Chapter 2 Operations On Decimal Numbers: Fill & Download for Free

To echo Tim Morgan's epic answer: all of them? There are lots and lots of ways to represent real numbers. I'll try to cover some of the more important ones, and survey a few of the exotic.Before we talk about representing real numbers, we should remind ourselves about natural numbers and their representations, rational numbers and their representations, and finally we will get to real numbers and their representations.Remember that natural numbers are also real numbers, so we're not wasting our time when we talk about natural numbers - it's just a special case. It's also a warm-up exercise.To learn a lot more about representing numbers and doing arithmetic using those representations, please do read Chapter 4 of one of the best books ever written, Knuth's The Art of Computer Programming.Natural NumbersThe natural numbers are simply the ordinary counting numbers 1, 2, 3 and so on. They already provide a good case study for overcoming the greatest psychological hurdle this answer tries to clear up:The same object can have many representations. The object and its representations are related but distinct entities.We're so used to the decimal notation of natural numbers that we tend to forget that the number 23 and the representation "23" are two different things.The number represented in decimal notation as "23" tells us how many circles there are here:oooooooooooooooooooooooThis number has lots of properties. It is odd, it is prime, in fact it is a regular prime, it is positive, it is greater than (the number represented by) 10, it is two less than a square and four less than a cube. All of these are properties of the number itself, in whichever form we choose to represent it.One thing this number doesn't have is any special relationship with the numbers 2 and 3. We use those numbers to represent it in decimal notation because this is our favorite notation that we've been using for centuries, but the number 23 and the numbers 2 and 3 have not much to do with each other. The reason the digits 2 and 3 are used is because[math]23 = [/math][math]2[/math][math] \times 10 + 3[/math]and this is how the decimal notation works: the number on the right counts 1's, the number next to it counts 10's, the next number counts 100's, and so on. But "10" doesn't have any special meaning in math or real life, so those 2 and 3 aren't especially important for 23.We use the decimal notation probably because we have ten fingers. It is also a fairly useful notation, because it only requires ten digits (as opposed to base 60, for instance, which requires sixty), and the representations are fairly compact (the year 2015 in binary is 11111011111 - a palindrome, but a long one), and the positional representation makes addition and multiplication fairly easy.What are other ways to represent natural numbers?We can use unary notation. Here we only use one digit, say 1, and we repeat it just as many times as the number we wish to represent:1: 12: 115: 1111123: 111111111111111111111112015: Er...no.One advantage of the unary system is that it makes it super easy to do addition: you just put one number next to the other.11111111 + 11111=1111111111111.Compare this to 8+5=13, which is rather tricky. Where did the "1" and the "3" come from?Different ways to represent numbers make certain tasks easier or harder.We can use binary notation, which is just like decimal notation only using powers of 2 instead of powers of 10. The various digits now represent ones, twos, fours, eights, sixteens, etc., instead of ones, tens, hundreds etc.23: 10111because if you take one 1, one 2, one 4 and one 16 (but no 8) and add them together you get the number which in decimal notation we call 23.We can similarly use ternary (base 3) notation, base 4, base 12, base 16, base 900 or any other base we want. The principle is always the same. In base 16, for example, twenty three is denoted "17". Yes! See how "2" and "3" are unimportant? The very same number now carries the digits "1" and "7".And in base 15 (fifteen), the same number is represented as 18. Wait, is it now an even number? No, of course not. Being even or odd is a property of the number. If you take 23 dots and try to arrange them in two rows, there'll be one left over. The number is odd, no matter what base you use.oooooooooooooooooooooooIt's just that in base 15, having an even digit on the right isn't the right way to tell if the number is even or odd. Just like in base 10 you can't tell if a number is divisible by 3 just by looking at the last digit.The ancient Babylonians used a system that was roughly positional (like our decimal system) but relied on base 60. This is why we still count seconds and minutes (for time) and seconds and minutes (in navigation, as angles) in sixties.The Romans used a system based on letters, as you probably know. 1 is I, 2 is II, 3 is III, 5 is V, 10 is X, 50 is L, 100 