Rectangle

Triangle
Square
Cube
triangular prism
 Shape
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For other uses, see Shape (disambiguation).
"Geometric shape" redirects here. For the Unicode symbols, see Geometric Shapes.

A children's toy used for learning various shapes

A shape or figure is the graphical representation of an object or its external boundary,
outline, or external surface, as opposed to other properties such as color, texture,
or material type. A plane shape or plane figure is constrained to lie on a plane, in contrast
to solid 3D shapes. A two-dimensional shape or two-dimensional figure (also: 2D
shape or 2D figure) may lie on a more general curved surface (a non-Euclidean two-
dimensional space).

Contents

 1Classification of simple shapes

 2In geometry
o 2.1Properties
 3Equivalence of shapes
o 3.1Congruence and similarity
o 3.2Homeomorphism
o 3.3Shape analysis
o 3.4Similarity classes
 4Human perception of shapes
 5See also
 6References
 7External links

Classification of simple shapes[edit]

Main article: Lists of shapes
A variety of polygonal shapes.
Some simple shapes can be put into broad categories. For instance, polygons are classified
according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these
is divided into smaller categories; triangles can
be equilateral, isosceles, obtuse, acute, scalene, etc. while quadrilaterals can
be rectangles, rhombi, trapezoids, squares, etc.
Other common shapes are points, lines, planes, and conic sections such as ellipses, circles,
and parabolas.
Among the most common 3-dimensional shapes are polyhedra, which are shapes with flat
faces; ellipsoids, which are egg-shaped or sphere-shaped objects; cylinders; and cones.
If an object falls into one of these categories exactly or even approximately, we can use it to
describe the shape of the object. Thus, we say that the shape of a manhole cover is a disk,
because it is approximately the same geometric object as an actual geometric disk.

In geometry[edit]

Geometric shapes in 2 dimensions: parallelogram, triangle & circle

Geometric shapes in 3 dimensions: pyramid, Sphere & cube
A geometric shape consists of the geometric information which remains
when location, scale, orientation and reflection are removed from the description of
a geometric object.[1] That is, the result of moving a shape around, enlarging it, rotating it, or
reflecting it in a mirror is the same shape as the original, and not a distinct shape.
Many two-dimensional geometric shapes can be defined by a set
of points or vertices and lines connecting the points in a closed chain, as well as the resulting
interior points. Such shapes are called polygons and include triangles, squares,
and pentagons. Other shapes may be bounded by curves such as the circle or the ellipse.
Many three-dimensional geometric shapes can be defined by a set of vertices, lines
connecting the vertices, and two-dimensional faces enclosed by those lines, as well as the
resulting interior points. Such shapes are called polyhedrons and include cubes as well
as pyramids such as tetrahedrons. Other three-dimensional shapes may be bounded by curved
surfaces, such as the ellipsoid and the sphere.
A shape is said to be convex if all of the points on a line segment between any two of its
points are also part of the shape.
Properties[edit]

Figures shown in the same color have the same shape as each other and are said to be similar.
There are several ways to compare the shapes of two objects:

 Congruence: Two objects are congruent if one can be transformed into the other by a
sequence of rotations, translations, and/or reflections.
 Similarity: Two objects are similar if one can be transformed into the other by a uniform
scaling, together with a sequence of rotations, translations, and/or reflections.
 Isotopy: Two objects are isotopic if one can be transformed into the other by a sequence
of deformations that do not tear the object or put holes in it.
Sometimes, two similar or congruent objects may be regarded as having a different shape if a
reflection is required to transform one into the other. For instance, the letters "b" and "d" are
a reflection of each other, and hence they are congruent and similar, but in some contexts
they are not regarded as having the same shape. Sometimes, only the outline or external
boundary of the object is considered to determine its shape. For instance, a hollow sphere
may be considered to have the same shape as a solid sphere. Procrustes analysis is used in
many sciences to determine whether or not two objects have the same shape, or to measure
the difference between two shapes. In advanced mathematics, quasi-isometry can be used as a
criterion to state that two shapes are approximately the same.
Simple shapes can often be classified into basic geometric objects such as a point, a line,
a curve, a plane, a plane figure (e.g. square or circle), or a solid figure (e.g. cube or sphere).
However, most shapes occurring in the physical world are complex. Some, such as plant
structures and coastlines, may be so complicated as to defy traditional mathematical
description – in which case they may be analyzed by differential geometry, or as fractals.

