus -rsb- -rsb- " ; this may have influenced the spelling when transcribing arabic ' ' ṣifr " . < ref name = " ifrah " > georges ifrah .
' ' the universal history of numbers : from prehistory to the invention of the computer " .
wiley -lrb- 2000 -rrb- .
isbn 0-471-39340-1 . < / ref > the italian mathematician -lsb- -lsb- fibonacci -rsb- -rsb- -lrb- c.1170-1250 -rrb- , who grew up in arab north africa and is credited with introducing the decimal system to europe , used the term ' ' zephyrum " .
this became ' ' zefiro " in italian , which was contracted to ' ' zero " in venetian .
<p>
as the decimal zero and its new mathematics spread from the arab world to europe in the -lsb- -lsb- middle ages -rsb- -rsb- , words derived from ' ' ṣifr " and ' ' zephyrus " came to refer to calculation , as well as to privileged knowledge and secret codes .
according to ifrah , " in thirteenth-century paris , a ' worthless fellow ' was called a " ... cifre en algorisme " , i.e. , an " arithmetical nothing " . " < ref name = " ifrah " / > from ' ' ṣifr " also came french ' ' chiffre " = " digit " , " figure " , " number " , ' ' chiffrer " = " to calculate or compute " , ' ' chiffré " = " encrypted " .
today , the word in arabic is still ' ' ṣifr " , and cognates of ' ' ṣifr " are common in the languages of europe and southwest asia .
<p>
= = history = = = = = early history = = = by the middle of the -lsb- -lsb- 2nd millennium bce -rsb- -rsb- , the -lsb- -lsb- babylonian mathematics -rsb- -rsb- had a sophisticated -lsb- -lsb- sexagesimal -rsb- -rsb- positional numeral system .
the lack of a positional value -lrb- or zero -rrb- was indicated by a ' ' space " between sexagesimal numerals .
by -lsb- -lsb- 300 bce -rsb- -rsb- , a punctuation symbol -lrb- two slanted wedges -rrb- was co-opted as a -lsb- -lsb- free variables and bound variables -pipe- placeholder -rsb- -rsb- in the same -lsb- -lsb- babylonian numerals -pipe- babylonian system -rsb- -rsb- .
in a tablet unearthed at -lsb- -lsb- kish -lrb- sumer -rrb- -pipe- kish -rsb- -rsb- -lrb- dating from about 700 bce -rrb- , the scribe bêl-bân-aplu wrote his zeros with three hooks , rather than two slanted wedges . < ref name = " multiref1 " > kaplan , robert .
-lrb- 2000 -rrb- .
' ' the nothing that is : a natural history of zero " .
oxford : oxford university press . < / ref >
<p>
the babylonian placeholder was not a true zero because it was not used alone .
nor was it used at the end of a number .
thus numbers like 2 and 120 -lrb- 2 × 60 -rrb- , 3 and 180 -lrb- 3 × 60 -rrb- , 4 and 240 -lrb- 4 × 60 -rrb- , looked the same because the larger numbers lacked a final sexagesimal placeholder .
only context could differentiate them .
<p>
records show that the -lsb- -lsb- ancient greece -pipe- ancient greeks -rsb- -rsb- seemed unsure about the status of zero as a number .
they asked themselves , " how can nothing be something ? " , leading to -lsb- -lsb- philosophy -pipe- philosophical -rsb- -rsb- and , by the medieval period , religious arguments about the nature and existence of zero and the -lsb- -lsb- vacuum -rsb- -rsb- .
the -lsb- -lsb- zeno 's paradoxes -pipe- paradoxes -rsb- -rsb- of -lsb- -lsb- zeno of elea -rsb- -rsb- depend in large part on the uncertain interpretation of zero .
<p>
the concept of zero as a number and not merely a symbol for separation is attributed to -lsb- -lsb- india -rsb- -rsb- where by the -lsb- -lsb- 9th century -rsb- -rsb- ce practical calculations were carried out using zero , which was treated like any other number , even in case of division . < ref name = " bourbaki46 " > bourbaki , nicolas -lrb- 1998 -rrb- .
' ' elements of the history of mathematics " .
berlin , heidelberg , and new york : springer-verlag .
46 .
isbn 3540647678 . < / ref > < ref name = " ebcal " > ' ' britannica concise encyclopedia " -lrb- 2007 -rrb- , entry ' ' algebra " < / ref > the indian scholar -lsb- -lsb- pingala -rsb- -rsb- -lrb- circa -lsb- -lsb- 5th century bce -pipe- 5th -rsb- -rsb- - -lsb- -lsb- 2nd century bce -rsb- -rsb- -rrb- used -lsb- -lsb- binary numeral system -pipe- binary numbers -rsb- -rsb- in the form of short and long syllables -lrb- the latter equal in length to two short syllables -rrb- , making it similar to -lsb- -lsb- morse code -rsb- -rsb- . < ref > -lsb- http : / / home.ica.net / ~ roymanju / binary.htm binary numbers in ancient india -rsb- < / ref > < ref > -lsb- http : / / www.sju.edu / ~ rhall / rhythms / poets / arcadia.pdf math for poets and drummers -rsb- -lrb- pdf , 145kb -rrb- < / ref > he and his contemporary indian scholars used the sanskrit word " -lsb- -lsb- śūnyatā -pipe- śūnya -rsb- -rsb- " to refer to zero or ' ' void " .
<p>
= = = history of zero = = = < ! - this section is linked from -lsb- -lsb- olmec -rsb- -rsb- - > -lsb- -lsb- image : estela c de tres zapotes.jpg -pipe- thumb -pipe- the back of olmec stela c from -lsb- -lsb- tres zapotes -rsb- -rsb- , the second oldest long count date yet discovered .
the numerals 7.16.6.16.18 translate to september , 32 bce -lrb- julian -rrb- .
the glyphs surrounding the date are thought to be one of the few surviving examples of -lsb- -lsb- isthmian script -pipe- epi-olmec script -rsb- -rsb- . -rsb- -rsb-
<p>
the -lsb- -lsb- mesoamerican long count calendar -rsb- -rsb- developed in south-central -lsb- -lsb- mexico -rsb- -rsb- and -lsb- -lsb- central america -rsb- -rsb- required the use of zero as a place-holder within its -lsb- -lsb- vigesimal -rsb- -rsb- -lrb- base-20 -rrb- positional numeral system .
many different glyphs , including this partial -lsb- -lsb- quatrefoil -rsb- -rsb- — -lsb- -lsb- image : maya-g-num-0-inc-v1.svg -rsb- -rsb- — were used as a zero symbol for these long count dates , the earliest of which -lrb- on stela 2 at chiapa de corzo , -lsb- -lsb- chiapas -rsb- -rsb- -rrb- has a date of 36 bce . < ref > no long count date actually using the number 0 has been found before the 3rd century ce , but since the long count system would make no sense without some placeholder , and since mesoamerican glyphs do not typically leave empty spaces , these earlier dates are taken as indirect evidence that the concept of 0 already existed at the time . < / ref > since the eight earliest long count dates appear outside the maya homeland , < ref > diehl , p .
