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Division (mathematics)

Division (mathematics)

:This article is about the arithmetic operation. For other uses, see Division (disambiguation). In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication, and sometimes it can be interpreted as repeated subtraction. Specifically, if :

c \times b = a

where b is not zero, then :

\frac ab = c

that is, a divided by b equals c. For instance,

\frac 63 = 2

since

2 \times 3 = 6\,

. In the above expression, a is called the dividend, b the divisor and c the quotient. Division by zero (i.e. where the divisor is zero) is usually not defined.

Notation

Division is most often shown by placing the dividend over the divisor with a horizontal line between them. For example, a divided by b is written

\frac ab

. This can be read out loud as "a divided by b". A way to express division all on one line is to write the dividend, then a slash, then the divisor, like this:

a/b\,

. This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of characters. A typographical variation which is halfway between these two forms uses a slash but elevates the dividend, and lowers the divisor: ^a⁄_b Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the numerator and denominator), and there is no implication that the division needs to be evaluated further. A less common way to show division is to use the obelus (or division sign) in this manner:

a \div b

. This form is infrequent except in elementary arithmetic. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator. In some non-English-speaking cultures, "a divided by b" has sometimes been written a : b. However, in English usage the colon is restricted to expressing the related concept of ratios.

Computing division

With a knowledge of multiplication tables, two integers can be divided on paper using the method of long division. If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction. Division can be calculated with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset. In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus. In such a case, division can be calculated by multiplication. This approach is useful in computers that do not have a fast division instruction.

Division of integers

Division of integers is not closed. Apart from division by zero being undefined, the quotient will not be an integer unless the dividend is an integer multiple of the divisor; for example 26 cannot be divided by 10 to give an integer. In such a case there are four possible approaches. # Say that 26 cannot be divided by 10. # Give the answer as a decimal fraction or a mixed number, so

\frac = 2.6

26/10 = 2 \frac 35

. This is the approach usually taken in mathematics. # Give the answer as a quotient and a remainder, so

\frac = 2

remainder 6. # Give the quotient as the answer, so

\frac = 2

. This is sometimes called integer division. One has to be careful when performing division of integers in a computer program. Some programming languages, such as C, will treat division of integers as in case 4 above, so the answer will be an integer. Other languages, such as MATLAB, will first convert the integers to real numbers, and then give a real number as the answer, as in case 2 above.

Division of rational numbers

The result of dividing two rational numbers is another rational number when the divisor is not 0. We may define division of two rational numbers p/q and r/s by :

= (p \times s)/(q \times r).

All four quantities are integers, and only p may be 0. This definition ensures that division is the inverse operation of multiplication.

Division of real numbers

Division of two real numbers results in another real number when the divisor is not 0. It is defined such a/b = c if and only if a = cb and b ≠ 0.

Division of complex numbers

Dividing two complex numbers results in another complex number when the divisor is not 0, defined thus: :

=  + i.

All four quantities are real numbers. r and s may not both be 0. Division for complex numbers expressed in polar form is simpler and easier to remember than the definition above: :

= e^.

Again all four quantities are real numbers. r may not be 0.

Division of polynomials

One can define the division operation for polynomials. Then, as in the case of integers, one has a remainder. See polynomial long division.

Division in abstract algebra

In abstract algebras such as matrix algebras and quaternion algebras, fractions such as

are typically defined as

a \cdot

a \cdot b^

where

b

is presumed to be an invertible element (i.e. there exists a multiplicative inverse

b^

such that

bb^ = b^b = 1

where

1

is the multiplicative identity). In an integral domain where such elements may not exist, division can still be performed on equations of the form

ab = ac

ba = ca

by left or right cancellation, respectively. More generally "division" in the sense of "cancellation" can be done in any ring with the aforementioned cancellation properties. By a theorem of Wedderburn, all finite division rings are fields, hence every nonzero element of such a ring is invertible, so division by any nonzero element is possible in such a ring. To learn about when algebras (in the technical sense) have a division operation, refer to the page on division algebras. In particular Bott periodicity can be used to show that any real normed division algebra must be isomorphic to either the real numbers R, the complex numbers C, the quaternions H, or the octonions O.

Division and calculus

The derivative of the quotient of two functions is given by the quotient rule: :

' = \frac

There is no general method to integrate the quotient of two functions.

External links

- [http://www.mathsisfun.com/dividing-decimals.html Method for Dividing Decimals]
-
- [http://webhome.idirect.com/~totton/abacus/pages.htm#Division1 Division on a Japanese abacus] selected from [http://webhome.idirect.com/~totton/abacus/ Abacus: Mystery of the Bead]
- [http://webhome.idirect.com/~totton/suanpan/sh_div/ Chinese Short Division Techniques on a Suan Pan] Category:Elementary arithmetic Category:Arithmetic ja:除法 simple:Division th:การหาร

Division (disambiguation)

Division may mean:
- Division (mathematics), the opposite operation to multiplication.
- Division (electronics), digital implementation of Division (mathematics).
- Division (military), a unit typically consisting of from 10,000 to 20,000 troops.
- Division (sport), a group of comparable teams in organised sport who compete amongst themselves for a Divisional title.
- Division (botany), a classification of plants (the botanical counterpart to zoology's phylum).
- Division (vote), a manner by which the votes of legislators are recorded.
- Division (organisation), a subsidiary of a larger organisation.
- Division (subnational entity), a subnational entity similar to a state or perfecture.
- Cell division, the process in which biological cells multiply.
- A county constituency or Parliamentary seat in the United Kingdom Parliament.
- A subpart of a hundred (division), former administrative unit in England and Wales.
- Police division, a large territorial unit (now often referred to as a Basic Command Unit) of the British police.
- Segmentation (disambiguation)
- Continental divide, the geographical term for separation between land masses.
- North-South divide, the geopolitical term for the division between wealthy nations and poor nations.
- In pipe organs a division is a grouping of pipe ranks, often with its own case, windchest, and keyboard.

Arithmetic

Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as a synonym for number theory. It is the oldest and simplest branch of mathematics, used widely by almost everyone from simple daily counting to more advanced science and business.

Arithmetic operations

The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations. The arithmetic of natural numbers, integers, rational numbers (in the form of vulgar fractions), and real numbers (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic. However, in adult life, many people prefer to use tools such as calculators, computers, or the abacus to perform the more complex arithmetical computations.

Number theory

The term arithmetic is also used to refer to number theory. This includes the properties of integers related to primality, divisibility, and the solution of equations by integers, as well as modern research which is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. A Course in Arithmetic by Serre reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry.

Subtraction

Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. Subtraction is denoted by an minus sign in infix notation. The traditional names for the parts of the formula :c − b = a are minuend (c) − subtrahend (b) = difference (a). The words "minuend" and "subtrahend" are virtually absent from modern usage, while "difference" is very common. Subtraction is used to model several closely related processes: #From a given collection, take away (subtract) a given number of objects. #Combine a given measurement with an opposite measurement, such as a movement right followed by a movement left, or a deposit and a withdrawal. #Compare two objects to find their difference. For example, the difference between $800 and $600 is $800 − $600 = $200. In mathematics, it is often useful to view or even define subtraction as a kind of addition, the addition of the opposite. We can view 7 − 3 = 4 as the sum of two terms: seven and negative three. This perspective allows us to apply to subtraction all of the familiar rules and nomenclature of addition. Subtraction is not associative or commutative— in fact, it is anticommutative— but addition of signed numbers is both.

