Conservapedia - User contributions [en]

Automorphism

2011-11-08T03:16:30Z

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An '''automorphism''' of a mathematical structure A is an [[Isomorphism|isomorphism]] from A to itself.{{citation needed}}

[[Category:Mathematics]]

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Asymptote

2011-11-08T03:16:26Z

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An '''asymptote''' is a [[line]] or [[curve]] that approaches another curve and is used to describe the behavior of that curve.

[[Category:Mathematics]]
[[Category:Geometry]]

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Arrow's Theorem

2011-11-08T03:16:23Z

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'''Arrow's Theorem''', also known as '''Arrow's impossibility theorem''' or '''Arrow's paradox''', is a seemingly [[paradox]]ical [[result]] in the [[mathematics|mathematical]] [[science]] of [[voting theory]]. The theorem was invented by the economist [[Kenneth Arrow]] in 1951, and applied the following year in the [[1952 Presidential Election]].

The theorem states, quite simply, that no [[representative]] voting system can simultaneously be "[[fair]]" and satisfy the following five [[condition]]s:

# ''Unrestricted domain.'' This condition means that the entire set of leaders (or whatever is being voted upon) must be decided by the voting system. It would clearly be both fair and satisfactory if the election, instead of determining the next [[President]], were to determine a Cabinet of 350 million people, a system known technically as [[direct democracy]] or [[referendum]]. This is not the situation being considered when we talk about Arrow's Theorem.
# ''Non-imposition.'' Non-imposition means, quite simply, that nobody is ''imposed upon'' to vote (or, contrariwise, not to vote). In the [[United States]], this criterion is satisfied by the [[secret ballot]].
# ''Non-dictatorship.'' Clearly this condition is satisfied as well.
# ''Positive association,'' also known as the ''Peter principle.'' Now we come to one of the trickier principles. The criterion of positive association means that if one voter — call him Alice — prefers Bush to Obama, then the final ranking of candidates will reflect that preference in some way. This condition was intended by Arrow to reflect real-world concerns regarding the protection of minority rights.
# ''Independence of irrelevant alternatives'' (known jocularly as the ''Nader corollary''). This condition means that a third-party candidate (such as [[Ralph Nader]] in the 2000 primaries) should not be able to "swing" the election to one side or another. In particular, a third party should be unable to draw votes away from the primary candidates, as Nader did to [[Al Gore]] in 2000, or as [[Pat Buchanan]] did to [[George W. Bush|Bush]]. This condition is perhaps the most relevant to the state of elections in the [[United States]] today.

In 1951, Kenneth Arrow proved that no system of voting, however so fair, can possibly satisfy all five of these constraints. For example, the [[Electoral College]] system used in the United States satisfies all of them except the last two (and in fact even the first, unrestricted domain, is questionable, since [[federal judge]]s are not elected by the people, but rather ''appointed'' by the bench).

The 1952 Presidential election demonstrated beyond doubt the correctness of Arrow's vision; incumbent [[Dwight D. Eisenhower]] beat his "egghead" Democratic challenger [[Adlai Stevenson]] in the largest landslide in human history. Nevertheless, the [[West Virginia]] Democratic primary was won by [[Averell Harriman]], who had never before held any elected office. This anomaly led to closer investigation of Arrow's results. For his work on modern voting methodology, Arrow was awarded the 1972 [[Nobel Prize]] in [[Economics]].

[[Category:Mathematics]]

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Arithmetic progression

2011-11-08T03:16:21Z

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An '''arithmetic progression''' is a sequence of numbers in which the difference between any number and its predecessor is constant, as in 2, 4, 6, 8, 10....

[[Dirichlet's theorem]] states that there are infinitely many [[prime number]]s in any arithmetic progression.

==See also==
*[[Geometric progression]]
[[Category:Mathematics]]

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Archimedean

2011-11-08T03:16:19Z

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A [[ring (mathematics)|ring]] '''R''' is said to be '''Archimedean''' if the ring is ordered, has a [[metric (mathematics)|metric]] <math>| |</math> and for all <math>x,y</math> in '''R''', x non-zero, there exists <math>n</math> in the natural numbers such that <math>n|x| > y </math>. Here concatentation with <math> n </math> denotes adding <math>n</math> times. Informally, a ring is Archimedean if it has no infinitely small or infinitely large elements. Examples of Archimedean rings include the [[real number]]s and the [[rational number]]s. Examples of non-Archimedean are less simple.