186 < / ref > it is assumed that the use of zero in the americas predated the maya and was possibly the invention of the -lsb- -lsb- olmec -rsb- -rsb- s .
many of the earliest long count dates were found within the olmec heartland , although the olmec civilization ended by the 4th century bce , several centuries before the earliest known long count dates .
<p>
although zero became an integral part of -lsb- -lsb- maya numerals -rsb- -rsb- , it did not influence -lsb- -lsb- old world -rsb- -rsb- numeral systems .
<p>
-lsb- -lsb- quipu -rsb- -rsb- , a knotted cord device , used in the -lsb- -lsb- inca empire -rsb- -rsb- and its predecessor societies in the -lsb- -lsb- andes -pipe- andean -rsb- -rsb- region to record accounting and other digital data , is encoded in a -lsb- -lsb- decimal -pipe- base ten -rsb- -rsb- positional system .
zero is represented by the absence of a knot in the appropriate position .
<p>
the use of a blank on a counting board to represent 0 dated back in india to 4th century bce . < ref > robert temple , ' ' the genius of china , a place for zero " ; isbn 1-85375-292-4 < / ref >
<p>
in -lsb- -lsb- china -rsb- -rsb- , -lsb- -lsb- counting rods -rsb- -rsb- were used for calculation since the -lsb- -lsb- 4th century bc -pipe- 4th century bce -rsb- -rsb- .
chinese mathematicians understood negative numbers and zero , though they had no symbol for the latter , < ref > -lcb- -lcb- citation -pipe- title = sangi o koeta otoko -lrb- the man who exceeded counting rods -rrb- -pipe- last = wáng -pipe- first = qīngxiáng -pipe- isbn = 4-88595-226-3 -pipe- publisher = tōyō shoten -pipe- place = tokyo -pipe- year = 1999 -rcb- -rcb- < / ref > until the work of -lsb- -lsb- song dynasty -rsb- -rsb- mathematician -lsb- -lsb- qin jiushao -rsb- -rsb- in 1247 established a symbol for zero in china . < ref > needham , joseph -lrb- 1986 -rrb- .
' ' science and civilization in china : volume 3 , mathematics and the sciences of the heavens and the earth " .
taipei : caves books , ltd .
page 43 . < / ref > " -lsb- -lsb- the nine chapters on the mathematical art -rsb- -rsb- ' ' , which was mainly composed in the -lsb- -lsb- 1st century -pipe- 1st century ce -rsb- -rsb- , stated " -lsb- when subtracting -rsb- subtract same signed numbers , add differently signed numbers , subtract a positive number from zero to make a negative number , and subtract a negative number from zero to make a positive number . " < ref > the statement in chinese , found in chapter 8 of ' ' the nine chapters on the mathematical art " is 正負術曰 : 同名相除 ， 異名相益 ， 正無入負之 ， 負無入正之 。 其異名相除 ， 同名相益 ， 正無入正之 ， 負無入負之 。 the word 無入 used here , for which ' ' zero " is the standard translation by mathematical historians , literally means : ' ' no entry " .
the full chinese text can be found at -lsb- -lsb- : wikisource : zh : 九章算術 -rsb- -rsb- . < / ref >
<p>
by -lsb- -lsb- 130 -rsb- -rsb- ce , -lsb- -lsb- ptolemy -rsb- -rsb- , influenced by -lsb- -lsb- hipparchus -rsb- -rsb- and the babylonians , was using a symbol for zero -lrb- a small circle with a long overbar -rrb- within a sexagesimal numeral system otherwise using alphabetic -lsb- -lsb- greek numerals -rsb- -rsb- .
because it was used alone , not just as a placeholder , this -lsb- -lsb- greek numerals # hellenistic zero -pipe- hellenistic zero -rsb- -rsb- was perhaps the first documented use of a ' ' number " zero in the old world .
however , the positions were usually limited to the fractional part of a number -lrb- called minutes , seconds , thirds , fourths , etc . -rrb- & mdash ; they were not used for the integral part of a number .
in later -lsb- -lsb- byzantine empire -pipe- byzantine -rsb- -rsb- manuscripts of ptolemy 's ' ' syntaxis mathematica " -lrb- also known as the ' ' almagest " -rrb- , the hellenistic zero had morphed into the greek letter -lsb- -lsb- omicron -rsb- -rsb- -lrb- otherwise meaning 70 -rrb- .
<p>
another zero was used in tables alongside -lsb- -lsb- roman numerals # zero -pipe- roman numerals -rsb- -rsb- by -lsb- -lsb- 525 -rsb- -rsb- -lrb- first known use by -lsb- -lsb- dionysius exiguus -rsb- -rsb- -rrb- , but as a word , ' ' nulla " meaning " nothing , " not as a symbol .
when division produced zero as a remainder , ' ' nihil " , also meaning " nothing , " was used .
these medieval zeros were used by all future medieval -lsb- -lsb- computus -pipe- computists -rsb- -rsb- -lrb- calculators of -lsb- -lsb- easter -rsb- -rsb- -rrb- .
an isolated use of the initial , n , was used in a table of roman numerals by -lsb- -lsb- bede -rsb- -rsb- or a colleague about -lsb- -lsb- 725 -rsb- -rsb- , a zero symbol .