is C, 500 is D and 1,000 is M. To save space (sorta) they used IV for 4 instead of IIII (although IIII was also used early on), and similarly IX for 9, XC for 90 and so on. The year 1999 would be written MCMXCIX, which is about as confusing as it gets. For some reason, to this day movie credits often indicate the year in Roman numerals, as if this has more gravitas. The results are sometimes disastrous:As you can imagine, doing arithmetic with Roman numerals is torture, and it's not even clear how to represent large numbers. When the world gradually shifted to the modern base-10 positional system, many people had a hard time letting go of Roman numerals. Conway and Guy point out that there are records of European merchants writing things like M50iv for the year 1504.Are there other ways to represent natural numbers? Sure there are.We can use negative bases, such as (-10).[math]123_{-10} = 1 \times (-10)^2 + [/math][math]2[/math][math] \times (-10) + 3 = 83_{10}[/math](here I'm using the subscript 10 or -10 to indicate which base is being used). One cute feature of negative bases is that you can represent any whole number, either positive or negative, without a sign. In 1955, the inimitable Don Knuth wrote a paper on negative-radix systems (and complex-radix ones) in a science contest paper. One of his systems, the Quater-imaginary base, can easily be used to produce images like this Dragon fractal.In The Art of Computer Programming, Knuth discusses many other positional systems. One of his favorites is balanced ternary notation, which uses base 3 but with the digits [math]1, 0, \bar{1}[/math] instead of [math]0,1,2[/math], where [math]\bar{1}[/math] represents negative unity (-1). So for instance[math]1\bar{1}1_{\mbox{bt}} = 9 - 3 + 1 = 7[/math].We can also use systems where the value of each digit is not a power of some fixed base. For example, in this answer a system is discussed where the value of the [math]k[/math]th digit is [math]k![/math]. This means that the allowed values for each digit change along the number: the [math]k[/math]th digit can be anything between 0 and [math]k[/math]. For example,[math]221_! = 1 \times 1! + [/math][math]2[/math][math] \times [/math][math]2[/math][math]! + [/math][math]2[/math][math] \times 3! = 17[/math].In 1972, Edouard Zeckendorf proved that every natural number can be represented uniquely as a sum of non-consecutive Fibonacci numbers. For example,[math]100 = 89 + 8 + 3[/math]which can be used to represent 100 as [math]1000010100_{Z}[/math], where the 1's indicate which Fibonacci numbers to use. This representation never uses two consecutive 1's.Some exotic representations of natural numbers have actual utility in mathematics. An important result in the combinatorics of convex polytopes (McMullen's conditions for [math]f[/math]-vectors of simplicial polytopes) is phrased and proved using a representation relying on binomial coefficients.Rational NumbersRational numbers are ratios of whole numbers, such as [math]\frac{1}{2}[/math] or [math]\frac{355}{113}[/math].By far the very best way to represent a rational number is just like this: use any way to represent the numerator and denominator as natural numbers (with a possible sign), and indicate that the rational number is their ratio.Why is this the best way? Because it captures the rational number precisely, as it is. When people use things like [math]33.3\% = 0.333[/math] to represent "one third", they are only using an approximation, which introduces errors,Many computers use a Floating point system to represent numbers, and this system typically only supports denominators which are powers of 10 (or 2, or some other base, but the damage is similar). This is quite terrible, since many numbers which could be represented completely accurately are only represented approximately. Whenever you actually need to work with rational numbers, you should use packages such as Python's fractions, which represent rational numbers as pairs of integers.One disadvantage of the ratio representations is that it's not unique. The same number, say [math]\frac{1}{3}[/math], can also be represented as [math]\frac{7}{21}[/math] or [math]\frac{100}{300}[/math]. In other words, two representations can be equal even if they look different. Beginning students often find this confusing: they are used to having two things equal only when they look exactly the same. But that's just not how things work.In school we learn to use "decimal notation" to represent numbers, like [math]\frac{1}{4} = 0.25[/math]. We tend to use this a lot, particularly in the form of percentages (a quarter is 25%). This is a fine thing to do, but let's remember a few things:First, the decimal representation is sometimes infinite, though for rational numbers the sequence of digits is always repeating. For instance,[math]\frac{1}{6} = 0.16666\ldots[/math]Second, some (not all) numbers have non-unique decimal representations:[math]1 = 