Equivalence of shapes[edit]
In geometry, two subsets of a Euclidean space have the same shape if one can be transformed
to the other by a combination of translations, rotations (together also called rigid
transformations), and uniform scalings. In other words, the shape of a set of points is all the
geometrical information that is invariant to translations, rotations, and size changes. Having
the same shape is an equivalence relation, and accordingly a precise mathematical definition
of the notion of shape can be given as being an equivalence class of subsets of a Euclidean
space having the same shape.
Mathematician and statistician David George Kendall writes:[2]
In this paper ‘shape’ is used in the vulgar sense, and means what one would normally expect
it to mean. [...] We here define ‘shape’ informally as ‘all the geometrical information that
remains when location, scale[3] and rotational effects are filtered out from an object.’
Shapes of physical objects are equal if the subsets of space these objects occupy satisfy the
definition above. In particular, the shape does not depend on the size and placement in space
of the object. For instance, a "d" and a "p" have the same shape, as they can be perfectly
superimposed if the "d" is translated to the right by a given distance, rotated upside down and
magnified by a given factor (see Procrustes superimposition for details). However, a mirror
image could be called a different shape. For instance, a "b" and a "p" have a different shape,
at least when they are constrained to move within a two-dimensional space like the page on
which they are written. Even though they have the same size, there's no way to perfectly
superimpose them by translating and rotating them along the page. Similarly, within a three-
dimensional space, a right hand and a left hand have a different shape, even if they are the
mirror images of each other. Shapes may change if the object is scaled non-uniformly. For
example, a sphere becomes an ellipsoid when scaled differently in the vertical and horizontal
directions. In other words, preserving axes of symmetry (if they exist) is important for
preserving shapes. Also, shape is determined by only the outer boundary of an object.
Congruence and similarity[edit]
Main articles: Congruence (geometry) and Similarity (geometry)
Objects that can be transformed into each other by rigid transformations and mirroring (but
not scaling) are congruent. An object is therefore congruent to its mirror image (even if it is
not symmetric), but not to a scaled version. Two congruent objects always have either the
same shape or mirror image shapes, and have the same size.
Objects that have the same shape or mirror image shapes are called geometrically similar,
whether or not they have the same size. Thus, objects that can be transformed into each other
by rigid transformations, mirroring, and uniform scaling are similar. Similarity is preserved
when one of the objects is uniformly scaled, while congruence is not. Thus, congruent objects
are always geometrically similar, but similar objects may not be congruent, as they may have
different size.
Homeomorphism[edit]
Main article: Homeomorphism
A more flexible definition of shape takes into consideration the fact that realistic shapes are
often deformable, e.g. a person in different postures, a tree bending in the wind or a hand
with different finger positions.
One way of modeling non-rigid movements is by homeomorphisms. Roughly speaking, a
homeomorphism is a continuous stretching and bending of an object into a new shape. Thus,
a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An
often-repeated mathematical joke is that topologists cannot tell their coffee cup from their
donut,[4] since a sufficiently pliable donut could be reshaped to the form of a coffee cup by
creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's
handle.
A described shape has external lines that you can see and make up the shape. If you were
putting you coordinates on and coordinate graph you could draw lines to show where you can
see a shape, however not every time you put coordinates in a graph as such you can make a
shape. This shape has a outline and boundary so you can see it and is not just regular dots on
a regular paper.
Shape analysis[edit]
Main article: Statistical shape analysis
The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the
field of statistical shape analysis. In particular, Procrustes analysis is a technique used for
comparing shapes of similar objects (e.g. bones of different animals), or measuring the
deformation of a deformable object. Other methods are designed to work with non-rigid
(bendable) objects, e.g. for posture independent shape retrieval (see for example Spectral
shape analysis).
Similarity classes[edit]
All similar triangles have the same shape. These shapes can be classified using complex
numbers u, v, w for the vertices, in a method advanced by J.A. Lester[5] and Rafael Artzy. For
example, an equilateral triangle can be expressed by the complex numbers 0, 1, (1 + i √3)/2
representing its vertices. Lester and Artzy call the ratio
the shape of triangle (u, v, w). Then the shape of the equilateral triangle is
(0–(1+ √3)/2)/(0–1) = ( 1 + i √3)/2 = cos(60°) + i sin(60°) = exp( i π/3).
For any affine transformation of the complex plane, a triangle is transformed but
does not change its shape. Hence shape is an invariant of affine geometry. The
shape p = S(u,v,w) depends on the order of the arguments of function S,
but permutations lead to related values. For instance,
 Also
Combining these permutations gives Furthermore,
These relations are "conversion rules" for shape of a triangle.
The shape of a quadrilateral is associated with two complex numbers p,q. If
the quadrilateral has vertices u,v,w,x, then p = S(u,v,w) and q = S(v,w,x).
Artzy proves these propositions about quadrilateral shapes:
1. If then the quadrilateral is a parallelogram.
2. If a parallelogram has | arg p | = | arg q |, then it is a rhombus.
3. When p = 1 + i and q = (1 + i)/2, then the quadrilateral is square.
4. If and sgn r = sgn(Im p), then the quadrilateral is a trapezoid.
A polygon has a shape defined by n – 2 complex numbers The polygon
bounds a convex set when all these shape components have imaginary
components of the same sign.[6]