Basic subtraction: integers

anticommutative Imagine a line segment of length b with the left end labeled a and the right end labeled c. Starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition: :a + b = c. From c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction: :c − b = a. addition Now, imagine a line segment labelled with the numbers 1, 2, and 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3. It takes 2 steps to the left to get to position 1, so 3 − 2 = 1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended. To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, ...). From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0. But 3 − 4 is still invalid since it again leaves the line. The natural numbers are not a useful context for subtraction. The solution is to consider the integer number line (…, −3, −2, −1, 0, 1, 2, 3, …). From 3, it takes 4 steps to the left to get to −1, so :3 − 4 = −1.

External links

Printable Worksheets: [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1214&CurriculumID=2&Method=Worksheet&NQ=24&NQ4P=3 One Digit Subtraction], [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1202&CurriculumID=2&Method=Worksheet&NQ=24&NQ4P=3 Two Digit Subtraction], and [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1273&CurriculumID=3&Method=Worksheet&NQ=24&NQ4P=3 Four Digit Subtraction]
- [http://www.cut-the-knot.org/Curriculum/Arithmetic/SubtractionGame.shtml Subtraction Game] at cut-the-knot
- [http://webhome.idirect.com/~totton/abacus/pages.htm#Subtraction1 Subtraction on a Japanese abacus] selected from [http://webhome.idirect.com/~totton/abacus/ Abacus: Mystery of the Bead] Category:Arithmetic ko:뺄셈 ja:減法 simple:Subtraction th:การลบ

0 (number)

:This page is about the number and numeral 0. For other uses of 0 or "zero", see 0 (disambiguation) 0 (zero), alternatively called naught or nought, is both a number and a numeral. It was the last numeral to be created in most numerical systems, as it is not a counting number (which is to say, one begins counting at the number 1) and was in many eras and places represented only by a gap or mark very different from the other numerals.

0 as a number

0 is the integer that precedes the positive 1, and all positive integers, and follows -1, and all negative integers. In most (if not all) numerical systems, 0 was identified before the idea of 'negative integers' was accepted. Zero is a number which means nothing, null, void or an absence of value. For example, if the number of one's brothers is zero, then that person has no brothers. If the difference between the number of pieces in two piles is zero, it means the two piles have an equal number of pieces. Almost all historians omit the year zero from the proleptic Gregorian and Julian calendars, but astronomers include it in these same calendars. However, the phrase Year Zero may be used to describe any event considered so significant that it virtually starts a new time reckoning.

0 as a numeral

Year Zero The modern numeral 0 is normally written as a circle or (rounded) rectangle. On the seven-segment displays of calculators, watches, etc., 0 is usually written with six line segments (at right), though on some historical calculator models it was written with four line segments. This variant glyph has not caught on. It is important to distinguish the number zero (as in the "zero brothers" example above) from the numeral or digit zero, used in numeral systems where the position of a digit signifies its value. Successive positions of digits have higher values, so the digit zero is used to skip a position and give appropriate value to the preceding and following digits. The Babylonian numeral system used two narrow slanting wedges, similar to \\, for the equivalent of a positional zero numeral starting in about 400BC. A zero digit is not always necessary in a positional number system: decimal without a zero provides a possible counterexample. In fonts with text figures, 0 is usually the same height as a lowercase X, for example, Image:TextFigs036.png.

History

Etymology

The word zero comes ultimately from the Arabic sifr (صفر) meaning empty or vacant, a literal translation of the Sanskrit meaning void or empty. Through transliteration this became zephyr or zephyrus in Latin. The word zephyrus already meant "west wind" in Latin; the proper noun Zephyrus was the Roman god of the west wind (after the Greek god Zephyros). With its new use for the concept of zero, zephyr came to mean a light breeze—"an almost nothing" (Ifrah 2000; see References). The word zephyr survives with this meaning in English today. The Italian mathematician Fibonacci (c.1170-1250), who grew up in Arab North Africa and is credited with introducing the Arabic decimal system to Europe, used the term zephyrum. This became zefiro in Italian, which was contracted to zero in the Venetian dialect, giving the modern English word. As the decimal zero and its new mathematics spread through a Europe that was still in the Middle Ages, words derived from sifr and zephyrus came to refer to calculation, as well as to privileged knowledge and secret codes. According to Ifrah (2000), "in thirteenth-century Paris, a 'worthless fellow' was called a... cifre en algorisme, i.e., an 'arithmetical nothing.' " (algorithm is also a borrowing from the Arabic, in this case from the name of the 9th-century mathematician al-Khwarizmi.) The Arabic root gave rise to the modern French chiffre, which means digit, figure, or number; chiffrer, to calculate or compute; and chiffré, encrypted; as well as to the English word cipher. Today, the word in Arabic is still sifr, and cognates of sifr are common throughout the languages of Europe. A few additional examples follow.
- Polish: cyfra, digit; szyfrować, to encrypt
- German: Ziffer, digit, figure, numeral, cypher
- French: zéro, zero
- Spanish: cifra, figure, numeral, cypher, code; cero, zero
- Swedish: siffra, numeral, sum, digit Note that zero in Greek is translated as Μηδέν (Meithen).

Babylonians and Greeks

By the mid second millennium BC, Babylonians had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. However, "... a tablet found at Kish ... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place" ([http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html] and natural number). Records show that the Ancient Greeks seemed unsure about the status of zero as a number: they asked themselves "how can 'nothing' be something?", leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero. (The ancient Greeks even questioned that 1 was a number.)

First use of the number

An early use of zero by the Indian mathematician Pingala (possibly 5th-3rd century BC), implied at first glance by [http://home.ica.net/~roymanju/Binary.htm Binary Numbers in Ancient India], is only the modern binary representation using 0 and 1 of Pingala's binary system, which used short and long syllables (the latter equal in length to two short syllables) as described in [http://www.sju.edu/~rhall/Rhythms/Poets/arcadia.pdf Math for Poets and Drummers] (pdf), making it similar to Morse code. In Pingala's system, four short syllables meant one, not zero. Nevertheless, he does use the Sanskrit word Shunya to refer to the concept of void, which was fairly similar to the concept of zero [http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html]. The late Olmec people of south-central Mexico began to use a true zero (a shell glyph) in the New World possibly by the 4th century BC but certainly by 40 BC, which became an integral part of Maya numerals, but did not influence Old World numeral systems. By 130, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70). Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), 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 computists (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol.

Zero as a decimal digit

The earliest known decimal digit zero is thought to have been introduced by Indian mathematicians sometime around the 3rd century. It was written in the shape of a dot, and consequently called "dot". An early documented use of the zero by Brahmagupta dates to 628. He treated zero as a number and discussed operations involving it. By this time (7th century) the concept had clearly reached Cambodia, and documentation shows the idea later spreading to China and the Islamic world. The Hindu-Arabic number system reached Europe in the late 11th century, via Andalusia, together with knowledge of astronomy and instruments like the astrolabe. The Italian mathematician Fibonacci was instrumental in bringing the system into European mathematics around 1200, though he spoke of the "sign" zero, not as a number. It was not until the 1600s that decimal notation began to come into widespread use in the Occident.