0.9999\ldots[/math][math]0.25 = 0.249999\ldots[/math]Those are not major problems, but they do get many children (and adults) confused.Once again, we have an opportunity to distinguish properties of a number from properties of the representation of the number.A rational number may be an integer (or not), it may be smaller than 1 (or not), it may or may not be a square (of a rational number), and so on.A rational number cannot be "terminating" or "have period 7". Those are properties of various representations of the number.Real NumbersReal numbers are limits of certain sequences of rational numbers, called Cauchy sequences. This definition may seem unfamiliar, but it's important to get a sense of what it means. If you have a sequence of rational numbers, such as[math]1, \frac{14}{10}, \frac{141}{100}, \frac{1414}{1000}, \ldots[/math]you can check if those numbers keep getting closer to one another. This means that if someone challenges you make them close to 1/100, you can skip some of the first few numbers and get to a place where all of them are no more than 1/100 apart, and you can do this with a challenge of 1/1000 by skipping more initial numbers, or 1/1,000,000, or any such challenge. This is a Cauchy sequence.If you have such a sequence, there's a real number which is the limit of that sequence. It is the number - which may itself be rational, or not - which the sequence gets closer and closer to. This is one of the main ways real numbers are defined in modern math.And this is, in some ways, bad news.If I have a real number and I want to tell you what it is, I need to give you infinitely many rational numbers. Since communicating a rational number takes at least some minimal amount of time, this will require an infinitely long time. In most civilized societies it is considered very rude to take up an infinitely long amount of someone's time.You may have seen things like this written down:[math]\pi = 3.14159265...[/math]The number on the left, [math]\pi[/math], is a name we gave to some specific real number. But the number on the right is not fully specified. It gives some information about [math]\pi[/math], but not all of it - and in fact, for most real numbers, you can't get all of it.All this equation tells you is that [math]3.14159265 \leq \pi \leq 3.14159266[/math]. This is good information, and for many purposes it is sufficient: if you need to calculate the length of the circumference of some circular thing, you probably can get by with much less precise info. But just writing "three point one four one five nine dot dot dot" does not pinpoint a real number.In some cases, we can describe an infinite sequence of rational numbers using a finite description. For example, consider this sequence:[math]\frac{1}{1}, \frac{2}{1}, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \ldots[/math]The rule here is very simple: in each fraction, the denominator is the numerator of the previous one, and the numerator is the sum of the previous numerator and denominator. So the next fraction is going to be [math]\frac{8+5}{8}=\frac{13}{8}[/math].This is a Cauchy sequence: the numbers get closer and closer. Its limit is a familiar number, the golden ratio [math]\frac{1+\sqrt{5}}{2}[/math]. So in this particular case we have a real number that does accommodate a finite description, in fact two of them: one as a limit of a finitely-describable sequence of rationals, and one as an expression involving arithmetical operations and square roots.Many familiar real numbers such as [math]\pi[/math], [math]e[/math], [math]\log_2(5)[/math] and [math]\sin(\pi/17)[/math] allow such finite descriptions, but they are outliers. Most real numbers cannot be finitely described in any way. If I have a secret real number [math]\zeta[/math], and I tell you that[math]\zeta = 4.669201609...[/math]I didn't really tell you what [math]\zeta[/math] is. You don't know if it's rational or irrational, you can't tell if it's a algebraic or not, all you know is its rough magnitude. If I ask you what is the nearest integer to [math]10^{100}\zeta[/math], you have no idea. You don't know what my number [math]\zeta[/math] is, and unless some very special circumstances are in place, there's no way for me to help you even if I wanted to.Therefore, a "representation" of a real number must be one of two things:Partial, providing some incomplete information about the number; orInfinitely long, and allowing a finite presentation only in some very special cases (i.e. for computable numbers.)So how do we represent real numbers, in either of these forms?Just like with natural numbers, we can use a positional number system in some base. When we write[math]\pi = 3.14\ldots[/math]We mean to say that[math]\pi = 3 \times 1 + 1 \times \frac{1}{10} + 4 \times \frac{1}{100} + [/math] a whole lot of things which we're not telling you, but they're altogether smaller than [math]\frac{1}{100}[/math].Of course, we can do the same