[[Category:Mathematics]]

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Arc

2011-11-08T03:16:17Z

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An '''arc''' a segment of the [[circumference]] of a [[circle]] between two points. It is also is a line in a [[network]] that connects two [[node]]s.

[[Category:Plane Geometry]]
[[Category:Mathematics]]

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Arabic numerals

2011-11-08T03:16:15Z

CoelAcant: /* References */AskAFly, your site has been fucked by the Zalgo Crew

'''Arabic numerals''' are the number system most commonly used in the world: 0, 1, 2, 3, 4, 5, 6, etc.

The term "Arabic numeral" is a misnomer that originated in 1847. The more accurate term is "Hindu-Arabic numeral", as the origin of the numerals is [[Hinduism]] in [[India]] between 400 B.C. and A.D. 400. The only connection with Arabs is that they communicated this system to [[Europe]] in the A.D. 900s, through Arabian [[mathematics|mathematicians]]. Most Arabs did not use these Indian numerals.

Note that Hindu-Arabic numerals include zero (0): Asian Indians were the first to discover and use this concept.

Westerners such as King John of England learned about Hindu-Arabic numerals as early as A.D. 1200, but there was resistance to converting from the Roman numeral system. It was not until the early 1500s, around the same time as the [[Reformation]], that Hindu-Arabic numerals became widely used in Western Europe.

==Numerals in the Arab World==
Despite the name as "Arabic numerals", the numerals used in the Arabic writing system are notably different, except for the "1" and "9".<ref name="Apic">
"Numeral pictograms" (of 10 Arab digit symbols), image:
[http://www.arabicnumerals.net/images/numrightleft.jpg ANjpg].
</ref>
In fact, an Arab 5 is designated by "o" which looks like zero, while zero is designated by a heavy dot "•". The basic ten digit symbols are as follows: <ref name=Apic/>
{| cellpadding=0
|
:: Number: 
| 0  || 1  || 2  || 3  || 4  || 5  || 6  || 7  || 8  || 9  || (the Arabic numerals)
|-
|
:: Arabic:
|•|| I|| Γ|| Ґ|| <big>ε</big>|| o|| 7|| <small>V</small>|| <big>^</big>|| <font face="Century Gothic, Symbol">9</font> || (numerals in the Arab World)
|}
Note how the Arab 6 looks like a seven ("7"). In typical usage, the Arab numeric symbols are more curved (than shown above), similar to the curvature of letters in the Arabic alphabet.<ref name=Apic/>
Some examples:  6004 = 7 • • <big>ε</big>  or  year 2010 = Γ • I • . Those numerals are believed to be derived from hand-signing of numbers, such as 5 (symbol "o") being the circle of touching the thumb to the fingertips.<ref name="ANnet">
"Origin of the Arabic Numerals", Adel S. Bishtawi,
[http://www.arabicnumerals.net/chapter_1.html AN-chap1].
</ref>

It is also worth noting that in Arabic writing, numbers are written left-to-right (as they are in the west), whilst words are written right to left.

==References==
{{reflist}}

[[Category:Mathematics]]

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Applied mathematics

2011-11-08T03:16:12Z

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'''Applied mathematics''' is the field of [[mathematics]] where mathematical ideas are put to immediate use in the real world. Applied mathematics receives lots of [[funding]] from the [[government]] and [[industry]] because it is clear what good comes from the money. Compare with [[pure mathematics]].

[[Category:Mathematics]]

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Apophysis

2011-11-08T03:16:09Z

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'''Apophysis''' is an open source, [[fractal]] flame generating software. A fractal flame is an image created by using an [[Iterated Function System]]; that is, the fractal is created by using repeating mathematical functions, which create a [[self-similar]] appearance. Fractal flames were created by Scott Draves in 1992.
[[category:Software]]
[[category:Mathematics]]

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Annulus

2011-11-08T03:16:07Z

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The '''annulus''', or '''annular region''', is the ringlike area bounded by two [[circle]]s, which are usually concentric (have the same center).
[[Category:geometry]]
[[Category:mathematics]]

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Analytic

2011-11-08T03:16:05Z

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'''Analytic Philosophy''' may refer to the study of the categories of logical propositions and philosophies known as analytics, which are defined by [[Logical positivism|logical positivists]]. Usually, logical [[positivism|positivists]] study the mechanisms of philosophy by applying the [[scientific method]] to solve philosophical problems. Analytical propositions are based on a priori knowledge, self-evident and generally [[tautology|tautalogical]]; they are definitively true or false by definition. Their truth or falsehood is often, but not always, obvious. Analytic propositions also encompass mathematical reasoning.