In mathematics

Zero (0) is both a number and a numeral. The natural number following zero is one and no natural number precedes zero. Zero may or may not be counted as a natural number, depending on the definition of natural numbers. Zero is neither prime nor composite. In set theory, the number zero is the cardinality of the empty set: if one does not have any apples, then one has zero apples. In fact, in certain axiomatic developments of mathematics from set theory, zero is defined to be the empty set. The following are some basic rules for dealing with the number zero, first described in Brahmasphutasiddhanta. These rules apply for any complex number x, unless otherwise stated.
- Addition: x + 0 = x and 0 + x = x. (That is, 0 is an identity element with respect to addition.)
- Subtraction: x − 0 = x and 0 − x = − x.
- Multiplication: x · 0 = 0 · x = 0.
- Division: 0 / x = 0, for nonzero x. But x / 0 is undefined, because 0 has no multiplicative inverse, a consequence of the previous rule. For positive x, as y in x / y approaches zero from positive values, its quotient increases toward positive infinity, but as y approaches zero from negative values, the quotient increases toward negative infinity. The different quotients confirms that division by zero is undefined.
- Exponentiation: x⁰ = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0^x = 0. The expression "0/0" is an "indeterminate form". That does not simply mean that it is undefined; rather, it means that if f(x) and g(x) both approach 0 as x approaches some number, then f(x)/g(x) could approach any finite number or ∞ or −∞; it depends on which functions f and g are. See L'Hopital's rule. The sum of 0 numbers is 0, and the product of 0 numbers is 1.

Extended use of zero in mathematics

- Zero is the identity element in an additive group or the additive identity of a ring.
- A zero of a function is a point in the domain of the function whose image under the function is zero. See zero (complex analysis).
- In geometry, the dimension of a point is 0.
- In analytic geometry, 0 is the origin.
- In nonstandard analysis the number zero is taken as an infinitesimal element of a non-principal ultrafilter.
- The concept of "almost" impossible in probability. More generally, the concept of almost nowhere in measure theory.
- A zero function is a constant function with 0 as its only possible output value; i.e.,

f(x) = 0

. A particular zero function is a zero morphism. A zero function is the identity in the additive group of functions.
- The zero of a function is a preimage of zero, also called the root of a function.
- Zero is one of three possible return values of the Möbius function. Passed an integer x² or x²y, the Möbius function returns zero.
- It is the number of n×n magic squares for n = 2.
- It is the number of n-queens problem solutions for n = 2, 3.
- Zero is neither a prime nor a composite number.

In physics

The value zero plays a special role for a large number of physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, where as it for others is more or less arbitrarily chosen. For example, on the kelvin temperature scale, zero is the coldest possible temperature (so that negative temperatures are non-existent), where as on the celsius scale, zero is arbitrarily defined to be at the freezing point of water. Measuring sound intensity in decibels or phons, the zero level is arbitrarily set at a reference value, e.g. at a value for the threshold of hearing.

In computer science

Numbering from 1 or 0?

Human beings usually number things starting from one, not zero. Yet in computer science zero has become the popular indication for a starting point. For example, in almost all old programming languages, an array starts from 1 by default, which is natural for humans. As programming languages have developed, it has become more common that an array starts from zero by default (zero-based). One reason for this convention is that modular arithmetic normally describes a set of N numbers as containing 0,1,2,...N-1 in order to contain the additive identity. Because of this, many arithmetic concepts (such as hash tables) are less elegant to express in code unless the array starts at zero. Another reason to use zero-based array indices is that it can improve efficiency under certain circumstances. To illustrate, suppose a is the memory address 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: :

a + s \times (i-1)

where s is the size of each element. In contrast, if the index numbers count from 0, the expression becomes this: :

a + s \times i

This simpler expression can be more efficient to compute in certain situations. 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: :

a' + s \times i

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. 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". For this reason, the first element is often referred to as the zeroth element to eliminate any possible doubt (though, strictly speaking, this is unnecessary and arguably incorrect, since the meanings of the ordinal numbers are not ambiguous).

Null value

In databases a field can have a null value. 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 three-valued logic. 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. This is owing to the notion that records in a relational database are a set of key/value tuples. A null value, notionally, indicates not that the record has some particular value – "null" – for a given column, but rather that the record has no value at all for that particular column.

Null pointer

A null pointer 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 compile time 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 (possibly different values for different data types), and has no particular association with zero.

Negative zero

In some signed number representations (but not the two's complement representation predominant today) and most floating point 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 negative zero. Representations with negative zero can be troublesome, because the two zeroes will compare equal but may be treated differently by some operations.

Distinguishing zero from O

negative zero The oval-shaped zero (appearing like a rugby ball stood on end) and circular letter O together came into use on modern character displays. The zero with a dot in the centre seems to have originated as an option on IBM 3270 controllers (this has the problem that it looks like the Greek letter Theta). The slashed zero, looking identical to the letter O other than the slash, is used in old-style ASCII graphic sets descended from the default typewheel on the venerable ASR-33 Teletype. This format causes problems for certain Scandinavian languages which use Ø as a letter. The convention which has the letter O with a slash and the zero without was used at IBM and a few other early mainframe makers; this is even more problematic for Scandinavians because it means two of their letters collide. Some Burroughs/Unisys equipment displays a zero with a reversed slash. And yet another convention common on early line printers left zero unornamented but added a tail or hook to the letter-O so that it resembled an inverted Q or cursive capital letter-O. The typeface used on some European number plates for cars distinguish the two symbols by making the O rather egg-shaped and the zero more circular, but most of all by opening the zero on the upper right side, so here the circle is not closed any more (as in German plates). In paper writing one may not distinguish the 0 and O at all, or may add a slash across it in order to show the difference, although this sometimes causes ambiguity in regard to the symbol for the null set.

In other fields

- In some countries, 0 on a telephone calls for operator assistance. On the BlackBerry the 0 key also functions as a spacebar.
- In Braille, the numeral 0 has the same dot configuration as the letter J.
- DVDs that can be played in any region are sometimes referred to as being "region 0".

References

- [http://www.amazon.com/exec/obidos/ASIN/0471393401/qid=1124292648/sr=2-2/ref=pd_bbs_b_2_2/102-7275474-2228915 The Universal History of Numbers: From Prehistory to the Invention of the Computer.] Georges Ifrah. Wiley (2000)
- [http://www.mediatinker.com/blog/archives/008821.html A Brief History of Zero] - Kristen McQuillin, July 1997 (revised January 2004)
- [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Zero.html A history of Zero]
- [http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM Zero Saga]
- [http://www.neo-tech.com/zero/part6.html The Discovery of the Zero]
- Charles Seife (2000). [http://www.amazon.com/exec/obidos/tg/detail/-/0140296476/qid=1111606043/sr=8-1/ref=sr_8_xs_ap_i1_xgl14/104-6166861-2891133?v=glance&s=books&n=507846 "Zero: The Biography of a Dangerous Idea".] Publisher: Penguin USA (Paper). ISBN 0140296476 Category:Elementary arithmetic 0 Category:Integers ko:0 ja:0 simple:Zero th:0 (จำนวน)

Slash (punctuation)

A solidus, oblique or slash, /, is a punctuation mark. It is also called a diagonal, separatrix, shilling mark, stroke, virgule, slant, or forward slash.