things using other bases:[math]\pi = 11.001001000011_2\ldots[/math][math]\pi = 10.010211_3\ldots[/math][math]\pi = 3.243F6A88_{16}\ldots[/math](note that in base 16, or hexadecimal, we need 16 digits, so we typically use the numerals 0-9 and then the letters A-F). These examples serve as a good reminder that the "mystical" digits 3, 1, 4, 1, 5, etc. have nothing in particular to do with the essence of the number [math]\pi[/math].None of those representations will terminate, or repeat. Why? Because we know that [math]\pi[/math] is an irrational number - in fact, a transcendental one. Remember, this is a property of the number, having nothing to do with any particular representation. For that reason it makes no sense to ask if [math]\pi[/math] has a "rational representation" - it's not a rational number.Another very important representation of real numbers is the continued fraction representation.Pick up your calculator - or actually, your smartphone, and then launch the calculator app. If you're using an iPhone, turn it sideways so that the app becomes actually useful. Then hit the button labeled "[math]\pi[/math]".The value is 3 point something, so now let's subtract 3 to get a number less than 1.Now hit the "1/x" button to get a big number again.Now repeat! You have 7 point something something, so take away 7......and hit 1/x.If you retrace your steps, you'll see that so far you've shown that[math]\pi = 3 + \frac{1}{7 + \frac{1}{15 + \ldots}}[/math]and of course you can keep going:[math]\pi = 3 + \frac{1}{7 + \frac{1}{15 + \frac{1}{1 + \frac{1}{292 + \ldots}}}}[/math]This is the continued fraction representation of [math]\pi[/math], and to save clutter it is often written like this:[math]\pi = [3; 7, 15, 1, 292, 1, 1, 1, [/math][math]2[/math][math], 1, 3, 1, \ldots][/math]Since [math]\frac{1}{292}[/math] is a pretty small number, we know that we can get something pretty close to [math]\pi[/math] if we stop right there:[math]\pi \approx 3+\frac{1}{7+\frac{1}{15+\frac{1}{1}}} = \frac{355}{113}[/math].This is a really good rational approximation of [math]\pi[/math]. In general the continued fraction produces the "best" rational approximations of a number, in some precise sense.It's really fun to try this with some other numbers on your calculator. If you try square roots, you'll get continued fractions which are repeating, just like the base-[math]d[/math] representation of rational numbers.[math]\sqrt{2} = [1; [/math][math]2[/math][math], [/math][math]2[/math][math], [/math][math]2[/math][math], [/math][math]2[/math][math], \ldots][/math][math]\frac{1+\sqrt{5}}{2} = [1; 1, 1, 1, 1, \ldots][/math][math]\sqrt{23}=[4;1,3,1,8,1,3,1,8,\ldots][/math]Now try [math]e[/math].[math]e = [2;1,\, 2,1,1, \, 4,1,1,\, 6,1,1, \, 8,1,1, \ldots][/math]Lovely isn't it? The continued fraction isn't repeating (only quadratic irrationalities are repeating) but it does have a very simple pattern.No such pattern is known for [math]\pi[/math]. In fact we know very little about the continued fraction expansion of [math]\pi[/math], and that of most other real numbers. Remember though that we know precious little about the decimal expansion of [math]\pi[/math] as well. Nobody knows if it ever contains a run of 30 consecutive 7's in a row, for example.Most of the time we don't actually represent real numbers in decimal notation or continued fraction form. We often work with real numbers that have some concrete representation as a limit (or an infinite series, which is the same thing), or an algebraic combination of other numbers.[math]\pi = 4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\ldots)[/math][math]e = \frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\ldots[/math][math]x = e^{\pi \sqrt{163}}[/math]Many people feel that those representations are insufficient in some sense; you find many questions in the classroom and online to the effect of "Yes but how do you compute that?" or "but this doesn't tell you what the number actually is". It's useful to understand that such representations are in no way inferior to the decimal representation we're so used to. In fact, [math]\pi = 3.1415926...[/math] is a great deal less informative than[math]\pi=2 \sum_{k=0}^\infty \frac{2^k k!^2}{(2k+1)!}[/math]Those are some of the ways real numbers can be represented.