Human perception of shapes[edit]

Human vision relies on a wide range of shape representations.[7][8] Some
psychologists have theorized that humans mentally break down images into
simple geometric shapes (e.g., cones and spheres) called geons.[9] Others have
suggested shapes are decomposed into features or dimensions that describe
the way shapes tend to vary, like
their segmentability, compactness and spikiness.[10] When comparing shape
similarity, however, at least 22 independent dimensions are needed to
account for the way natural shapes vary. [7]
There is also clear evidence that shapes guide human attention.[11][12

Decimal
The decimal numeral system (also called the base-ten positional numeral
system and denary /ˈdiːnəri/[1] or decanary) is the standard system for denoting integer and
non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral
system.[2] The way of denoting numbers in the decimal system is often referred to as decimal
notation.[3]
 A decimal numeral (also often just decimal or, less correctly, decimal number), refers
generally to the notation of a number in the decimal numeral system. Decimals may
sometimes be identified by a decimal separator (usually "." or "," as in 25.9703 or 3,1415).
[4]
Decimal may also refer specifically to the digits after the decimal separator, such as in
"3.14 is the approximation of π to two decimals". Zero-digits after a decimal separator serve
the purpose of signifying the precision of a value.
The numbers that may be represented in the decimal system are the decimal fractions. That
is, fractions of the form a/10n, where a is an integer, and n is a non-negative integer.
The decimal system has been extended to infinite decimals for representing any real number,
by using an infinite sequence of digits after the decimal separator (see decimal
representation). In this context, the decimal numerals with a finite number of non-zero digits
after the decimal separator are sometimes called terminating decimals. A repeating
decimal is an infinite decimal that, after some place, repeats indefinitely the same sequence of
digits (e.g., 5.123144144144144... = 5.123144).[5] An infinite decimal represents a rational
number, the quotient of two integers, if and only if it is a repeating decimal or has a finite
number of non-zero digits.

Contents

 1Origin
 2Decimal notation
 3Decimal fractions
 4Real number approximation
 5Infinite decimal expansion
o 5.1Rational numbers
 6Decimal computation
 7History
o 7.1History of decimal fractions
o 7.2Natural languages
o 7.3Other bases
 8See also
 9Notes
 10References

Origin[edit]
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Ten fingers on two hands, the possible origin of decimal counting

Many numeral systems of ancient civilizations use ten and its powers for representing
numbers, possibly because there are ten fingers on two hands and people started counting by
using their fingers. Examples are firstly the Egyptian numerals, then the Brahmi
numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals. Very
large numbers were difficult to represent in these old numeral systems, and only the best
mathematicians were able to multiply or divide large numbers. These difficulties were
completely solved with the introduction of the Hindu–Arabic numeral system for
representing integers. This system has been extended to represent some non-integer numbers,
called decimal fractions or decimal numbers, for forming the decimal numeral system.