Usage

History

This symbol goes back to the days of ancient Rome. In the early modern period, in the Fraktur script, which was widespread through Europe in the Middle Ages, one slash (/) represented a comma, while two slashes (//) represented a Dash. The two slashes eventually evolved into a sign similar to the Equals sign (=), then being further simplified to a single dash (-).

English

The most common use is to replace the hyphen to make clear a strong joint between words or phrases, such as "the Ernest Hemingway/William Faulkner generation". Yet very often it is used to represent the concept or, especially in instruction books. The symbol also appears in the phrase and/or, a prose representation of the logical concept of inclusive or. The State Legislature of Georgia, however, has banned this usage as cumbersome. The slash is often used (incorrectly) to separate the letters in a two-letter initialism, such as R/C (short for radio control) or even w/e (an internet slang abbreviation for whatever). Purists strongly discourage this misuse of the symbol, however, because it could potentially create confusion about its meaning. The virgule is also used to indicate a line break when quoting multiple lines from a poem, play, or headline. For a specialized use of the slash in the titles of fan fiction stories, see slash fiction. Note that the solidus and virgule are distinctly different typographic symbols with decidedly different uses. The solidus is significantly more oblique than the virgule. The character found on standard keyboards is the virgule and while most people lump the two characters together, (and when there is no alternative it is acceptable to use the virgule in place of the solidus,) they are different. The solidus is used in the display of ratios and fractions as in constructing a fraction using superscript and subscript as in “¹²³⁄₄₅₆”; the virgule is used for essentially any other textual purpose.

Linguistics

Slashes are used to enclose a phonemic transcription of speech.

Arithmetic

A solidus is used to separate the numerator and denominator in a vulgar fraction, or as a division operator in general. :3/8 (three eighths) :x = a / b (x equals a divided by b) Note that the special character Fraction slash U+2044, character ⁄ (the solidus or shilling mark proper), can be used instead of a virgule, and is preferred whenever possible. It is also found in many legacy Apple Macintosh character sets. Systems capable of fine typography should display the result as a true fraction with smaller numbers. Unicode also distinguishes the Division Slash U+2215 (∕) which may be more oblique than the normal solidus character.

Computing

Files

On Unix-like systems, the slash carries two distinct meanings. Its primary use, as with URLs, is to separate directory and file components of a path: :pictures/image.jpg :http://en.wikipedia.org/wiki/Slash_%28punctuation%29 A leading slash however represents the root directory of the Virtual file system; it is used when specifying absolute paths: :/home/joe/pictures/image.jpg It is sometimes called a "forward slash" to contrast with the backslash \, which is the path delimiter on MS-DOS and Microsoft Windows systems. These operating systems use the backslash rather than the slash because in the early days of CP/M—before directories were supported —the slash was chosen as the command-line option indicator: :dir /w /ogn Note however that the "forward slash" will be translated into a backslash by most versions of DOS and Windows, in contexts where there is little ambiguity with command-line options. Some people incorrectly refer to a slash as a "backslash", for instance when reading URLs out loud.

Chat

Many Internet Relay Chat and in-game chat clients use the slash to distinguish commands, such as the ability to join or part a chat room or send a private message to a certain user. :/join #services :/me sings a song about birds.

Programming

In computer programming, the solidus corresponds to Unicode and ASCII character 47, or 0x002F. It is used in the following settings:
- In most programming languages, / is used as a division operator,
- Comments in C and JAVA begin with / - (a slash and an asterisk), and ended with - / (the same characters in the opposite order).
- C++ and C99 introduced comments that begin with // (two slashes) and span a single line.
- In HTML and XML, a slash is used to indicate a closing tag. For example, in HTML, </em> ends a section of emphasized text that had been started with <em>.

Dates

Certain shorthand date formats use / as a delimiter, for example "9/16/2003" (in United States usage) or "16/9/2003" (in many other countries) means September 16, 2003. In Britain there was a specialized use in prose: 7/8 May referred to the night which starts the evening of 7 May and ends the morning of 8 May, totalling about 12 hours depending on the season. This was used to list night-bombing air-raids which would carry past midnight. Some police units in the US use this notation for night disturbances or chases. Contrariwise, the form with a hyphen, 7-8 May, would refer to the two-day period, at most 49 hours. This would commonly be used for meetings. The International Standard ISO 8601, in attempting to resolve this ambiguity, introduced problems of its own. According to this norm, dates must be written year-month-day using hyphens, but time periods are written as two standard dates separated by a slash: 1939-09-01/1945-05-08, for example, would be the duration of the Second World War in the European theatre, while 09-03/12-22 might be used for a fall term of a Western school, from September third to December twenty-second.

British money

Before decimalisation in the UK, / was used to separate pounds, shillings, and pence values. Notice how the dash is used to represent zero.

Alternative names

In the UK, the usual term for the mark is an oblique, although slash is gaining currency with increasing use of computers and also through the media, such as BBC Radio and mainstream television. Sometimes this is called stroke (and oblique stroke) , although that may be confused with the hyphen. Stroke is most commonly used among the North American amateur radio community. Among American telephone technical support representatives in the computer industry, the double-slash in a URL address is commonly nicknamed "whack-whack." For example, the representative may tell the user to go to a support website: "OK, open up a browser and type in the following address: http colon whack-whack, www dot supportsite dot com." The origins of this slang term are not known but may be related to the slang for the exclamation point (!) which support engineers, especially in password resets, often refer to as "bang." —C.J. Newton Category:Punctuation Category:Typography ja:スラッシュ (記号)

Fraction (mathematics)

: In mathematics, a fraction is a way of expressing a quantity based on an amount that is divided into a number of equal-sized parts. For example, each part of a cake split into four equal parts is called a quarter (and represented numerically as ¹⁄₄); two quarters is half the cake, and eight quarters would make two cakes. Mathematically, a fraction is a quotient of numbers, like ³⁄₄, or more generally, an element of a quotient field. In our cake example above, where a quarter is represented numerically as ¹⁄₄ the bottom number, (called the denominator) is the total number of equal parts making up the cake as a whole, and the top number (called the numerator) is the number of these parts we have. For example, the fraction ³⁄₄ represents three quarters. The numerator and denominator are the terms of the fraction. The word "numerator" is related to the word "enumerate." To enumerate means to "tell how many"; thus the numerator tells us how many fractional parts we have in the indicated fraction. To denominate means to "give a name" or "tell what kind"; thus the denominator tells us what kind of parts we have (halves, thirds, fourths, etc.). The word is also used in related expressions, like continued fraction, see Special cases below.

Arithmetic

There are four basic arithmetic operations, which in order of simplicity for fractions, includes (1) Multiplication (2) Addition (3) Subtraction (4) Division.