The term '''analytic''' may also describe a particular type of function in mathematics. Roughly speaking, a function is analytic if it is determined by its [[Series (mathematics)|series]]. A function on the complex numbers is analytic if it is [[derivative|differentiable]].

[[Category:Mathematics]]
[[Category:Logic]]

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Analysis

2011-11-08T03:16:02Z

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'''Analysis''' is the branch of [[mathematics]] concerned particularly with the concepts of function and limit. The subject has its origins in the quest to put [[calculus]] on a rigorous footing, and it is to this end that concepts like [[continuous]] and [[limit]] were first defined rigorously by [[Karl Weierstrass]] and [[Augustin-Louis Cauchy]]. Weierstrass gave the now-familiar "epsilon-delta" definition of a limit and worked to elaborate its basic properties. He demonstrated, for example, that there exist functions which are continuous functions which are not differentiable at any point. The possibility of such pathological functions could not have been imagined by [[Isaac Newton]] and others who had worked on calculus with less formal underpinnings.

Since its origins in the calculus, analysis has expanded into a vast subject with applications to every other branch of mathematics. It now includes other familiar topics like [[integral|Riemann integration]] and [[Lebesgue integral|Lebesgue integration]], picking up entire fields like measure theory along the way. Analysis also plays an important role in [[applied mathematics]], where it provides the machinery which make methods like [[Fourier series|Fourier analysis]] possible, and many deep results about solutions of [[differential equations]] may be proved by analytic methods. Besides these well-known subjects, there are numerous other subfields of analysis dealing with more specialized subjects:

* [[Real analysis]], the study of functions of real variables. Real analysis includes most of basic calculus.
* [[Complex analysis]], the study of functions of [[holomorphic]] functions of complex variables.
* [[Functional analysis]], the study of spaces of functions, a critical ingredient much of physics.
* [[Harmonic analysis]], dealing with Fourier series and their generalizations.
* [[Numerical analysis]], examining algorithms used for a variety of computations.

In addition to these branches, there are other less familiar branches of analysis, including ''p-adic'' and ''non-standard'' analysis.

== Sources ==

The New American Desk Encyclopedia, Penguin Group, 1989

[[Category: Mathematics]]

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Analog

2011-11-08T03:15:57Z

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An '''analog''' system is a [[mathematics|mathematical]] system in which numbers vary continuously instead of shifting between discrete states as a [[digital]] system does. For example, in an analog clock, the hour, minute, and second hands all move continuously. Halfway between 2:00 and 3:00, the hour hand will be pointing halfway between the 2 and the 3. In a digital clock, the numbers simply shift between discrete states: halfway between 2:00 and 3:00, the hour digit will still read 2.

A slide rule is an analog calculator whereas an abacus is a digital calculator because the slide rule's numbers vary continuously whereas the abacus' numbers are either all the way up or all the way down, there is no meaning for a bead on an abacus that is half way up or down. Analog [[television]]s are operated by machinery where the numbers vary continuously, whereas in digital television the numbers shift between discrete states; this is also true of analog computers and digital computers.

==Disadvantages of Analog Systems==
The need to make fine measurements when reading analog devices severely limits their practical accuracy, and digital devices are usually much more accurate in practice.

== Advantages of Analog Systems ==

On the other hand, analog devices usually have better resolution than digital ones, because digital systems must round the data to the nearest discrete state. For example, suppose the analog signal has values 2.91, 2.99, and 3.04. If the digital system only has room to store integers, all three values will be stored as 3. A digital system with more possible states - for example, each tenth rather than simply each integer - can mitigate this problem, as 2.91 will round down to 2.9 instead of up to 3. However, it cannot completely solve this problem: 2.99 and 3.04 will both still round to 3.