Decimal notation[edit]
For writing numbers, the decimal system uses ten decimal digits, a decimal mark, and,
for negative numbers, a minus sign "−". The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
[6]
the decimal separator is the dot "." in many countries (mostly English-speaking),[7] and a
comma "," in other countries.[4]
For representing a non-negative number, a decimal numeral consists of
 either a (finite) sequence of digits (such as "2017"), where the entire sequence represents
an integer,

 or a decimal mark separating two sequences of digits (such as "20.70828")

.
If m > 0, that is, if the first sequence contains at least two digits, it is generally assumed
that the first digit am is not zero. In some circumstances it may be useful to have one or
more 0's on the left; this does not change the value represented by the decimal: for
example, 3.14 = 03.14 = 003.14. Similarly, if the final digit on the right of the decimal
mark is zero—that is, if bn = 0—it may be removed; conversely, trailing zeros may be
added after the decimal mark without changing the represented number; [note 1] for
example, 15 = 15.0 = 15.00 and 5.2 = 5.20 = 5.200.
For representing a negative number, a minus sign is placed before am.
The numeral represents the number
.
The integer part or integral part of a decimal numeral is the integer written to the left
of the decimal separator (see also truncation). For a non-negative decimal numeral, it
is the largest integer that is not greater than the decimal. The part from the decimal
separator to the right is the fractional part, which equals the difference between the
numeral and its integer part.
When the integral part of a numeral is zero, it may occur, typically in computing, that
the integer part is not written (for example, .1234, instead of 0.1234). In normal
writing, this is generally avoided, because of the risk of confusion between the
decimal mark and other punctuation.
In brief, the contribution of each digit to the value of a number depends on its
position in the numeral. That is, the decimal system is a positional numeral system.

Decimal fractions[edit]
Decimal fractions (sometimes called decimal numbers, especially in contexts
involving explicit fractions) are the rational numbers that may be expressed as
a fraction whose denominator is a power of ten.[8] For example, the
decimals represent the
fractions 8/10 , 1489/100 , 24/100000 , +1618/1000 and +314159/100000 , and are therefore
decimal numbers.
More generally, a decimal with n digits after the separator (a point or comma)
represents the fraction with denominator 10n, whose numerator is the integer obtained
by removing the separator.
It follows that a number is a decimal fraction if and only if it has a finite decimal
representation.
Expressed as a fully reduced fraction, the decimal numbers are those whose
denominator is a product of a power of 2 and a power of 5. Thus the smallest
denominators of decimal numbers are

Real number approximation[edit]

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Decimal numerals do not allow an exact representation for all real numbers, e.g.
for the real number π. Nevertheless, they allow approximating every real number
with any desired accuracy, e.g., the decimal 3.14159 approximates the real π,
being less than 10−5 off; so decimals are widely used in science, engineering and
everyday life.
More precisely, for every real number x and every positive integer n, there are
two decimals L and u with at most n digits after the decimal mark such
that L ≤ x ≤ u and (u − L) = 10−n.
Numbers are very often obtained as the result of measurement. As measurements
are subject to measurement uncertainty with a known upper bound, the result of a
measurement is well-represented by a decimal with n digits after the decimal
mark, as soon as the absolute measurement error is bounded from above by 10−n.
In practice, measurement results are often given with a certain number of digits
after the decimal point, which indicate the error bounds. For example, although
0.080 and 0.08 denote the same number, the decimal numeral 0.080 suggests a
measurement with an error less than 0.001, while the numeral 0.08 indicates an
absolute error bounded by 0.01. In both cases, the true value of the measured
quantity could be, for example, 0.0803 or 0.0796 (see also significant figures).

Infinite decimal expansion[edit]