Bash, itself, no — but, as usual, the POSIX toolkit has something for you: dateTL;DR →mv "$old_name" "$new_name.$(date +%Y-%m-%d).xyzzy" The manual for date (POSIX):DATE(1P) -- 2013 -- IEEE/The Open Group -- POSIX Programmer's ManualPROLOGThis manual page is part of the POSIX Programmer's Manual. The Linux implementation of this interface may differ (consult the correspondingLinux manual page for details of Linux behavior), or the interface may not be implemented on Linux.NAMEdate -- write the date and timeSYNOPSISdate [-u] [+format]date [-u] mmddhhmm[[cc]yy]DESCRIPTIONThe date utility shall write the date and time to standard output or attempt to set the system date and time. By default, the current dateand time shall be written. If an operand beginning with '+' is specified, the output format of date shall be controlled by the conversionspecifications and other text in the operand.OPTIONSThe date utility shall conform to the Base Definitions volume of POSIX.1‐2008, Section 12.2, Utility Syntax Guidelines.The following option shall be supported:-u Perform operations as if the TZ environment variable was set to the string \(dqUTC0\(dq, or its equivalent historical value of\(dqGMT0\(dq. Otherwise, date shall use the timezone indicated by the TZ environment variable or the system default if thatvariable is unset or null.OPERANDSThe following operands shall be supported:+format When the format is specified, each conversion specifier shall be replaced in the standard output by its corresponding value.All other characters shall be copied to the output without change. The output shall always be terminated with a <newline>.Conversion Specifications%a Locale's abbreviated weekday name.%A Locale's full weekday name.%b Locale's abbreviated month name.%B Locale's full month name.%c Locale's appropriate date and time representation.%C Century (a year divided by 100 and truncated to an integer) as a decimal number [00,99].%d Day of the month as a decimal number [01,31].%D Date in the format mm/dd/yy.%e Day of the month as a decimal number [1,31] in a two-digit field with leading <space> character fill.%h A synonym for %b.%H Hour (24-hour clock) as a decimal number [00,23].%I Hour (12-hour clock) as a decimal number [01,12].%j Day of the year as a decimal number [001,366].%m Month as a decimal number [01,12].%M Minute as a decimal number [00,59].%n A <newline>.%p Locale's equivalent of either AM or PM.%r 12-hour clock time [01,12] using the AM/PM notation; in the POSIX locale, this shall be equivalent to %I:%M:%S %p.%S Seconds as a decimal number [00,60].%t A <tab>.%T 24-hour clock time [00,23] in the format HH:MM:SS.%u Weekday as a decimal number [1,7] (1=Monday).%U Week of the year (Sunday as the first day of the week) as a decimal number [00,53]. All days in a new year preceding thefirst Sunday shall be considered to be in week 0.%V Week of the year (Monday as the first day of the week) as a decimal number [01,53]. If the week containing January 1 hasfour or more days in the new year, then it shall be considered week 1; otherwise, it shall be the last week of theprevious year, and the next week shall be week 1.%w Weekday as a decimal number [0,6] (0=Sunday).%W Week of the year (Monday as the first day of the week) as a decimal number [00,53]. All days in a new year preceding thefirst Monday shall be considered to be in week 0.%x Locale's appropriate date representation.%X Locale's appropriate time representation.%y Year within century [00,99].%Y Year with century as a decimal number.%Z Timezone name, or no characters if no timezone is determinable.%% A <percent-sign> character.See the Base Definitions volume of POSIX.1‐2008, Section 7.3.5, LC_TIME for the conversion specifier values in the POSIX locale.Modified Conversion SpecificationsSome conversion specifiers can be modified by the E and O modifier characters to indicate a different format or specification as specifiedin the LC_TIME locale description (see the Base Definitions volume of POSIX.1‐2008, Section 7.3.5, LC_TIME). If the corresponding keyword(see era, era_year, era_d_fmt, and alt_digits in the Base Definitions volume of POSIX.1‐2008, Section 7.3.5, LC_TIME) is not specified ornot supported for the current locale, the unmodified conversion specifier value shall be used.%Ec Locale's alternative appropriate date and time representation.%EC The name of the base year (period) in the locale's alternative representation.%Ex Locale's alternative date representation.%EX Locale's alternative time representation.%Ey Offset from %EC (year only) in the locale's alternative representation.%EY Full alternative year representation.%Od Day of month using the locale's alternative numeric symbols.%Oe Day of month using the locale's alternative numeric symbols.%OH Hour (24-hour clock) using the locale's alternative numeric symbols.%OI Hour (12-hour clock) using the locale's alternative numeric symbols.%Om Month using the locale's alternative numeric symbols.%OM Minutes using the locale's alternative numeric symbols.%OS Seconds using the locale's alternative numeric symbols.%Ou Weekday as a number in the locale's alternative representation (Monday = 1).%OU Week number of the year (Sunday as the first day of the week) using the locale's alternative numeric symbols.%OV Week number of the year (Monday as the first day of the week, rules corresponding to %V), using the locale's alternativenumeric symbols.%Ow Weekday as a number in the locale's alternative representation (Sunday = 0).%OW Week number of the year (Monday as the first day of the week) using the locale's alternative numeric symbols.