Digital systems can also only measure data at discrete times. We do not know what the signal - for example, the sound wave - is doing between the measurements. More than one possible wave can fit. This problem is called [[aliasing]]; it can also be mitigated by taking more frequent measurements but never totally solved.

[[Category:Computer Science]]
[[Category: Mathematics]]

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Alternate Angles

2011-11-08T03:15:56Z

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Alternate Angles are a pair of nonadjacent [[angle]]s on opposite sides of a transversal that cuts two [[line]]s.

The angles can be both exterior or both interior to the two lines. '''Interior alternate angles''' are inside the [[parallel]] lines, and on opposite sides of the transversal. '''Exterior alternate angles''' are outside the parralel lines and on opposite sides of the transversal.

[[Category:Geometry terms]]
[[Category:Geometry]]
[[Category:Mathematics]]

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Algorithm

2011-11-08T03:15:51Z

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An '''algorithm''' is a procedure for carrying out a task which, given an initial state, will terminate in a clearly defined end-state. It can be thought of as a recipe, or a step-by-step method, so that in following the steps of the algorithm one is guaranteed to find the solution or the answer. One commentator describes it as:<ref>David Berlinski, ''The Advent of the Algorithm: The Idea that Rules the World'' (Harcourt Inc.: 2000).</ref>

{{cquote|a finite procedure, written in a fixed symbolic vocabulary, governed by precise instructions, moving in discrete Steps 1, 2, 3, ..., whose execution requires no insight, cleverness, intuition, intelligence, or perspicuity, and that sooner or later comes to an end.}}

The name is derived from a corruption of [[Muhammad ibn Mūsā al-Khwārizmī|Al-Khwārizmī]], the Persian astronomer and mathematician who wrote a treatise in Arabic in 825 AD entitled: ''On Calculation with Hindu Numerals''. This was translated into Latin in the 12th century as ''Algoritmi de numero Indorum''. The title translates as "Al-Khwārizmī on the numbers of the Indians", with ''Algoritmi'' being the translator's rendition of the original author's name in Arabic. Later writers misunderstood the title, and treated ''Algoritmi'' as a Latin plural of the neologism ''Algoritmus'' or ''Algorithmus'', and took its meaning to be 'calculation method'. In the Middle Ages there were many variations of this, such as alkorisms, algorisms, algorhythms etc.

There are two main current usages:
# In elementary education, where an 'algorithm' is used for calculation, such as the decomposition algorithm, or the equal addition method of subtraction. Many of these algorithms are, in fact derivations from methods in Al-Khwārizmī's original treatise.
# In computing, where an algorithm is the methodology which underlies a computer program. This tells the computer what specific steps to perform (in what specific order) in order to carry out a specified task, such as calculating employees’ paychecks or printing students’ report cards.

All algorithms must adhere to two rules -

*They must work for any given input or network.
*They must have a single start point, and a single finish point.

There is a second type of algorithm - whereas most algorithms provably evaluate the most desirable end-state (for example, it is possible to mathematically prove that Dijkstra's algorithm gives the shortest route from one point to another) - others known as 'heuristic algorithms' cannot provably give the best solution (although they do give a fairly good result). While this may seem inferior, some problems are very difficult or even impossible to map out using normal algorithms, so heuristic ones are superior in these cases.

==Examples of Algorithms==

===Dijkstra's algorithm===

Dijkstra's algorithm finds the shortest path through a [[network]], from one point to another.

#Start by labelling the first [[node]] (1,0,0). The first number, the ordinal value, denotes the order in which the arcs were labelled, the second, the label value (the distance travelled thus far), while the third, the working value, denotes the possible distance to that point.
#Then, update the working values for nodes separated to the current node(s) by one [[arc]], by adding the weight of the arc to that node to the label value of the node at the other end. This value may decrease during the progression of the algorithm, as new, shorter, routes are found.
#Choose the node with the lowest working value, and promote its working value to the label value, and write in its new ordinal value. For example, if this was the second node labelled, its complete annotation would now read (2, x, x), where 'x' would be the distance between it and the first node.
#Now return to the second step. If there are no more nodes to be added, the algorithm has finished.