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 Main article: Decimal representation
For a real number x and an integer n ≥ 0, let [x]n denote the (finite) decimal
expansion of the greatest number that is not greater than x that has
exactly n digits after the decimal mark. Let di denote the last digit of [x]i. It is
straightforward to see that [x]n may be obtained by appending dn to the right
of [x]n−1. This way one has
[x]n = [x]0.d1d2...dn−1dn,
and the difference of [x]n−1 and [x]n amounts to
,
which is either 0, if dn = 0, or gets arbitrarily small as n tends to infinity.
According to the definition of a limit, x is the limit of [x]n when n tends
to infinity. This is written asor
x = [x]0.d1d2...dn...,
which is called an infinite decimal expansion of x.
Conversely, for any integer [x]0 and any sequence of digits the
(infinite) expression [x]0.d1d2...dn... is an infinite decimal expansion of
a real number x. This expansion is unique if neither all dn are equal to
9 nor all dn are equal to 0 for n large enough (for all n greater than
some natural number N).
If all dn for n > N equal to 9 and [x]n = [x]0.d1d2...dn, the limit of the
sequence is the decimal fraction obtained by replacing the last digit
that is not a 9, i.e.: dN, by dN + 1, and replacing all subsequent 9s by
0s (see 0.999...).
Any such decimal fraction, i.e.: dn = 0 for n > N, may be converted to
its equivalent infinite decimal expansion by replacing dN by dN −
1 and replacing all subsequent 0s by 9s (see 0.999...).
In summary, every real number that is not a decimal fraction has a
unique infinite decimal expansion. Each decimal fraction has exactly
two infinite decimal expansions, one containing only 0s after some
place, which is obtained by the above definition of [x]n, and the other
containing only 9s after some place, which is obtained by
defining [x]n as the greatest number that is less than x, having
exactly n digits after the decimal mark.
Rational numbers[edit]
Main article: Repeating decimal
Long division allows computing the infinite decimal expansion of
a rational number. If the rational number is a decimal fraction, the
division stops eventually, producing a decimal numeral, which may
be prolongated into an infinite expansion by adding infinitely many
zeros. If the rational number is not a decimal fraction, the division
may continue indefinitely. However, as all successive remainders are
less than the divisor, there are only a finite number of possible
remainders, and after some place, the same sequence of digits must
 be repeated indefinitely in the quotient. That is, one has a repeating
decimal. For example,
1/81 = 0. 012345679 012... (with the group 012345679 indefinitely repeating).
The converse is also true: if, at some point in the decimal
representation of a number, the same string of digits starts
repeating indefinitely, the number is rational.
For example, if x is 0.4156156156...
then 10,000x is 4156.156156156...
and 10x is 4.156156156...
so 10,000x − 10x, i.e. 9,990x, is 4152.000000000...
and x is 4152/9990
or, dividing both numerator and denominator by 6, 692/1665 .

Decimal computation[edit]

Diagram of the world's earliest known multiplication table (c. 305 BCE) from
the Warring States period

Most modern computer hardware and software systems

commonly use a binary representation internally (although many
early computers, such as the ENIAC or the IBM 650, used
decimal representation internally).[9] For external use by computer
specialists, this binary representation is sometimes presented in
the related octal or hexadecimal systems.
For most purposes, however, binary values are converted to or
from the equivalent decimal values for presentation to or input
from humans; computer programs express literals in decimal by
default. (123.1, for example, is written as such in a computer
program, even though many computer languages are unable to
encode that number precisely.)
Both computer hardware and software also use internal
representations which are effectively decimal for storing decimal
values and doing arithmetic. Often this arithmetic is done on data
which are encoded using some variant of binary-coded decimal,[10]
[11]
especially in database implementations, but there are other
decimal representations in use (including decimal floating
point such as in newer revisions of the IEEE 754 Standard for
Floating-Point Arithmetic).[12]
Decimal arithmetic is used in computers so that decimal
fractional results of adding (or subtracting) values with a fixed
length of their fractional part always are computed to this same
length of precision. This is especially important for financial
calculations, e.g., requiring in their results integer multiples of the
smallest currency unit for book keeping purposes. This is not
possible in binary, because the negative powers of have no finite
binary fractional representation; and is generally impossible for
multiplication (or division).[13][14] See Arbitrary-precision
arithmetic for exact calculations.

History[edit]

The world's earliest decimal multiplication table was made from bamboo slips,
dating from 305 BCE, during the Warring States period in China.