<p>
in 498 ce , indian mathematician and astronomer -lsb- -lsb- aryabhata -rsb- -rsb- stated that " sthanam sthanam dasa gunam " or place to place in ten times in value , which may be the origin of the modern decimal-based place value notation . < ref > ' ' aryabhatiya of aryabhata " , translated by walter eugene clark . < / ref >
<p>
the oldest known text to use a decimal -lsb- -lsb- positional notation -pipe- place-value system -rsb- -rsb- , including a zero , is the jain text from india entitled the ' " lokavibhâga "' , dated 458 ce .
this text uses sanskrit numeral words for the digits , with words such as the sanskrit word for ' ' void " for zero . < ref > ifrah , georges -lrb- 2000 -rrb- , p . & nbsp ; 416 . < / ref > the first known use of special -lsb- -lsb- glyph -rsb- -rsb- s for the decimal digits that includes the indubitable appearance of a symbol for the digit zero , a small circle , appears on a stone inscription found at the -lsb- -lsb- chaturbhuja temple -rsb- -rsb- at -lsb- -lsb- gwalior -rsb- -rsb- in india , dated 876 ce . < ref > -lsb- http : / / www.ams.org / featurecolumn / archive / india-zero.html feature column from the ams < ! - bot generated title - > -rsb- < / ref > < ref > ifrah , georges -lrb- 2000 -rrb- , p . & nbsp ; 400 . < / ref > there are many documents on copper plates , with the same small ' ' o " in them , dated back as far as the sixth century ce , but their authenticity may be doubted . < ref name = " multiref1 " / >
<p>
the hindu-arabic numerals and the positional number system were introduced to the -lsb- -lsb- islamic golden age -pipe- islamic civilization -rsb- -rsb- by -lsb- -lsb- al-khwarizmi -rsb- -rsb- . < ref > -lsb- http : / / www.archive.org / details / ageoffaithahisto012288mbp will durant , < i > ' the story of civilization ' < / i > , volume 4 , < i > ' the age of faith ' < / i > , pp. 241 . -rsb- < / ref > al-khwarizmi 's book on arithmetic synthesized greek and hindu knowledge and also contained his own fundamental contribution to mathematics and science including an explanation of the use of zero .
<p>
it was only centuries later , in the 12th century , that the arabic numeral system was introduced to the -lsb- -lsb- western world -rsb- -rsb- through -lsb- -lsb- latin -rsb- -rsb- translations of his ' ' arithmetic " .
<p>
= = = rules of brahmagupta = = = the rules governing the use of zero appeared for the first time in -lsb- -lsb- brahmagupta -rsb- -rsb- 's book " -lsb- -lsb- brahmasphutasiddhanta -pipe- brahmasputha siddhanta -rsb- -rsb- -lrb- the opening of the universe -rrb- ' ' , < ref name = " brahmagupta " > -lsb- http : / / books.google.com / books ? id = a3caaaaamaaj & printsec = frontcover & dq = brahmagupta ' ' algebra with arithmetic of brahmagupta and bhaskara " -rsb- , translated to english by henry thomas colebrooke , london1817 < / ref > written in -lsb- -lsb- 628 -rsb- -rsb- .
here brahmagupta considers not only zero , but negative numbers , and the algebraic rules for the elementary operations of arithmetic with such numbers .
in some instances , his rules differ from the modern standard .
here are the rules of brahmagupta : < ref name = " brahmagupta " / >
<p>
* the sum of zero and a negative number is negative . * the sum of zero and a positive number is positive . * the sum of zero and zero is zero . * the sum of a positive and a negative is their difference ; or , if their absolute values are equal , zero . * a positive or negative number -lsb- -lsb- division by zero -pipe- when divided by zero -rsb- -rsb- is a fraction with the zero as denominator . * zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator . * zero divided by zero is zero .
<p>
in saying zero divided by zero is zero , brahmagupta differs from the modern position .
mathematicians normally do not assign a value to this , whereas computers and calculators sometimes assign -lsb- -lsb- nan -rsb- -rsb- , which means " not a number . "
moreover , non-zero positive or negative numbers when divided by zero are either assigned no value , or a value of unsigned infinity , positive infinity , or negative infinity .
once again , these assignments are not numbers , and are associated more with computer science than pure mathematics , where in most contexts no assignment is done .
<p>
= = = zero as a decimal digit = = = : ' ' see also : -lsb- -lsb- history of the hindu-arabic numeral system -rsb- -rsb- . "
<p>
positional notation without the use of zero -lrb- using an empty space in tabular arrangements , or the word ' ' kha " " emptiness " -rrb- is known to have been in use in india from the -lsb- -lsb- 6th century -rsb- -rsb- .
the earliest certain use of zero as a ' ' decimal " positional digit dates to the -lsb- -lsb- 5th century -rsb- -rsb- mention in the text -lsb- -lsb- lokavibhaga -rsb- -rsb- .
the glyph for the zero digit was written in the shape of a dot , and consequently called " -lsb- -lsb- bindu -rsb- -rsb- " -lrb- " dot " -rrb- .
the dot had been used in greece during earlier ciphered numeral periods .
<p>
the -lsb- -lsb- hindu-arabic numeral system -rsb- -rsb- -lrb- base 10 -rrb- reached europe in the 11th century , via the -lsb- -lsb- iberian peninsula -rsb- -rsb- through -lsb- -lsb- spain -pipe- spanish -rsb- -rsb- -lsb- -lsb- muslim -rsb- -rsb- s , the -lsb- -lsb- moors -rsb- -rsb- , together with knowledge of -lsb- -lsb- astronomy -rsb- -rsb- and instruments like the -lsb- -lsb- astrolabe -rsb- -rsb- , first imported by -lsb- -lsb- pope sylvester ii -pipe- gerbert of aurillac -rsb- -rsb- .
for this reason , the numerals came to be known in europe as " -lsb- -lsb- arabic numerals -rsb- -rsb- " .
the italian mathematician -lsb- -lsb- fibonacci -rsb- -rsb- or leonardo of pisa was instrumental in bringing the system into european mathematics in 1202 , stating :
<p>
<blockquote>
after my father 's appointment by his homeland as state official in the customs house of bugia for the pisan merchants who thronged to it , he took charge ; and in view of its future usefulness and convenience , had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days .
there , following my introduction , as a consequence of marvelous instruction in the art , to the nine digits of the hindus , the knowledge of the art very much appealed to me before all others , and for it i realized that all its aspects were studied in egypt , syria , greece , sicily , and provence , with their varying methods ; and at these places thereafter , while on business .
i pursued my study in depth and learned the give-and-take of disputation .
but all this even , and the algorism , as well as the art of pythagoras , i considered as almost a mistake in respect to the method of the -lsb- -lsb- hinduism -pipe- hindus -rsb- -rsb- -lrb- modus indorum -rrb- .