Addition

Adding Fractions

Adding fractions can be a little tricky, since you cannot simply add the numerators and denominators. For example, if we had a cake divided into three pieces, each piece would be 1/3. Then, if we try to add one piece from the cake divided into four pieces, and one piece from the cake divided into three pieces, what would be have? Well, we would have, um, 1/4 + 1/3 = ??? You can see this is NOT equal to 1/7 or 2/7 !! To add fractions together, they must be changed to equivalent values having the same fractional unit -- the same denominator -- in this case 1/12. How do we do this? By multiplying each fraction by 1. By one? Yes. 1 = 3/3 and 1 = 4/4. Now watch: 1/4 = 1/4 x 1 = 1/4 x 3/3 = 3/12. And 1/3 = 1/3 x 1 = 1/3 x 4/4 = 4/12. So now 1/4 + 1/3 = 3/12 + 4/12 = 7/12 and we have the correct result. Notice that we only add the numerators together. The denominator does not change, since we are working with the same fractional unit. Another way to see this is: 1/4 + 1/3 = 3/12 + 4/12 = 1/12 x (3 + 4) = 1/12 x 7 = 7/12. Lets take another example. If you add a half dollar to a quarter, what will you get? You know it's 75 cents, right? When we say 75 cents we have automatically, in our mind, changed each coin into cents (pennies): One half dollar = 50 cents; one quarter = 25 cents; so 1/2 + 1/4 = 50/100 + 25/100 = 75/100 or 75 cents. Of course, we could use a smaller denominator since we know one half dollar equals two quarters. I.e., 1/2 + 1/4 = 2/4 + 1/4 = 3/4. In words, one half plus one quarter equals two quarters plus one quarter equals three quarters, or 75 cents. So the trick is to find a common fractional unit -- a common denominator -- that will let us simply add the numerators together. Let's take one more example. Find 2/3 + 1/2. We see that the denominators are 3 and 2. We need to find a value that each denominator can be multiplied by to give a common value. Well, it's easy to see that we can multiply 3 by 2, and 2 by 3, to give a common denominator of 6. But remember, you cannot change the value of each fraction, so we must multiply both numerator and denominator by the same number. We now have: 2/3 + 1/2 = 2/2 x 2/3 + 3/3 x 1/2 = 4/6 + 3/6 = 7/6 or 1 + 1/6.

Multiplication

By whole numbers

If you consider the cake example above, if you have a quarter of the cake, and you multiple the amount by three, then you end up with three quarters. We can write this numerically as follows: :

3 \times  =

As another example, suppose that five people work for three hours out of a seven hour day (ie. for three seventh of the work day). In total, they will have worked for 15 hours (5 x 3 hours each), or 15 sevenths of a day. Since 7 seventh of a day is a whole day, 14 sevenths is two days, then in total, they will have worked for 2 days and a seventh of day. Numerically: :

5 \times  =  = 2

By fractions

If you consider the cake example above, if you have a quarter of the cake, and you multiple the amount by a third, then you end up with a twelfth of the cake. In other words, a third of a quarter (or a third times a quarter), is a twelfth. Why? Because we are splitting each quarter into three pieces, and four quarters times three makes 12 parts (or twelfths). We can write this numerically as follows: :

\times  =

As another example, suppose that five people do an equal amount work that totals three hours out of a seven hour day. Each person will have done a fifth of the work, so they will have worked for a fifth of three sevenths of a day. Numerically: :

\times  =

General rule

You may have noticed that when we multiply fractions, we simply multiply the two numerators (the top numbers), and multiply the two denominators) (the bottom numbers). For example: :

\times  =  =

By mixed whole number/fractions

If we are multiplying fractions that include a whole number component, then it is best to convert the whole number into a fraction. For example: :

3 \times 2 = 3 \times \left ( \right ) = 3 \times  =  = 8

In other words,

2

is the same as

\left ( \right )

, making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total). And 33 quarters is

8

since 8 cakes, each made of quarters, is 32 quarters in total.

Commutativity

It is also worth recalling that multiplication is commutative which just means that the order of the numbers we are multiplying does not matter. In other words, three lots of a quarters is equivalent to a quarter of three; numerically: :

3 \times  =  \times 3 =

Note that when talking, we say "three times a quarter", but "a quarter of three", the implication being that in the latter example, we are talking about a fractional part of a larger number.

Special cases

- A vulgar fraction (or common fraction) is a rational number written as one integer (the numerator) divided by a non-zero integer (the denominator). The line that separates the numerator and the denominator is called the vinculum. Rational numbers are the quotient field of integers.

Particular vulgar fractions

- irreducible fraction: a vulgar fraction "in lowest terms", where the numerator is an integer, the denominator is a positive integer, and the highest common factor of the numerator and the denominator is 1;
- proper fraction: a vulgar fraction with (absolute) value between 0 and 1;
- improper fraction: a vulgar fraction with a (absolute) value greater than 1;
- unit fraction: a vulgar fraction with a numerator of 1;
- Egyptian fraction: the sum of distinct unit fractions;
- decimal fraction: a vulgar fraction where the denominator is a power of 10;
- dyadic fraction: a vulgar fraction in which the denominator is a power of two.

Other fractions

- A mixed fraction: A mixed fraction is an integer plus a proper fraction.
- A compound fraction is a fraction where the numerator or denominator (or both) contain fractions.
- Rational functions are the quotient field of polynomials (over some integral domain). Let us end with the only example on this page where the "fraction" is not an element of a quotient field:
- A continued fraction is an expression such as

a_0 + \frac

, where the a_i are integers. The term partial fraction is used in algebra, when decomposing rational functions. However, a partial fraction is an expression of a particular decomposition, and so is more than just an element of a quotient field. The term irrational fraction is sometimes used to indicate a magnitude whose quotient with another fixed magnitude is irrational, e.g. "1 is an irrational fraction of 2π". "Fraction", in this sense, simply means "a part of the whole", not a strict ratio in the mathematical sense. Taking the latter meaning, the term is an oxymoron.

Pedagogical tools

In Primary Schools, fractions have been demonstrated through Cuisenaire rods. See also the external links below.

External links

- [http://www.mathfactcafe.com Curricula for Creating Fractions]
- [http://www.ericdigests.org/2000-2/fractions.htm Curricula for Teaching about Fractions]
- [http://www.ericdigests.org/2004-1/fractions.htm Teaching Fractions: New Methods, New Resources]
- [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1551&CurriculumID=4&Method=Worksheet Worksheets: Identifying Fractions]
- [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1364&CurriculumID=4&Method=Worksheet Worksheets: Improper Fractions to Mixed Numbers]
- [http://www.math-lessons.ca Curricula for Teaching about Equivalent Fractions]
- [http://www.quiz-tree.com/Fractions_Practice_main.html Free online quizzes about Fractions]
- Category:Mathematical disambiguation Category:Elementary arithmetic Category:Numbers ja:分数

Integer

The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. They are also known as the whole numbers, although that term is also used to refer only to the positive integers (with or without zero). Like the natural numbers, the integers form a countably infinite set. The set of all integers is usually denoted in mathematics by a boldface Z (or blackboard bold,

\mathbb

), which stands for Zahlen (German for "numbers"). The term rational integer is used, in algebraic number theory, to distinguish these 'ordinary' integers, in the rational numbers, from other concepts such as the Gaussian integers.

Algebraic properties

Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under the operation of division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer. The following table lists some of the basic properties of addition and multiplication for any integers a, b and c. In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, since every nonzero integer can be written as a finite sum 1 + 1 + ... 1 or (−1) + (−1) + ... + (−1). In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, note that not every integer has a multiplicative inverse; e.g. there is no integer x such that 2x = 1, because the left hand side is even, while the right hand side is odd. This means that Z under multiplication is not a group. All the properties from the above table taken together say that Z together with addition and multiplication is a commutative ring with unity. In fact, Z provides the motivation for defining such a structure. The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z is not a field. The smallest field containing the integers is the field of rational numbers. This process can be mimicked to form the quotient field of any integral domain, where an integral domain is a commutative ring with unity such that whenever ab = 0, either a = 0 or b = 0. Although ordinary division is not defined on Z, it does possess an important property called the division algorithm: that is, given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ r < |b|, where |b| denotes the absolute value of b. The integer q is called the quotient and r is called the remainder, resulting from division of a by b. This is the basis for the Euclidean algorithm for computing greatest common divisors. Again, in the language of abstract algebra, the above says that Z is a Euclidean domain. This implies that Z is a principal ideal domain and any positive integer can be written as the products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.