Many ancient cultures calculated with numerals based on ten,

sometimes argued due to human hands typically having ten
fingers/digits.[15] Standardized weights used in the Indus Valley
Civilization (c. 3300–1300 BCE) were based on the ratios: 1/20,
1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, while their
standardized ruler – the Mohenjo-daro ruler – was divided into
ten equal parts.[16][17][18] Egyptian hieroglyphs, in evidence since
around 3000 BCE, used a purely decimal system,[19] as did
the Cretan hieroglyphs (c. 1625−1500 BCE) of
the Minoans whose numerals are closely based on the Egyptian
model.[20][21] The decimal system was handed down to the
consecutive Bronze Age cultures of Greece, including Linear
A (c. 18th century BCE−1450 BCE) and Linear B (c. 1375−1200
BCE) – the number system of classical Greece also used powers
of ten, including, Roman numerals, an intermediate base of 5.
[22]
Notably, the polymath Archimedes (c. 287–212 BCE) invented
a decimal positional system in his Sand Reckoner which was
based on 108[22] and later led the German mathematician Carl
Friedrich Gauss to lament what heights science would have
already reached in his days if Archimedes had fully realized the
potential of his ingenious discovery.[23] Hittite hieroglyphs (since
15th century BCE) were also strictly decimal.[24]
 Some non-mathematical ancient texts such as the Vedas, dating
back to 1700–900 BCE make use of decimals and mathematical
decimal fractions.[25]
The Egyptian hieratic numerals, the Greek alphabet numerals, the
Hebrew alphabet numerals, the Roman numerals, the Chinese
numerals and early Indian Brahmi numerals are all non-positional
decimal systems, and required large numbers of symbols. For
instance, Egyptian numerals used different symbols for 10, 20 to
90, 100, 200 to 900, 1000, 2000, 3000, 4000, to 10,000.[26] The
world's earliest positional decimal system was the Chinese rod
calculus.[27]

The world's earliest positional decimal system

Upper row vertical form
Lower row horizontal form

History of decimal fractions[edit]

counting rod decimal fraction 1/7

Decimal fractions were first developed and used by the Chinese

in the end of 4th century BCE,[28] and then spread to the Middle
East and from there to Europe.[27][29] The written Chinese decimal
fractions were non-positional.[29] However, counting rod
fractions were positional.[27]
Qin Jiushao in his book Mathematical Treatise in Nine
Sections (1247[30]) denoted 0.96644 by
寸

, meaning
寸
096644
J. Lennart Berggren notes that positional decimal fractions appear for the first time in a book
by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century.[31] The Jewish
mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon
Stevin, but did not develop any notation to represent them.[32] The Persian
mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the
15th century.[31] Al Khwarizmi introduced fraction to Islamic countries in the early 9th
century; a Chinese author has alleged that his fraction presentation was an exact copy of
traditional Chinese mathematical fraction from Sunzi Suanjing.[27] This form of fraction with
numerator on top and denominator at bottom without a horizontal bar was also used by al-
Uqlidisi and by al-Kāshī in his work "Arithmetic Key".[27][33]

A forerunner of modern European decimal notation was introduced by Simon Stevin in the
16th century.[34]
John Napier introduced using the period (.) to separate the integer part of a decimal number
from the fractional part in his book on constructing tables of logarithms, published
posthumously in 1620.[35]: p. 8, archive p. 32)
Natural languages[edit]
A method of expressing every possible natural number using a set of ten symbols emerged in
India. Several Indian languages show a straightforward decimal system. Many Indo-
Aryan and Dravidian languages have numbers between 10 and 20 expressed in a regular
pattern of addition to 10.[36]
The Hungarian language also uses a straightforward decimal system. All numbers between 10
and 20 are formed regularly (e.g. 11 is expressed as "tizenegy" literally "one on ten"), as with
those between 20 and 100 (23 as "huszonhárom" = "three on twenty").
A straightforward decimal rank system with a word for each order (10 十, 100 百, 1000 千,
10,000 万), and in which 11 is expressed as ten-one and 23 as two-ten-three, and 89,345 is
expressed as 8 (ten thousands) 万 9 (thousand) 千 3 (hundred) 百 4 (tens) 十 5 is found
in Chinese, and in Vietnamese with a few irregularities. Japanese, Korean, and Thai have
imported the Chinese decimal system. Many other languages with a decimal system have
special words for the numbers between 10 and 20, and decades. For example, in English 11 is
"eleven" not "ten-one" or "one-teen".
Incan languages such as Quechua and Aymara have an almost straightforward decimal
system, in which 11 is expressed as ten with one and 23 as two-ten with three.
Some psychologists suggest irregularities of the English names of numerals may hinder
children's counting ability.[37]
Some cultures do, or did, use other bases of numbers.