therefore , embracing more stringently that method of the hindus , and taking stricter pains in its study , while adding certain things from my own understanding and inserting also certain things from the niceties of euclid 's geometric art .
i have striven to compose this book in its entirety as understandably as i could , dividing it into fifteen chapters .
almost everything which i have introduced i have displayed with exact proof , in order that those further seeking this knowledge , with its pre-eminent method , might be instructed , and further , in order that the latin people might not be discovered to be without it , as they have been up to now .
if i have perchance omitted anything more or less proper or necessary , i beg indulgence , since there is no one who is blameless and utterly provident in all things .
the nine indian figures are : 9 8 7 6 5 4 3 2 1 .
with these nine figures , and with the sign 0 ... any number may be written . < ref > sigler , l. , ' ' fibonacci ’ s liber abaci " .
english translation , springer , 2003 . < / ref > < ref > grimm , r.e. , " the autobiography of leonardo pisano " , " -lsb- -lsb- fibonacci quarterly -rsb- -rsb- " ' "11 "' / 1 -lrb- february 1973 -rrb- , pp. 99-104 . < / ref >
</blockquote>
here leonardo of pisa uses the phrase " sign 0 , " indicating it is like a sign to do operations like addition or multiplication .
from the 13th century , manuals on calculation -lrb- adding , multiplying , extracting roots , etc . -rrb- became common in europe where they were called " -lsb- -lsb- algorism -pipe- algorimus -rsb- -rsb- " after the persian mathematician al-khwarizmi .
the most popular was written by -lsb- -lsb- johannes de sacrobosco -rsb- -rsb- , about 1235 and was one of the earliest scientific books to be ' ' printed " in -lsb- -lsb- 1488 -rsb- -rsb- .
until the late 15th century , hindu-arabic numerals seem to have predominated among mathematicians , while merchants preferred to use the -lsb- -lsb- roman numerals -rsb- -rsb- .
in the -lsb- -lsb- 16th century -rsb- -rsb- , they became commonly used in europe .
<p>
= = in mathematics = = = = = elementary algebra = = = the number 0 is the least -lsb- -lsb- negative and non-negative numbers -pipe- non-negative -rsb- -rsb- integer .
the -lsb- -lsb- natural number -rsb- -rsb- following 0 is 1 and no natural number precedes 0 .
the number 0 -lsb- -lsb- natural number # history of natural numbers and the status of zero -pipe- may or may not be considered a natural number -rsb- -rsb- , but it is a whole number and hence a rational number and a real number -lrb- as well as an algebraic number and a complex number -rrb- .
<p>
the number 0 is neither positive nor negative , neither a -lsb- -lsb- prime number -rsb- -rsb- nor a -lsb- -lsb- composite number -rsb- -rsb- , nor is it a -lsb- -lsb- unit -lrb- ring theory -rrb- -pipe- unit -rsb- -rsb- .
it is , however , -lsb- -lsb- parity -lrb- mathematics -rrb- -pipe- even -rsb- -rsb- -lrb- see -lsb- -lsb- evenness of zero -rsb- -rsb- -rrb- .
<p>
the following are some basic -lrb- elementary -rrb- rules for dealing with the number 0 .
these rules apply for any real or -lsb- -lsb- complex number -rsb- -rsb- ' ' x " , unless otherwise stated . * addition : ' ' x " + 0 = 0 + ' ' x " = ' ' x " .
that is , 0 is an -lsb- -lsb- identity element -rsb- -rsb- -lrb- or neutral element -rrb- with respect to -lsb- -lsb- addition -rsb- -rsb- . * subtraction : ' ' x " − 0 = ' ' x " and 0 − ' ' x " = − ' ' x " . * multiplication : ' ' x " · 0 = 0 · ' ' x " = 0 . * division : -lcb- -lcb- frac -pipe- 0 -pipe- ' ' x " -rcb- -rcb- = 0 , for nonzero ' ' x " .
but -lcb- -lcb- frac -pipe- ' ' x " -pipe- 0 -rcb- -rcb- is -lsb- -lsb- defined and undefined -pipe- undefined -rsb- -rsb- , because 0 has no multiplicative inverse , a consequence of the previous rule ; see -lsb- -lsb- division by zero -rsb- -rsb- . * exponentiation : ' ' x " < sup > 0 < / sup > = < sup > ' ' x " < / sup > / < sub > ' ' x " < / sub > = 1 , except that the case ' ' x " = 0 may be left undefined in some contexts ; see -lsb- -lsb- exponentiation # zero to the zero power -pipe- zero to the zero power -rsb- -rsb- .
for all positive real ' ' x " , 0 < sup > ' ' x " < / sup > = 0 .
<p>
the expression -lcb- -lcb- frac -pipe- 0 -pipe- 0 -rcb- -rcb- , which may be obtained in an attempt to determine the limit of an expression of the form -lcb- -lcb- frac -pipe- ' ' f " -lrb- ' ' x " -rrb- -pipe- ' ' g " -lrb- ' ' x " -rrb- -rcb- -rcb- as a result of applying the -lsb- -lsb- limit of a function -pipe- lim -rsb- -rsb- operator independently to both operands of the fraction , is a so-called " -lsb- -lsb- indeterminate form -rsb- -rsb- " .
that does not simply mean that the limit sought is necessarily undefined ; rather , it means that the limit of -lcb- -lcb- frac -pipe- ' ' f " -lrb- ' ' x " -rrb- -pipe- ' ' g " -lrb- ' ' x " -rrb- -rcb- -rcb- , if it exists , must be found by another method , such as -lsb- -lsb- l 'hôpital 's rule -rsb- -rsb- .
<p>
-lsb- -lsb- empty sum -pipe- the sum of 0 numbers -rsb- -rsb- is 0 , and -lsb- -lsb- empty product -pipe- the product of 0 numbers -rsb- -rsb- is 1 .
the -lsb- -lsb- factorial -rsb- -rsb- 0 ! evaluates to 1 .