Order-theoretic properties

Z is a totally ordered set without upper or lower bound. The ordering of Z is given by : ... < −2 < −1 < 0 < 1 < 2 < ... An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: # if a < b and c < d, then a + c < b + d # if a < b and 0 < c, then ac < bc. (From this fact, one can show that if c < 0, then ac > bc.)

Integers in computing

An integer (sometimes known as an "int", from the name of a datatype in the C programming language) is often a primitive datatype in computer languages. However, integer datatypes can only represent a subset of all integers, since practical computers are of finite capacity. Variable-length representations of integers, such as bignums, can store any integer that fits in the computers memory. Other integer datatypes are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10). In contrast, theoretical models of digital computers, such as Turing machines, typically do have infinite (but only countable) capacity.

Quotations

God invented the integers, all else is the work of man. Kronecker

External links

- [http://www.positiveintegers.org The Positive Integers - divisor tables and numeral representation tools] Category:Elementary mathematics Category:Group theory Category:Integers Category:Elementary number theory Category:Set theory ko:정수 ja:整数 th:จำนวนเต็ม

Calculator

A calculator is a device for performing numerical calculations. The type is considered distinct from both a calculating machine and a computer in that the calculator is a special-purpose device that may not qualify as a Turing machine. Although modern calculators often incorporate a general purpose computer, the device as a whole is designed for ease of use to perform specific operations, rather than for flexibility. The complexity of calculators varies with the intended purpose. A simple one with only four functions (addition, subtraction, multiplication and division and perhaps a single-number memory) may be useful for everyday activities such as shopping or checking a bill. More complex ones may include complex mathematical functions suitable to engineering or accounting as well as a substantial memory and the ability to execute moderately complex programs. Since the late-1980's, it has become common to incorporate simple calculators in other small devices, such as mobile phones, pagers or wrist watches. In most developed countries, students use calculators for schoolwork. There was some initial resistance to the idea out of fear that basic arithmetic skills would suffer. There remains disagreement about the importance of the ability to perform calculations by hand or "in the head", with some curricula restricting calculator use until a certain level of proficiency has been obtained, while others concentrate more on teaching estimation techniques and problem-solving.

Overview

Modern calculators are electrically powered, most often by battery, and are made by numerous manufacturers, in countless shapes and sizes varying from cheap, give-away, credit-card sized models to more sturdy adding machine-like models with built-in printers. Only a very few companies develop and make modern professional engineering and finance calculators: The most well-known are Casio, Sharp, Hewlett-Packard (HP) and Texas Instruments (TI). Such calculators are good examples of embedded systems. They are also often complex enough to be programmed; calculator applications include algebraic equation solvers, financial models and even games. In the near past, mechanical and clerical aids such as abacuses, comptometers, Napier's bones, books of mathematical tables, slide rules, adding machines, were used for serious numeric work, and the word "calculator" denoted a person (most often male) who did such work for a living using such aids as well as pen and paper. This semi-manual process of calculation was tedious and error-prone.

Electronic calculators

Today most calculators are handheld microelectronic devices, but in the past some calculators were as large as today's computers. The first mechanical calculators were mechanical desktop devices, which were soon replaced by electromechanical desktop calculators, and then by electronic devices using first thermionic valves, then transistors, then hard-wired integrated circuit logic. A pocket calculator is a small battery-powered or solar powered electronic digital computer made possible by integrated circuit and semiconductor technology. Typically they are limited to an 8–10 digit single-number display and a few basic functions of arithmetic, but some modern calculators have more of the features of a general-purpose computer. Pocket calculators rendered the slide rule obsolete. Calculators vary in their capabilities. Some are limited to only basic arithmetic; others support trigonometric, statistical and other mathematical functions. The most advanced modern calculators are programmable, can display graphics, and include features of computer algebra systems.

Personal computing

Personal computers and personal digital assistants can perform general calculations in a variety of ways:
- computers often have a separate calculator program, varying from one that just emulates a simple calculator, such as Microsoft Calculator, to advanced spreadsheet programs such as Excel or OpenOffice.org Calc
- for more advanced calculations one can use a computer algebra program, such as Mathematica, Maple or Matlab.
- browsers can perform calculations using client-side scripting, e.g. using Client-side JavaScript by entering "javascript:alert(12
- 13)" in the address bar (the answer 156 appears in a separate alert window) or "document.write (12
- 13)" in a HTML file, preceded with "<script type="text/javascript">" and followed by "</script>".
- an interpreter or compiler for a general programming language can be used
- calculations can also be performed server-side, e.g. with the calculator feature of the Google search engine

History

Origin: The Abacus

calculator feature of the Google search engine The first calculators were abacuses, and were often constructed as a wooden frame with beads sliding on wires. Abacuses were in use centuries before the adoption of the written Arabic numerals system and are still widely used by merchants and clerks in China and elsewhere.

The 17th century

Wilhelm Schickard built the first automatic calculator called the "Calculating Clock" in 1623. Some 20 years later, in 1645, French philosopher Blaise Pascal invented the calculation device later known as Pascal's calculator, which was used for taxes in France until 1799. The German philosopher G.W.v.Leibniz also produced a calculating machine.

1930s to 1960s

calculating machine From approximately the 1930s through the 1960s, mechanical calculators were often used (see Mechanical Calculator under History of computing hardware). These desktop devices were motor-driven and had multiple columns of keys for each digit. Addition and subtraction were performed in a single operation, as on a conventional adding machine, but multiplication and division were accomplished by repeated mechanical additions and subtractions. Handheld mechanical calculators such as the Curta continued to be used until they were displaced by electronic calculators in the 1970s. In 1954, IBM demonstrated a large all-transistor calculator and, in 1957, they released the first commercial all-transistor calculator (the IBM 608). In October 1961, the world's first all-electronic desktop calculator, the Bell Punch/Sumlock Comptometer ANITA Mk.VII was released. This British designed-and-built machine used vacuum tubes in its circuits and cold-cathode nixie tubes for its display. It was superseded, technologically, in 1964 when Sharp introduced the CS-10A—the world's first all-transistor desktop calculator—which weighed 25 kg (55 lb) and cost 500,000 yen (~US$2500). The first handheld electronic calculators went on sale in 1970 with models from Japanese manufacturers Sharp and Canon weighing around 770 g (1.7 lb).