%Oy Year (offset from %C) in alternative representation.mmddhhmm[[cc]yy]Attempt to set the system date and time from the value given in the operand. This is only possible if the user has appropriateprivileges and the system permits the setting of the system date and time. The first mm is the month (number); dd is the day(number); hh is the hour (number, 24-hour system); the second mm is the minute (number); cc is the century and is the first twodigits of the year (this is optional); yy is the last two digits of the year and is optional. If century is not specified, thenvalues in the range [69,99] shall refer to years 1969 to 1999 inclusive, and values in the range [00,68] shall refer to years 2000to 2068 inclusive. The current year is the default if yy is omitted.Note: It is expected that in a future version of this standard the default century inferred from a 2-digit year will change.(This would apply to all commands accepting a 2-digit year as input.)STDINNot used.INPUT FILESNone.ENVIRONMENT VARIABLESThe following environment variables shall affect the execution of date:LANG Provide a default value for the internationalization variables that are unset or null. (See the Base Definitions volume ofPOSIX.1‐2008, Section 8.2, Internationalization Variables for the precedence of internationalization variables used to determinethe values of locale categories.)LC_ALL If set to a non-empty string value, override the values of all the other internationalization variables.LC_CTYPE Determine the locale for the interpretation of sequences of bytes of text data as characters (for example, single-byte as opposedto multi-byte characters in arguments).LC_MESSAGESDetermine the locale that should be used to affect the format and contents of diagnostic messages written to standard error.LC_TIME Determine the format and contents of date and time strings written by date.NLSPATH Determine the location of message catalogs for the processing of LC_MESSAGES.TZ Determine the timezone in which the time and date are written, unless the -u option is specified. If the TZ variable is unset ornull and -u is not specified, an unspecified system default timezone is used.ASYNCHRONOUS EVENTSDefault.STDOUTWhen no formatting operand is specified, the output in the POSIX locale shall be equivalent to specifying:date "+%a %b %e %H:%M:%S %Z %Y"STDERRThe standard error shall be used only for diagnostic messages.OUTPUT FILESNone.EXTENDED DESCRIPTIONNone.EXIT STATUSThe following exit values shall be returned:0 The date was written successfully.>0 An error occurred.CONSEQUENCES OF ERRORSDefault.The following sections are informative.APPLICATION USAGEConversion specifiers are of unspecified format when not in the POSIX locale. Some of them can contain <newline> characters in some locales,so it may be difficult to use the format shown in standard output for parsing the output of date in those locales.The range of values for %S extends from 0 to 60 seconds to accommodate the occasional leap second.Although certain of the conversion specifiers in the POSIX locale (such as the name of the month) are shown with initial capital letters,this need not be the case in other locales. Programs using these fields may need to adjust the capitalization if the output is going to beused at the beginning of a sentence.The date string formatting capabilities are intended for use in Gregorian-style calendars, possibly with a different starting year (oryears). The %x and %c conversion specifications, however, are intended for local representation; these may be based on a different,non-Gregorian calendar.The %C conversion specification was introduced to allow a fallback for the %EC (alternative year format base year); it can be viewed as thebase of the current subdivision in the Gregorian calendar. The century number is calculated as the year divided by 100 and truncated to aninteger; it should not be confused with the use of ordinal numbers for centuries (for example, ``twenty-first century''.) Both the %Ey and%y can then be viewed as the offset from %EC and %C, respectively.The E and O modifiers modify the traditional conversion specifiers, so that they can always be used, even if the implementation (or thecurrent locale) does not support the modifier.The E modifier supports alternative date formats, such as the Japanese Emperor's Era, as long as these are based on the Gregorian calendarsystem. Extending the E modifiers to other date elements may provide an implementation-defined extension capable of supporting othercalendar systems, especially in combination with the O modifier.The O modifier supports time and date formats using the locale's alternative numerical symbols, such as Kanji or Hindi digits or ordinalnumber representation.Non-European locales, whether they use Latin digits in computational items or not, often have local forms of the digits for use in dateformats. This is not totally unknown even in Europe; a variant of dates uses Roman numerals for the months: the third day of September 1991would be written as 3.IX.1991. In Japan, Kanji digits are regularly used for dates; in Arabic-speaking countries, Hindi digits are used.The %d, %e, %H, %I, %m, %S, %U, %w, %W, and %y conversion specifications always return the date and time field in Latin digits (that is,0 to 9). The %O modifier was introduced to support the use for display purposes of non-Latin digits. In the LC_TIME category in localedef,the optional alt_digits keyword is intended for this purpose. As an example, assume the following (partial) localedef source:alt_digits "";"I";"II";"III";"IV";"V";"VI";"VII";"VIII" \"IX";"X";"XI";"XII"d_fmt "%e.