<p>
= = = other branches of mathematics = = = * in -lsb- -lsb- set theory -rsb- -rsb- , 0 is the -lsb- -lsb- cardinality -rsb- -rsb- of the empty set : if one does not have any apples , then one has 0 apples .
in fact , in certain axiomatic developments of mathematics from set theory , 0 is " -lsb- -lsb- definition -pipe- defined -rsb- -rsb- " to be the empty set .
when this is done , the empty set is the -lsb- -lsb- von neumann cardinal assignment -rsb- -rsb- for a set with no elements , which is the empty set .
the cardinality function , applied to the empty set , returns the empty set as a value , thereby assigning it 0 elements . * also in set theory , 0 is the least -lsb- -lsb- ordinal number -rsb- -rsb- , corresponding to the empty set viewed as a -lsb- -lsb- well-order -pipe- well-ordered set -rsb- -rsb- . * in -lsb- -lsb- propositional calculus -pipe- propositional logic -rsb- -rsb- , 0 may be used to denote the -lsb- -lsb- truth value -rsb- -rsb- false . * in -lsb- -lsb- abstract algebra -rsb- -rsb- , 0 is commonly used to denote a -lsb- -lsb- algebraic structure -pipe- zero element -rsb- -rsb- , which is a -lsb- -lsb- identity element -pipe- neutral element -rsb- -rsb- for addition -lrb- if defined on the structure under consideration -rrb- and an -lsb- -lsb- absorbing element -rsb- -rsb- for multiplication -lrb- if defined -rrb- . * in -lsb- -lsb- lattice -lrb- order -rrb- -pipe- lattice theory -rsb- -rsb- , 0 may denote the -lsb- -lsb- greatest element -pipe- bottom element -rsb- -rsb- of a -lsb- -lsb- lattice -lrb- order -rrb- -pipe- bounded lattice -rsb- -rsb- . * in -lsb- -lsb- category theory -rsb- -rsb- , 0 is sometimes used to denote an -lsb- -lsb- initial and terminal objects -pipe- initial object -rsb- -rsb- of a -lsb- -lsb- category -lrb- mathematics -rrb- -pipe- category -rsb- -rsb- .
<p>
= = = related mathematical terms = = = * a -lsb- -lsb- root of a function -pipe- zero of a function -rsb- -rsb- ' ' f " is a point ' ' x " in the domain of the function such that < span style = " white-space : nowrap ; " > ' ' f " -lrb- ' ' x " -rrb- = 0 < / span > .
when there are finitely many zeros these are called the -lsb- -lsb- root of a function -pipe- roots -rsb- -rsb- of the function .
see also -lsb- -lsb- zero -lrb- complex analysis -rrb- -rsb- -rsb- for zeros of a -lsb- -lsb- holomorphic function -rsb- -rsb- . * the zero function -lrb- or zero map -rrb- on a domain ' ' d " is the -lsb- -lsb- constant function -rsb- -rsb- with 0 as its only possible output value , i.e. , the function ' ' f " defined by < span style = " white-space : nowrap ; " > ' ' f " -lrb- ' ' x " -rrb- = 0 < / span > for all ' ' x " in ' ' d " .
a particular zero function is a -lsb- -lsb- zero morphism -rsb- -rsb- in category theory ; e.g. , a zero map is the identity in the additive group of functions .
the -lsb- -lsb- determinant -rsb- -rsb- on non-invertible -lsb- -lsb- matrix -lrb- mathematics -rrb- -pipe- square matrices -rsb- -rsb- is a zero map .
<p>
= = in science = = = = = physics = = = the value zero plays a special role for many physical quantities .
for some quantities , the zero level is naturally distinguished from all other levels , whereas for others it is more or less arbitrarily chosen .
for example , on the -lsb- -lsb- kelvin -rsb- -rsb- temperature scale , zero is the coldest possible temperature -lrb- -lsb- -lsb- negative temperature -rsb- -rsb- s exist but are not actually colder -rrb- , whereas on the -lsb- -lsb- celsius -rsb- -rsb- scale , zero is arbitrarily defined to be at the -lsb- -lsb- melting point -pipe- freezing point -rsb- -rsb- of water .
measuring sound intensity in -lsb- -lsb- decibel -rsb- -rsb- s or -lsb- -lsb- phon -rsb- -rsb- s , the zero level is arbitrarily set at a reference value — for example , at a value for the threshold of hearing .
in -lsb- -lsb- physics -rsb- -rsb- , the -lsb- -lsb- zero-point energy -rsb- -rsb- is the lowest possible -lsb- -lsb- energy -rsb- -rsb- that a -lsb- -lsb- quantum mechanics -pipe- quantum mechanical -rsb- -rsb- -lsb- -lsb- physical system -rsb- -rsb- may possess and is the energy of the -lsb- -lsb- stationary state -pipe- ground state -rsb- -rsb- of the system .
<p>
= = = chemistry = = = zero has been proposed as the -lsb- -lsb- atomic number -rsb- -rsb- of the theoretical element -lsb- -lsb- tetraneutron -rsb- -rsb- .
it has been shown that a cluster of four -lsb- -lsb- neutron -rsb- -rsb- s may be stable enough to be considered an -lsb- -lsb- atom -rsb- -rsb- in its own right .
this would create an -lsb- -lsb- chemical element -pipe- element -rsb- -rsb- with no -lsb- -lsb- proton -rsb- -rsb- s and no charge on its -lsb- -lsb- atomic nucleus -pipe- nucleus -rsb- -rsb- .
<p>
as early as 1926 , professor andreas von antropoff coined the term -lsb- -lsb- neutronium -rsb- -rsb- for a conjectured form of -lsb- -lsb- matter -rsb- -rsb- made up of neutrons with no protons , which he placed as the chemical element of atomic number zero at the head of his new version of the -lsb- -lsb- periodic table -rsb- -rsb- .
it was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements .
it is at the centre of the -lsb- -lsb- chemical galaxy -rsb- -rsb- -lrb- 2005 -rrb- .
<p>
= = = medical science = = = * -lsb- -lsb- patient zero -rsb- -rsb- is the initial -lsb- -lsb- patient -rsb- -rsb- in the -lsb- -lsb- sample -lrb- statistics -rrb- -pipe- population sample -rsb- -rsb- of an -lsb- -lsb- epidemiology -pipe- epidemiological -rsb- -rsb- investigation .
<p>
= = in computer science = = = = = numbering from 1 or 0 = = = the most common practice throughout human history has been to start counting at one .
nevertheless , in -lsb- -lsb- computer science -rsb- -rsb- zero is often used as the starting point .
for example , in the vast majority of -lsb- -lsb- programming language -rsb- -rsb- s , the elements of an -lsb- -lsb- array data type -pipe- array -rsb- -rsb- are numbered starting from 0 by -lsb- -lsb- default -lrb- computer science -rrb- -pipe- default -rsb- -rsb- .
the popularity of the -lsb- -lsb- c -lrb- programming language -rrb- -pipe- c -rsb- -rsb- programming language in the 1980s has made this approach common .