1970s to mid-1980s

In the early 1970s, the Monroe EPIC programmable calculator came on the market. A large desk-top unit, with an attached floor-standing logic tower, it was capable of being programmed to perform many computer-like functions. However, the only branch instruction was an implied unconditional branch (GOTO) at the end of the operation stack, returning the program to its starting instruction. Thus, it was not possible to include any conditional branch (IF-THEN-ELSE) logic. During this era, the absence of the conditional branch was sometimes used to distinguish a programmable calculator from a computer. The first pocket-sized calculator, the Bowmar 901B (popularly referred to as The Bowmar Brain), measuring 5.2×3.0×1.5 in (131×77×37 mm), came out in the fall of 1971, with four functions and an eight-digit red LED display, for $240, while in August 1972 the four-function Sinclair Executive became the first slimline pocket calculator measuring 5.4×2.2×0.35 in (138×56×9 mm) and weighing 2.5 oz (70g). It retailed for around $150 (GB£79). By the end of the decade, similar calculators were priced less than $10 (GB£5). The first pocket calculator with scientific functions, i.e. the first slide rule-replacing model, was the 1972 HP-35 from Hewlett Packard (HP); it, along with all later HP engineering calculators, used reverse Polish notation (RPN) (where a calculation like "6 – 2" is performed by pressing "6", "Enter↑", "2", and "–"; instead of algebraically: "6", "–", "2", "="). In 1973, Texas Instruments (TI) introduced the SR-10, (SR signifying slide rule) a hand-held algebraic notation calculator, which was later followed by the SR-11 and eventually the TI-30. The first programmable hand-held calculator was the HP-65, in 1974; it had a capacity of 100 instructions, and could store and retrieve programs with a built-in magnetic card reader. A year later the HP-25C introduced continuous memory, i.e. programs and data were retained in memory during power-off. In 1979, HP released the first alphanumeric, programmable, expandable calculator, the HP-41C. It could be expanded with RAM (memory) and ROM (software) modules, as well as peripherals like bar code readers, microcassette and floppy disk drives, paper-roll thermal printers, and miscellaneous communication interfaces (RS-232, HP-IL, HP-IB).

Mid-1980s to present

HP-IB HP-IB calculator.]] The two leading manufacturers, HP and TI, released steadily more feature-laden calculators during the 1980s and 90s. At the turn of the millennium, the line between a graphing calculator and a PDA/ handheld computer was not always clear (forgetting the keyboard for the sake of the argument), as some very advanced calculators such as the TI-89 and HP-49G could differentiate and integrate functions, run word processing and PIM software, and connect by wire or IR to other calculators/computers. In March 2002, HP announced that the company would no longer produce calculators, which was hard to fathom for some fans of the company's products; the HP-48 range in particular had an extremely loyal customer base. Nevertheless, HP restarted their production of calculators in late 2003. The new models, however, reportedly didn't have the mechanical quality and sober design HP's earlier calculators were famous for (instead featuring the more "youthful" look and feel of contemporary competing designs from TI). The business calculator HP-12C is still produced. It was introduced in 1981 and is still being made with nearly no changes. In 2003 several new models were released, including an improved version of the HP-12C, the "HP-12C platinum edition".

Drawbacks

- Built-in inaccuracy commonly due to arithmetic underflow is a drawback occurring in many ordinary digital calculators. To obtain an example of this potential problem, the following exercise may be performed: enter the number one, divide by three, to reach 0.333 (recurring, i.e. followed by a theoretically infinite number of 3s), and then multiply by three to get back to one. On some calculators this operation will not work correctly, in that the result is given as 0.999 (recurring)—roughly speaking, this anomaly happens because the calculator works with only a finite number of decimals. It is important to note, however, that with infinite precision, .999... repeating is equal to one.
- Another kind of "drawback" resulting from the use, rather than the construction, of calculators, is the tendency of users to carelessly rely on the calculator's output without double-checking the magnitude (in practice, the placement of the decimal separator) of the result. This problem was all but nonexistent in the era of slide rules and pencil-and-paper calculations, when the task of establishing the magnitudes of results had to be done by the (sufficiently meticulous) user.

Trivia

- The word "calculator" is occasionally used as a pejorative term to describe an inadequately capable general-purpose microcomputer. The synonym of this meaning is "bitty box", as discussed in the Jargon file.
- A curious episode of the mid 1970s involved the Melcor 635, a scientific calculator with a bug in its trigonometric functions. Because the CORDIC algorithms used in most calculators cannot compute the inverse trigonometric functions of zero, these need to be hardcoded — and some engineer at Melcor got it wrong. For any input other than exactly zero, even for instance 1.0E-99, the calculator worked correctly; the user simply had to remember not to compute the arc-cosine of zero. The company discovered this after making 50,000 calculators. The upshot was an advertisement in Scientific American headlined 'Somebody Goofed', offering these calculators for sale at half-price.
- As many schoolchildren and students know, some words and simple phrases can be written using an ordinary seven-segment display calculator; this involves entering certain numbers and then viewing the resulting words by turning the calculator display upside-down.

Patents

- – Complex computer – G. R. Stibitz (electromechanic device that would calculate, record, and print results)
- – Miniature electronic calculator – J. S. Kilby (TI electromechanic device)

External links

- [http://www.ti.com/corp/docs/company/history/calc.shtml On TI's US Patent No. 3819921] – From TI's own website
- [http://sharp-world.com/corporate/info/his/h_company/1994/ 30th Anniversary of the Calculator] – From Sharp's web presentation of its history; including a picture of the CS-10A desktop calculator
- [http://www.maths.hscripts.com/ Online Calculators and Converters]
- [http://web.peoriadesignweb.com/calculator Online Calculator Software]
- [http://www.satsig.net/seticalc.htm Online deep space SETI range calculator]
- [http://ostermiller.org/calc/calculator.html JavaScript Scientific Calculator] – Scientific notation, hex, octal, decimal, binary, and math functions; requires JavaScript (from ostermiller.org)
- [http://www.oldcalculatormuseum.com The Old Calculator Web Museum]
- [http://www.calculators.de Calculator Museum]
- [http://www.taswegian.com/MOSCOW/soviet.html Museum of Soviet Calculators]
- [http://www.rk86.com/frolov/calcolle.htm Soviet Calculators Collection]
- [http://www.vintagecalculators.com/index.html Vintage Calculators]
- [http://www.lendingok.com various calculators]
- [http://www.cut-the-knot.org/Curriculum/Arithmetic/BrokenCalculator.shtml Broken Calculator]
- [http://www.graphcalc.com GraphCalc – an Open Source graphing calculator program]
- [http://www.binarythings.com/hidigit/ HiDigit scientific calculator]
- [http://www.hpmuseum.org The Museum of HP Calculators] ([http://www.hpmuseum.org/prehp.htm slide rules/mech. section])
- [http://www.hydrix.com/wiki/ HP Calculator Wiki]
- [http://www.typeonline.co.uk/number_pad_lesson1.html Number pad typing tutorial]
- [http://www.casiocalc.org International Casio Calculator Community]
- [http://www.graph100.com French Casio Calculator Community]
- Calculator Category:Mathematical tools Category:Office equipment ja:電卓 th:เครื่องคิดเลข

Colon (punctuation)

A colon is a punctuation mark, with one dot above another: ":".