===Planarity algorithm===

The planarity algorithm is used to determine whether a given shape is 2-dimensional or not. It has found particular use in road and circuit board design to avoid cross-overs.

#Identify a [[Hamiltonian Cycle]] in the network.
#Redraw the network with the Hamiltonian cycle on the outside.
#Identify any crossings between lines.
#Choose an arc with crossings to stay inside the cycle. Move any arcs with crossings to the outside.
#Repeat from step 3 until there are no more crossings. If this step is completed, it shows that the shape is planar (2-d). If it cannot be completed, the shape is 3-dimensional and not planar.

===Critical path analysis===

Critical path analysis is used to determine the fastest/most efficient way of completing a series of tasks, which often depend upon each other.
===Kruskal's algorithm===

Kruskal's algorithm is used to find the minimum spanning tree in an network that you can see all of.
===Prim's algorithm===

Prim's algorithm is used to find the minimum spanning tree when only some of a network is visible.
===Bellman-Ford algorithm===

The Bellman-Ford algorithm is used to find the shortest path in a graph with negative weighted edges.
===Krim-Jacob algorithm===

The Krim-Jacob algorithm is used to determine if a problem can be solved with abstract time and memory.
===Bin-filling algorithm===

The Bin filling algorithm is used to find the most efficient way of combining several differently sized objects into a space(s) with a certain size. An example would be the problem of packing objects into the boot of a car; the bin-filling algorithm is in fact a mathematically formulated version of the rule of thumb 'put the big things in first'; but in can be used for many other problems - loading cars onto ferries, sending messages via [[routers]] on the [[internet]], etc. It is a heuristic algorithm, as it does not give a provably maximal solution (the only way to do this, until Vijay Vazirani invented a new form of [[approximation algorithm]], was to arrange the objects in every conceivable order).
==References==
<references/>
==See also==
*[[Computability]]
*[[Genetic algorithms]]
*[[Evolutionary algorithm]]
*[[Solved game]]
*[http://mathworld.wolfram.com/Algorithm.html MathWorld]
[[category:mathematics]]

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Algebra terms

2011-11-08T03:15:46Z

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{{Algebra Index}}
[[Category:Algebra Terms]]
[[Category:Algebra]]
[[Category:Mathematics]]

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Advanced calculus

2011-11-08T03:15:41Z

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'''Advanced calculus''' is a general term for courses in calculus beyond the introductory college-level course or advanced high-school level course. This is required as part of some college engineering programs, and can also serve as an introduction to advanced courses in mathematics departments. [[Actuary]] exams typically test knowledge of this material.

Topics included in advanced calculus are:

*[[Vector spaces]]
*[[Partial differentiation]]
*Maxima and minima
*Types of [[Integrals]]
*Infinite, Power and Fourier Series
*Laplace and Fourier Transforms
*Introduction to [[Differential Geometry]]
[[Category:mathematics]]

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Category:Advanced Mathematics

2011-11-08T03:15:39Z

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[[Category:Mathematics]]

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Additive property of equality

2011-11-08T03:15:37Z

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The '''additive property of [[equality]]''' states that:

:if a, b and c are real numbers such that a=b,

:then a+c = b+c

[[category:mathematics]]

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Additive inverse

2011-11-08T03:15:36Z

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The '''additive inverse''' of a [[complex number|complex]] or [[real number|real]] number x is the number y such that x and y add to equal the [[additive identity of addition|additive identity]], the number [[zero]]. The additive inverse is a [[function]] defined for all complex numbers, and is [[cyclical function|cyclical]] with period 2 ([[idempotent]]). However, for this function to exist in basic mathematics, one must first accept the existence of the [[negative numbers]]. This was a large impedence to early [[mathematics]], because early people had difficulty imagining something less than nothing.

[[Category:Mathematics]]

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Additive identity property

2011-11-08T03:15:33Z

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The '''Additive identity property''' is a [[Mathematics|mathematical]] property which states that if a number and zero are added, the answer will be the original number.

Example: 6+0=6
[[Category:Mathematics]]

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Additive identity

2011-11-08T03:15:27Z

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The '''additive identity''' for a [[ring]] or [[field]] is the [[element]], ''0'', of the set that, when added to any element ''a'' of the set, yields an answer of ''a''.

In the [[integers]], [[rational numbers]], and [[real numbers]], the additive identity is [[zero]].