<p>
one advantage of this convention is in the use of -lsb- -lsb- modular arithmetic -rsb- -rsb- .
every integer is -lsb- -lsb- congruence relation -pipe- congruent -rsb- -rsb- modulo ' ' n " to one of the numbers 0 , 1 , 2 , ... , -lcb- -lcb- nowrap -pipe- ' ' n " − 1 -rcb- -rcb- , where -lcb- -lcb- nowrap -pipe- ' ' n " ≥ 1 -rcb- -rcb- .
because of this , many arithmetic concepts -lrb- such as hash tables -rrb- are more elegantly expressed in code when the array starts at zero .
<p>
a second advantage of zero-based array indexes is that this can improve efficiency under certain circumstances .
to illustrate , suppose ' ' a " is the -lsb- -lsb- memory address -rsb- -rsb- of the first element of an array , and ' ' i " is the index of the desired element .
in this fairly typical scenario , it is quite common to want the address of the desired element .
if the index numbers count from 1 , the desired address is computed by this expression :
<p>
: < math > a + s \ times -lrb- i-1 -rrb- \ , \ ! < / math >
<p>
where ' 's " is the size of each element .
in contrast , if the index numbers count from 0 , the expression becomes this :
<p>
: < math > a + s \ times i \ , \ ! < / math >
<p>
this simpler expression can be more efficient to compute in certain situations .
<p>
note , however , that a language wishing to index arrays from 1 could simply adopt the convention that every " array address " is represented by ' ' a ′ " = ' ' a " – ' 's " ; that is , rather than using the address of the first array element , such a language would use the address of an imaginary element located immediately before the first actual element .
the indexing expression for a 1-based index would be the following :
<p>
: < math > a ' + s \ times i \ , \ ! < / math >
<p>
hence , the efficiency benefit of zero-based indexing is not inherent , but is an artifact of the decision to represent an array by the address of its first element .
<p>
a third advantage is that ranges are more elegantly expressed as the half-open -lsb- -lsb- interval -lrb- mathematics -rrb- -pipe- interval -rsb- -rsb- , -lsb- 0 , " n " -rrb- , as opposed to the closed interval , -lsb- 1 , " n " -rsb- , because empty ranges often occur as input to algorithms -lrb- which would be tricky to express with the closed interval without resorting to obtuse conventions like -lsb- 1,0 -rsb- -rrb- .
on the other hand , closed intervals occur in mathematics because it is often necessary to calculate the terminating condition -lrb- which would be impossible in some cases because the half-open interval is n't always a -lsb- -lsb- closed set -rsb- -rsb- -rrb- which would have a subtraction by 1 everywhere .
<p>
this situation can lead to some confusion in terminology .
in a zero-based indexing scheme , the first element is " element number zero " ; likewise , the twelfth element is " element number eleven " .
therefore , an analogy from the ordinal numbers to the quantity of objects numbered appears ; the highest index of ' ' n " objects will be -lcb- -lcb- nowrap -pipe- ' ' n " – 1 -rcb- -rcb- and referred to the ' ' n " th element .
for this reason , the first element is often referred to as the " -lsb- -lsb- zeroth -rsb- -rsb- " element to avoid confusion .
<p>
= = = null value = = = in databases a field can have a -lsb- -lsb- null -lrb- computer programming -rrb- -pipe- null value -rsb- -rsb- .
this is equivalent to the field not having a value .
for numeric fields it is not the value zero .
for text fields this is not blank nor the empty string .
the presence of null values leads to -lsb- -lsb- ternary logic -pipe- three-valued logic -rsb- -rsb- .
no longer is a condition either true or false , but it can be undetermined .
any computation including a null value delivers a null result .
asking for all records with value 0 or value not equal 0 will not yield all records , since the records with value null are excluded .
<p>
= = = null pointer = = = a " -lsb- -lsb- pointer -lrb- computing -rrb- -pipe- null pointer -rsb- -rsb- " is a pointer in a computer program that does not point to any object or function .
in c , the integer constant 0 is converted into the null pointer at -lsb- -lsb- compile time -rsb- -rsb- when it appears in a pointer context , and so 0 is a standard way to refer to the null pointer in code .
however , the internal representation of the null pointer may be any bit pattern -lrb- possibly different values for different data types -rrb- .
<p>
-lrb- note that on most common architectures , the null pointer is represented internally the same way an integer of the same byte width having a value of zero is represented , so c compilers on such systems perform no actual conversion . -rrb-
<p>
= = = negative zero = = = -lcb- -lcb- main -pipe- signed zero -rcb- -rcb- in mathematics < math > -0 = 0 = + 0 < / math > , both -lsb- -lsb- − 0 -lrb- number -rrb- -pipe- − 0 -rsb- -rsb- and + 0 represent the exact same number , i.e. , there is no “ negative zero ” distinct from zero .
in some -lsb- -lsb- signed number representations -rsb- -rsb- -lrb- but not the -lsb- -lsb- two 's complement -rsb- -rsb- representation used to represent integers in most computers today -rrb- and most -lsb- -lsb- floating point -rsb- -rsb- number representations , zero has two distinct representations , one grouping it with the positive numbers and one with the negatives ; this latter representation is known as -lsb- -lsb- − 0 -lrb- number -rrb- -pipe- negative zero -rsb- -rsb- .