%Om.%Y"With the above date, the command:date "+%x"would yield 3.IX.1991. With the same d_fmt, but without the alt_digits, the command would yield 3.9.1991.EXAMPLES1. The following are input/output examples of date used at arbitrary times in the POSIX locale:$ dateTue Jun 26 09:58:10 PDT 1990$ date "+DATE: %m/%d/%y%nTIME: %H:%M:%S"DATE: 11/02/91TIME: 13:36:16$ date "+TIME: %r"TIME: 01:36:32 PM2. Examples for Denmark, where the default date and time format is %a %d %b %Y %T %Z:$ LANG=da_DK.iso_8859-1 dateons 02 okt 1991 15:03:32 CET$ LANG=da_DK.iso_8859-1 \date "+DATO: %A den %e. %B %Y%nKLOKKEN: %H:%M:%S"DATO: onsdag den 2. oktober 1991KLOKKEN: 15:03:563. Examples for Germany, where the default date and time format is %a %d.%h.%Y, %T %Z:$ LANG=De_DE.88591 dateMi 02.Okt.1991, 15:01:21 MEZ$ LANG=De_DE.88591 date "+DATUM: %A, %d. %B %Y%nZEIT: %H:%M:%S"DATUM: Mittwoch, 02. Oktober 1991ZEIT: 15:02:024. Examples for France, where the default date and time format is %a %d %h %Y %Z %T:$ LANG=Fr_FR.88591 dateMer 02 oct 1991 MET 15:03:32$ LANG=Fr_FR.88591 date "+JOUR: %A %d %B %Y%nHEURE: %H:%M:%S"JOUR: Mercredi 02 octobre 1991HEURE: 15:03:56RATIONALESome of the new options for formatting are from the ISO C standard. The -u option was introduced to allow portable access to CoordinatedUniversal Time (UTC). The string \(dqGMT0\(dq is allowed as an equivalent TZ value to be compatible with all of the systems using the BSDimplementation, where this option originated.The %e format conversion specification (adopted from System V) was added because the ISO C standard conversion specifications did notprovide any way to produce the historical default date output during the first nine days of any month.There are two varieties of day and week numbering supported (in addition to any others created with the locale-dependent %E and %O modifiercharacters):* The historical variety in which Sunday is the first day of the week and the weekdays preceding the first Sunday of the year areconsidered week 0. These are represented by %w and %U. A variant of this is %W, using Monday as the first day of the week, but stillreferring to week 0. This view of the calendar was retained because so many historical applications depend on it and the ISO C standardstrftime() function, on which many date implementations are based, was defined in this way.* The international standard, based on the ISO 8601:2004 standard where Monday is the first weekday and the algorithm for the first weeknumber is more complex: If the week (Monday to Sunday) containing January 1 has four or more days in the new year, then it is week 1;otherwise, it is week 53 of the previous year, and the next week is week 1. These are represented by the new conversion specifications%u and %V, added as a result of international comments.FUTURE DIRECTIONSNone.SEE ALSOThe Base Definitions volume of POSIX.1‐2008, Section 7.3.5, LC_TIME, Chapter 8, Environment Variables, Section 12.2, Utility SyntaxGuidelinesThe System Interfaces volume of POSIX.1‐2008, fprintf(), strftime()COPYRIGHTPortions of this text are reprinted and reproduced in electronic form from IEEE Std 1003.1, 2013 Edition, Standard for InformationTechnology -- Portable Operating System Interface (POSIX), The Open Group Base Specifications Issue 7, Copyright (C) 2013 by the Instituteof Electrical and Electronics Engineers, Inc and The Open Group. (This is POSIX.1-2008 with the 2013 Technical Corrigendum 1 applied.) Inthe event of any discrepancy between this version and the original IEEE and The Open Group Standard, the original IEEE and The Open GroupStandard is the referee document. The original Standard can be obtained online at The UNIX System .Any typographical or formatting errors that appear in this page are most likely to have been introduced during the conversion of the sourcefiles to man page format. To report such errors, see Reporting man-pages bugs .