For any number: a + 0 = a and 0 + a = a.

==See also==
[[Identity Element]]

[[Category:Algebra Terms]]
[[Category:Algebra]]
[[Category:Mathematics]]

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Addition

2011-11-08T03:15:25Z

CoelAcant: /* Formal definition */AskAFly, your site has been fucked by the Zalgo Crew

Addition is the mathematical operation that combines two or more numbers to produce their [[sum]].

For simple addition of [[whole number]]s, it is easy to add by counting steps. If we consider 6 + 4, then we can start at 6, then add 1 four times: 7, 8, 9, 10. Thus, 6 + 4 is 10.

For more complex addition, [[long addition]] can be carried out.

Addition is [[commutative]], thus a + b = b + a.

==Formal definition==

It is helpful to see addition in terms of [[function]]s - addition adds one number (a) to another number (b) to make their sum (c).

Thus,

a + b = c.

Its reverse function, or [[inverse operation]], is [[subtraction]], which moves the other way:

c - b = a.

It is possible to add a negative number, but this is the same as subtracting that number:

a + (-b) = a - b.

[[category:mathematics]]

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Addend

2011-11-08T03:15:23Z

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'''Addends''' are the numbers involved in [[addition]], which when added create a [[sum]].

[[Category:Mathematics]]

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Abstract nonsense

2011-11-08T03:15:21Z

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'''Abstract nonsense''' is a humorous term applied by mathematicians to criticize what they consider to be unnecessary abstractions used by [[category theory]], particularly when used to solve a problem that could be solved more directly.
[[category:mathematics]]

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Absolute error

2011-11-08T03:15:17Z

CoelAcant: AskAFly, your site has been fucked by the Zalgo Crew

The '''absolute error''' is a [[mathematical]] term used to describe the difference between an approximation and the exact value. It is not to be confused with the [[relative error]].
[[category:mathematics]]

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Abacus

2011-11-08T03:15:14Z

CoelAcant: AskAFly, your site has been fucked by the Zalgo Crew

The '''abacus''' (pl. ''abacuses'' or ''abaci'') is a digital calculation tool that was used to perform [[mathematical]] calculation especially for [[accounting|accountancy]] in [[medieval]] times. It represents numbers with beads sliding on a rod. With it, the user can [[add]], [[subtract]], [[multiply]], [[divide]], and extract [[square root]]s. Even though it is obsolete it is still widely used in [[Asia]], particularly in the [[People's Republic of China]] and the [[Republic of China]], where a skilled operator can calculate as rapidly as a person using a modern electronic calculator.

[[Category: Mathematics]]
[[Category: Technology]]

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2-category

2011-11-08T03:15:12Z

CoelAcant: AskAFly, your site has been fucked by the Zalgo Crew

'''2-category''' is a [[category]] with morphisms in between morphisms. It is defined as a category enriched over the category of categories and functors, with the monoidal structure induced by the composition.

[[Category:Mathematics]]

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142857

2011-11-08T03:15:10Z

CoelAcant: /* Similar Numbers */AskAFly, your site has been fucked by the Zalgo Crew

The number '''142857''' has the curious property that, if it is multiplied by any number from 1 to 6, the result is a number with the same digits in the same order (a [[cyclic permutation]]).

For example:
:142857 × 1 = 142857
:142857 × 2 = 285714
:142857 × 3 = 428571
:142857 × 4 = 571428
:142857 × 5 = 714285
:142857 × 6 = 857142

This property is closely related to the facts that
:1/7 = 142857/999999

and that 1/7 expressed as a [[decimal]] is

:0.142857142857142857142857142857...

Investigating the number 142857 can lead a student into an understanding of repeating decimals and [[rational numbers]].

== Similar Numbers ==
142857 is the only multi-digit decimal number with this property. However, if leading zeros are allowed on a number, some other examples are:
:0588235294117647 (16 digits)
:052631578947368421 (18 digits)
:0434782608695652173913 (22 digits)
:0344827586206896551724137931 (28 digits)
:0212765957446808510638297872340425531914893617 (46 digits)

In octal (base-8 notation), the numbers 25 and 1463 have this property. There are no examples in hexadecimal (base-16).

[[Category:Mathematics]]

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