<p>
= = in other fields = = -lsb- -lsb- image : ics zero.svg -pipe- thumb -pipe- -lsb- -lsb- international maritime signal flags -pipe- international maritime signal flag -rsb- -rsb- for 0 -rsb- -rsb-
<p>
* in some countries and some company phone networks , dialing 0 on a telephone places a call for -lsb- -lsb- operator assistance -rsb- -rsb- . * in -lsb- -lsb- braille -rsb- -rsb- , the numeral 0 has the same dot configuration as the letter -lsb- -lsb- j -rsb- -rsb- . * -lsb- -lsb- dvd -rsb- -rsb- s that can be played in any region are sometimes referred to as being " -lsb- -lsb- region 0 -rsb- -rsb- " * in classical music , 0 is very rarely used as a number for a composition : -lsb- -lsb- anton bruckner -rsb- -rsb- wrote a -lsb- -lsb- symphony no. 0 -lrb- bruckner -rrb- -pipe- symphony no. 0 in d minor -rsb- -rsb- and a -lsb- -lsb- study symphony -pipe- symphony no. 00 -rsb- -rsb- ; -lsb- -lsb- alfred schnittke -rsb- -rsb- also wrote a symphony no. 0 . * -lsb- -lsb- roulette -rsb- -rsb- wheels usually feature a " 0 " space -lrb- and sometimes also a " 00 " space -rrb- , whose presence is ignored when calculating payoffs -lrb- thereby allowing the house to win in the long run -rrb- . * a chronological prequel of a series may be numbered as 0 . * in -lsb- -lsb- formula one -rsb- -rsb- , if the reigning -lsb- -lsb- list of formula one world drivers ' champions -pipe- world champion -rsb- -rsb- no longer competes in formula one in the year following their victory in the title race , 0 is given to one of the drivers of the team that the reigning champion won the title with .
this happened in -lsb- -lsb- 1993 -rsb- -rsb- and -lsb- -lsb- 1994 -rsb- -rsb- , with -lsb- -lsb- damon hill -rsb- -rsb- driving car 0 , due to the reigning world champion -lrb- -lsb- -lsb- nigel mansell -rsb- -rsb- and -lsb- -lsb- alain prost -rsb- -rsb- respectively -rrb- not competing in the championship . * in the educational series -lsb- -lsb- schoolhouse rock ! -rsb- -rsb- , the song ' ' my hero , zero " is about the use of zero as a placeholder .
the song explains that by appending zeros to a number , it is multiplied by 10 for each one added .
this enables mathematicians to create numbers as large as needed .
<p>
= = quotations = = -lcb- -lcb- cquote2 -pipe- the importance of the creation of the zero mark can never be exaggerated .
this giving to airy nothing , not merely a local habitation and a name , a picture , a symbol , but helpful power , is the characteristic of the hindu race from whence it sprang .
it is like coining the -lsb- -lsb- nirvana -rsb- -rsb- into -lsb- -lsb- dynamo -rsb- -rsb- s .
no single mathematical creation has been more potent for the general on-go of intelligence and power . -pipe- -lsb- -lsb- g. b. halsted -rsb- -rsb- -rcb- -rcb-
<p>
-lcb- -lcb- cquote2 -pipe- dividing by zero ... allows you to prove , mathematically , anything in the universe .
you can prove that 1 + 1 < nowiki > = < / nowiki > 42 , and from there you can prove that j. edgar hoover is a space alien , that william shakespeare came from uzbekistan , or even that the sky is polka-dotted .
-lrb- see appendix a for a proof that winston churchill was a carrot . -rrb- -pipe- -lsb- -lsb- charles seife -rsb- -rsb- -pipe- ' ' zero : the biography of a dangerous idea " -rcb- -rcb-
<p>
-lcb- -lcb- cquote2 -pipe- ... a profound and important idea which appears so simple to us now that we ignore its true merit .
but its very simplicity and the great ease which it lent to all computations put our arithmetic in the first rank of useful inventions . -pipe- -lsb- -lsb- pierre-simon laplace -rsb- -rsb- -rcb- -rcb-
<p>
-lcb- -lcb- cquote2 -pipe- the point about zero is that we do not need to use it in the operations of daily life .
no one goes out to buy zero fish .
it is in a way the most civilized of all the cardinals , and its use is only forced on us by the needs of cultivated modes of thought . -pipe- -lsb- -lsb- alfred north whitehead -rsb- -rsb- -rcb- -rcb-
<p>
-lcb- -lcb- cquote2 -pipe- ... a fine and wonderful refuge of the divine spirit – almost an amphibian between being and non-being . -pipe- -lsb- -lsb- gottfried leibniz -rsb- -rsb- -rcb- -rcb-
<p>
= = see also = = * -lsb- -lsb- grammatical number -rsb- -rsb- * -lsb- -lsb- number theory -rsb- -rsb- * -lsb- -lsb- peano axioms -rsb- -rsb-
<p>
= = notes = = -lcb- -lcb- reflist -pipe- 2 -rcb- -rcb-
<p>
= = references = = -lcb- -lcb- foldoc -rcb- -rcb- -lcb- -lcb- refbegin -rcb- -rcb- * -lsb- -lsb- john d. barrow -pipe- barrow , john d. -rsb- -rsb- -lrb- 2001 -rrb- ' ' the book of nothing " , vintage .
isbn 0-09-928845-1 . * diehl , richard a. -lrb- 2004 -rrb- ' ' the olmecs : america 's first civilization " , thames & hudson , london . * ifrah , georges -lrb- 2000 -rrb- ' ' the universal history of numbers : from prehistory to the invention of the computer " , wiley .
isbn 0-471-39340-1 . * kaplan , robert -lrb- 2000 -rrb- ' ' the nothing that is : a natural history of zero " , oxford : oxford university press . * -lsb- -lsb- charles seife -pipe- seife , charles -rsb- -rsb- -lrb- 2000 -rrb- ' ' zero : the biography of a dangerous idea " , penguin usa -lrb- paper -rrb- .
isbn 0-14-029647-6 . * -lsb- -lsb- nicolas bourbaki -pipe- bourbaki , nicolas -rsb- -rsb- -lrb- 1998 -rrb- .
' ' elements of the history of mathematics " .
berlin , heidelberg , and new york : springer-verlag .