Although this is a question whose answer can be found on the first page of chapter 2 of most any calculus textbook, it is also worth answering here because people need to see math as more than just facts on a page in a book you only read once. They need to see the personal relationship between mathematicians and our theories.The concept of a limit is basically physics; it captures the intuition of a “moving measurement”, like tracking an airplane in the sky with your eyes. Or like tracking a particle being swept down the drain of a bathtub. And, like any science, it consists of extrapolation from a data set: you see the plane heading towards a cloud and determine that it will pass behind it, and after how long; you see the particle spiraling into the drain and, even though it is actually moving quite actively and in many directions the whole time, you can see that at some point in the future it will go down the drain (if you do this particular experiment, you will appreciate that it can be hard to predict, exactly, how long this will take).Calculus takes this intuition and applies it to the same kind of situation, but stripped of all its physical features. The motion of the plane or of the particle is just some function of time; the plane’s path can also be considered as a function of position (measuring around the horizon). Mathematical limits concern abstract functions that accept real or natural numbers and return real numbers, functions that stand in for any possible measurement of a physical phenomenon.The notation [math]\lim_{x \to a} f(x)[/math] means “the value that [math]f(x)[/math] tends towards if [math]x[/math] moves onto a”; the similar notation [math]\lim_{n \to \infty} a_n[/math] means “the value that [math]a_n[/math] tends towards as [math]n[/math] increases beyond all bounds”. The first notation means that the inputs to [math]f[/math] move continuously towards [math]a[/math], while the second one means that the subscript of [math]a_n[/math] increases in discrete steps. The concept is the same; only the implementation differs.Limits were discovered many times in the past before becoming an accepted part of mathematics in the 17th century; typical reasons for their discovery included the ones that actually resulted in their gaining acceptance:Computing geometric quantities, like areas, associated with curved shapes, by taking the limit of the analogous computations associated with flat shapes that approximate the curved ones;Extending a concept, like velocity, that is directly defined only with reference to a nonzero duration of measurement, to a corresponding instantaneous measurement.Limits, in and of themselves, are not that interesting. If you take a calculus class you will see a number of examples that all seem pointless because they are limits of continuous functions towards inputs in their domain, and can (by a theorem you would also learn) therefore be replaced by direct evaluation. These examples only serve to test your understanding.The first interesting (and systematic) application of limits was the definition of instantanous velocity, as described above. It is the limit of mean velocities, taken over some period of time that decreases to zero. Simultaneously with this definition was the computation of areas of figures in the plane, most notably, figures that are the region bounded by the graph of a function and various horizontal and vertical lines; it is the limit of the sum of areas of rectangles closely fitting that region. The reason this particular application qualified as the moment when calculus was invented is that it leads to the “fundamental theorem of calculus” describing how, in effect, the two operations above are inverse to each other.The limit concept has had a more subtle effect on the nature of theoretical mathematics. It more or less introduced an entirely new method for reasoning about functions, that of approximation, in contrast with the existing exact algebra. Another way to look at this is that it is only through the concept of a limit that functions (from reals to reals) are understood as entire objects, rather than just a collection of isolated values.Another philosophical consequence is that the project of extrapolating “infinitary” concepts from “finitary” axioms got its first rigorous success (this project wasn’t actually described as such until Hilbert did it in the early 20th century). This applies to things like the definition of the real numbers (which are mostly inaccessible to direct description) from the rational numbers, which are themselves derived from integers, which are actually a simple extension of the natural numbers (nonnegative integers), which are just counting. And everyone can believe that counting has intrinsic meaning, even if a priori it is not clear that infinite decimals do. It turns out that the former implies the latter.In the end, limits are another descriptive tool. You can say that a number is positive or negative; you can say that a fraction is between two integers; you can do arithmetic on various kinds of numbers. Now you can also do this other operation, taking a limit, on functions, which if nothing else allows you to say more about them. Though I think I’ve shown that there is, indeed, something else.