Uses

- A colon can be used to start off a list when not using is or are and often with the following. :The major cities of the US include the following: New York, Chicago, and Los Angeles.
- A colon may be used to emphasize a word or phrase acting as an appositive. :John moved to a new state: Missouri.
- A colon can introduce a phrase which restates a previous statement. :The road was never ending: it seemed to go on forever.
- A colon can introduce a long quotation set off by indentation (this is not quoted). :John Smith stated: ::I could never live in the same place for more than a couple of years. I have wandering feet and I like to keep them happy.
- A colon can introduce a shorter quotation not set off and replaced a comma when such a quote is to be emphasized or is lengthy. Quotation marks are used. :The sign read: "Do not enter."
- Colons are also used after the salutation in a formal letter, though in the US this is falling out of favor.
- A colon is used between chapters and verses in many religious scriptures :John 3:16; The Quran, Sura 5:18
- A colon is used between the hour and the minutes when telling time (though a full stop is sometimes used instead). :The time is 10:45.
- A colon occurs between titles and subtitles :Star Wars Episode IV: A New Hope
- Note that a colon is never preceded by: "namely," "for example," "e.g.," or "that is." The original definiton for colon on this page read, "A colon is a punctuation mark, with one dot above another, e.g.: ":"." This definition is redundant because a colon implies "e.g." within its definition.

Mathematics

The colon is also used in mathematics to indicate ratio, and is also the standard sign for division in most non-English-speaking countries. In mathematical logic the colon is often used to represent "such that" in a relational phrase from predicate calculus. Unicode provides ratio U+2236 (∶, ∶) for such mathematical usage if the distinction is required.

Linguistics

A special triangular colon symbol is used in IPA to indicate a preceding long vowel. It is available in Unicode as Modifier letter triangular colon Unicode U+02D0 (). A regular colon is often used as a fallback when this character is not available.

Computer representation

In computer science, the colon character corresponds to the decimal value 58 (hexadecimal value 3A) in Unicode and ASCII character encodings.

The colon in foreign languages

In , the colon can appear inside words in a manner similar to the English apostrophe, between a word (or abbreviation) and its grammatical suffixes.

Other meanings

See Colon, the disambiguation page. Category:Punctuation Category:Typography ja:コロン (記号)

Ratio

: For the use of ratio as a human capacity, see reason. In algebra, a ratio is the relationship between two quantities. It is expressed as the quotient of two numbers, or as two numbers separated by a colon (pronounced "to"). A number that can be written as a ratio of two integers is a rational number. In physics, a ratio between two magnitudes of the same type of quantity gives a positive real number when the magnitudes are expressed relative to an absolute or natural zero. The ratio between a difference of two magnitudes to a third magnitude, such as a unit, gives a real number (i.e. positive or negative).

Examples

- If a school has a twenty-to-one student-teacher ratio, that means that there are twenty times as many students as teachers.
- The ratio of heights of the Eiffel Tower (300 m) and the Great Pyramid (137 m) is 300:137, so one structure is more than twice the height of the other (or more precisely, 2.2 times).
- The ratio of the mass of Jupiter to the mass of the Earth is approximately 317.8:1.
- The musical interval of a perfect fifth, the pitch ratio 3:2, consists of two pitches, one approximately 1.5 times the frequency of another.
- If two axles are connected by gear wheels, the number of times one axle turns for each turn of the other is known as the gear ratio. The best example being the number of turns of the pedals of a bicycle compared with number of turns of the bicycle's rear wheel.
- The ratio of hydrogen atoms to oxygen in water is 2:1, or two parts to one. Note the use of words such as "times", "parts", "number", etc. Because two objects are being compared using the same measure, ratios are unitless; the units cancel out of the ratio. For example, the ingredients in a recipe that required 500 grams and 300 grams of each, would be in the ratio of 5:3, with no units. Note also the difference between ratios and vulgar fractions. For example, if there are three raspberry candies and five blackcurrant candies, then the ratio of raspberry candies to blackcurrant candies is 3:5. This indicates that there are three fifths as many raspberry candies as blackcurrant candies. However the fraction of all the candies that are raspberry is three out of a total of all eight candies or 3/(3+5) = 3/8. Thus the chances of a randomly selected candy being raspberry are three in eight.

Long division

:For the album by Rustic Overtones, see Long Division. In arithmetic, long division is an algorithm for division of two real numbers. It requires only the means to write the numbers down, and is simple to perform even for large dividends because the algorithm separates a complex division problem into smaller problems. However, the procedure requires various numbers to be divided by the divisor: this is simple with single-digit divisors, but becomes harder with larger ones. A more generalized version of this method is used for dividing polynomials (sometimes using a shorthand version called synthetic division). In long division notation, 500 / 4 = 125 is denoted as follows: :

\begin
\quad 125\\
4\overline\\
\end

The method involves several steps: 1. Write the dividend and divisor in this form: :

4\overline

In this example, 500 is the dividend and 4 is the divisor. 2. Consider the leftmost digit of the dividend (5). Find the largest multiple of the divisor that is less than the leftmost digit: in other words, mentally perform "5 divided by 4". If this digit is too small, consider the first two digits. In this case, the largest multiple of 4 that is less than 5 is 4. Write this number under the leftmost digit of the dividend. Write the multiple divided by the divisor (4 divided by 4 = 1) above the line over the leftmost digit of the dividend. :

\begin
1\\
4\overline\\
4
\end

3. Subtract the digit under the dividend from the digit used in the dividend. Write the result (remainder) (5 − 4 = 1) under the bottom digit, then drop the zero (the second digit) to the right of it. :

\begin
1\\
4\overline\\
\underline\\
\;\,10
\end

4. Repeat steps 2 and 3, except use the number you just created to divide by, and write above and under the second digit. :

\begin
\,\,12\\
4\overline\\
\underline\\
\;\,10\\
\quad\underline\\
\quad\;\,20
\end

5. Repeat step 4 until there are no digits remaining in the dividend. The number written above the bar is the quotient, and the last remainder calculated is the remainder for the entire problem. :

\begin
\quad 125\\
4\overline\\
\underline\\
\;\,10\\
\quad\underline\\
\quad\;\,20\\
\quad\;\,\underline\\
\qquad0
\end

External links

- [http://www.mathsisfun.com/long_division.html Long Division] [http://www.mathsisfun.com/long_division2.html with Remainders]
- [http://www.mathsisfun.com/worksheets/long-division.php Practice Long Division with Printable Worksheets]

Fraction (mathematics)

Arithmetic

There are four basic arithmetic operations, which in or

Division (mathematics)

Notation

Computing division

Division of integers

Division of rational numbers

Division of real numbers

Division of complex numbers

Division of polynomials

Division in abstract algebra

Division and calculus

See also

External links

Division (disambiguation)

Arithmetic

Arithmetic operations

Number theory

See also

Subtraction

Basic subtraction: integers

See also

Algorithms

External links

0 (number)

0 as a number

0 as a numeral

History

Etymology

Babylonians and Greeks

First use of the number

Zero as a decimal digit

In mathematics

Extended use of zero in mathematics

In physics

In computer science

Numbering from 1 or 0?

Null value

Null pointer

Negative zero

Distinguishing zero from O

In other fields

See also

References

Slash (punctuation)

Usage

History

English

Linguistics

Arithmetic

Computing

Files

Chat

Programming

Dates

British money

Alternative names

Fraction (mathematics)

Arithmetic

Addition

Adding Fractions

Multiplication

By whole numbers

By fractions

General rule

By mixed whole number/fractions

Commutativity

Special cases

Particular vulgar fractions

Other fractions

Pedagogical tools

See also

External links

Integer

Algebraic properties

Order-theoretic properties

Integers in computing

Quotations

External links

Calculator

Overview

Electronic calculators