isbn 3540647678 . * isaac asimov article " nothing counts " in " asimov on numbers " pocket books , 1978 -lcb- -lcb- refend -rcb- -rcb-
<p>
= = external links = = -lcb- -lcb- wikisource1911enc -pipe- zero -rcb- -rcb- * -lsb- http : / / www-gap.dcs.st-and.ac.uk / ~ history / histtopics / zero.html a history of zero -rsb- * -lsb- http : / / home.ubalt.edu / ntsbarsh / zero / zero.htm zero saga -rsb- * -lsb- http : / / www.neo-tech.com / zero / part6.html the discovery of the zero -rsb- * -lsb- http : / / www.ucs.louisiana.edu / ~ sxw8045 / history.htm the history of algebra -rsb- * -lsb- -lsb- edsger w. dijkstra -rsb- -rsb- : -lsb- http : / / www.cs.utexas.edu / users / ewd / ewd08xx / ewd831.pdf why numbering should start at zero -rsb- , 192 -lrb- -lsb- -lsb- portable document format -pipe- pdf -rsb- -rsb- of a handwritten manuscript -rrb- * -lsb- http : / / www.schoolhouserock.tv / my.html " my hero zero " -rsb- educational children 's song in -lsb- -lsb- schoolhouse rock ! -rsb- -rsb-
<p>
-lsb- -lsb- category : zero -pipe- -rsb- -rsb- -lsb- -lsb- category : elementary arithmetic -rsb- -rsb- -lsb- -lsb- category : integers -pipe- 00 -rsb- -rsb- -lsb- -lsb- category : nothing -rsb- -rsb- -lsb- -lsb- category : indian inventions -rsb- -rsb-
<p>
-lsb- -lsb- ar : 0 -lrb- عدد -rrb- -rsb- -rsb- -lsb- -lsb- bg : нула -rsb- -rsb- -lsb- -lsb- ca : zero -rsb- -rsb- -lsb- -lsb- cs : nula -rsb- -rsb- -lsb- -lsb- da : 0 -lrb- tal -rrb- -rsb- -rsb- -lsb- -lsb- de : null -rsb- -rsb- -lsb- -lsb- et : null -rsb- -rsb- -lsb- -lsb- el : μηδέν -rsb- -rsb- -lsb- -lsb- myv : чаво -rsb- -rsb- -lsb- -lsb- es : cero -rsb- -rsb- -lsb- -lsb- eo : nulo -rsb- -rsb- -lsb- -lsb- eu : zero -rsb- -rsb- -lsb- -lsb- fa : ۰ -lrb- عدد -rrb- -rsb- -rsb- -lsb- -lsb- fr : zéro -rsb- -rsb- -lsb- -lsb- fur : 0 -lrb- numar -rrb- -rsb- -rsb- -lsb- -lsb- gl : cero -rsb- -rsb- -lsb- -lsb- gu : ૦ -lrb- શૂન્ય -rrb- -rsb- -rsb- -lsb- -lsb- xal : 0 -rsb- -rsb- -lsb- -lsb- ko : 0 -rsb- -rsb- -lsb- -lsb- hi : शून्य -rsb- -rsb- -lsb- -lsb- io : zero -rsb- -rsb- -lsb- -lsb- ig : zero -rsb- -rsb- -lsb- -lsb- id : 0 -lrb- angka -rrb- -rsb- -rsb- -lsb- -lsb- ia : 0 -lrb- numero -rrb- -rsb- -rsb- -lsb- -lsb- xh : iqanda -rsb- -rsb- -lsb- -lsb- is : núll -rsb- -rsb- -lsb- -lsb- it : zero -rsb- -rsb- -lsb- -lsb- he : 0 -lrb- מספר -rrb- -rsb- -rsb- -lsb- -lsb- ka : ნული -rsb- -rsb- -lsb- -lsb- ht : 0 -lrb- nonm -rrb- -rsb- -rsb- -lsb- -lsb- ku : sifir -lrb- hejmar -rrb- -rsb- -rsb- -lsb- -lsb- la : 0 -rsb- -rsb- -lsb- -lsb- lv : nulle -rsb- -rsb- -lsb- -lsb- lt : 0 -lrb- skaičius -rrb- -rsb- -rsb- -lsb- -lsb- lmo : nümar 0 -rsb- -rsb- -lsb- -lsb- hu : 0 -lrb- szám -rrb- -rsb- -rsb- -lsb- -lsb- mk : 0 -lrb- број -rrb- -rsb- -rsb- -lsb- -lsb- ml : പൂജ്യം -rsb- -rsb- -lsb- -lsb- ms : 0 -lrb- nombor -rrb- -rsb- -rsb- -lsb- -lsb- nl : 0 -lrb- getal -rrb- -rsb- -rsb- -lsb- -lsb- nds-nl : 0 -lrb- getal -rrb- -rsb- -rsb- -lsb- -lsb- new : शून्य -rsb- -rsb- -lsb- -lsb- ja : 0 -rsb- -rsb- -lsb- -lsb- no : null -rsb- -rsb- -lsb- -lsb- nn : 0 -rsb- -rsb- -lsb- -lsb- pl : 0 -lrb- liczba -rrb- -rsb- -rsb- -lsb- -lsb- pt : zero -rsb- -rsb- -lsb- -lsb- ro : 0 -lrb- cifră -rrb- -rsb- -rsb- -lsb- -lsb- qu : ch 'usaq yupay -rsb- -rsb- -lsb- -lsb- ru : 0 -lrb- число -rrb- -rsb- -rsb- -lsb- -lsb- simple : zero -rsb- -rsb- -lsb- -lsb- sk : 0 -lrb- číslo -rrb- -rsb- -rsb- -lsb- -lsb- sl : 0 -rsb- -rsb- -lsb- -lsb- srn : numro 0 -rsb- -rsb- -lsb- -lsb- sr : 0 -lrb- број -rrb- -rsb- -rsb- -lsb- -lsb- fi : 0 -lrb- luku -rrb- -rsb- -rsb- -lsb- -lsb- sv : 0 -lrb- tal -rrb- -rsb- -rsb- -lsb- -lsb- tl : 0 -lrb- bilang -rrb- -rsb- -rsb- -lsb- -lsb- ta : சூனியமும் இடமதிப்புத் திட்டமும் -rsb- -rsb- -lsb- -lsb- te : సున్న -rsb- -rsb- -lsb- -lsb- th : 0 -rsb- -rsb- -lsb- -lsb- tr : 0 -lrb- sayı -rrb- -rsb- -rsb- -lsb- -lsb- uk : 0 -lrb- число -rrb- -rsb- -rsb- -lsb- -lsb- ur : صفر -lrb- عدد -rrb- -rsb- -rsb- -lsb- -lsb- vi : 0 -rsb- -rsb- -lsb- -lsb- vls : 0 -lrb- getal -rrb- -rsb- -rsb- -lsb- -lsb- wo : tus -rsb- -rsb- -lsb- -lsb- yi : 0 -lrb- נומער -rrb- -rsb- -rsb- -lsb- -lsb- yo : 0 -lrb- nọ́mbà -rrb- -rsb- -rsb- -lsb- -lsb- zh-yue : 0 -rsb- -rsb- -lsb- -lsb- zh : 0 -rsb-