 Pre-Columbian Mesoamerican cultures such as

the Maya used a base-20 system (perhaps based on using
all twenty fingers and toes).
 The Yuki language in California and the Pamean
languages[38] in Mexico have octal (base-8) systems
because the speakers count using the spaces between
their fingers rather than the fingers themselves.[39]
 The existence of a non-decimal base in the earliest traces
of the Germanic languages is attested by the presence of
words and glosses meaning that the count is in decimal
(cognates to "ten-count" or "tenty-wise"); such would be
expected if normal counting is not decimal, and unusual
if it were.[40][41] Where this counting system is known, it is
based on the "long hundred" = 120, and a "long
thousand" of 1200. The descriptions like "long" only
appear after the "small hundred" of 100 appeared with the
 Christians. Gordon's Introduction to Old Norse p. 293,
gives number names that belong to this system. An
expression cognate to 'one hundred and eighty' translates
to 200, and the cognate to 'two hundred' translates to
240. Goodare details the use of the long hundred in
Scotland in the Middle Ages, giving examples such as
calculations where the carry implies i C (i.e. one
hundred) as 120, etc. That the general population were
not alarmed to encounter such numbers suggests common
enough use. It is also possible to avoid hundred-like
numbers by using intermediate units, such as stones and
pounds, rather than a long count of pounds. Goodare
gives examples of numbers like vii score, where one
avoids the hundred by using extended scores. There is
also a paper by W.H. Stevenson, on 'Long Hundred and
its uses in England'.[42][43]
 Many or all of the Chumashan languages originally used
a base-4 counting system, in which the names for
numbers were structured according to multiples of 4
and 16.[44]
 Many languages[45] use quinary (base-5) number systems,
including Gumatj, Nunggubuyu,[46] Kuurn Kopan
Noot[47] and Saraveca. Of these, Gumatj is the only true 5–
25 language known, in which 25 is the higher group of 5.
 Some Nigerians use duodecimal systems.[48] So did some
small communities in India and Nepal, as indicated by
their languages.[49]
 The Huli language of Papua New Guinea is reported to
have base-15 numbers.[50] Ngui means 15, ngui ki means
15 × 2 = 30, and ngui ngui means 15 × 15 = 225.
 Umbu-Ungu, also known as Kakoli, is reported to
have base-24 numbers.[51] Tokapu means 24, tokapu
talu means 24 × 2 = 48, and tokapu tokapu means 24 ×
24 = 576.
 Ngiti is reported to have a base-32 number system with
base-4 cycles.[45]
 The Ndom language of Papua New Guinea is reported to
have base-6 numerals.[52] Mer means 6, mer an thef means
6 × 2 = 12, nif means 36, and nif thef means 36×2 = 72.
Abacus
For other uses, see Abacus (disambiguation).
"Abaci" and "Abacuses" redirect here. For the Turkish Surname, see Abacı. For the
medieval book, see Liber Abaci.
Calculating-Table by Gregor Reisch: Margarita Philosophica, 1503. The woodcut
shows Arithmetica instructing an algorist and an abacist (inaccurately represented
as Boethius and Pythagoras). There was keen competition between the two from the
introduction of the Algebra into Europe in the 12th century until its triumph in the 16th.[1]
The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool
which has been used since ancient times. It was used in the ancient Near East, Europe, China,
and Russia, centuries before the adoption of the Hindu-Arabic numeral system.[1] The exact
origin of the abacus has not yet emerged. It consists of rows of movable beads, or similar
objects, strung on a wire. They represent digits. One of the two numbers is set up, and the
beads are manipulated to perform an operation such as addition, or even a square or cubic
root.
In their earliest designs, the rows of beads could be loose on a flat surface or sliding in
grooves. Later the beads were made to slide on rods and built into a frame, allowing faster
manipulation. Abacuses are still made, often as a bamboo frame with beads sliding on wires.
In the ancient world, particularly before the introduction of positional notation, abacuses were
a practical calculating tool. The abacus is still used to teach the fundamentals
of mathematics to some children, for example, in Russia.
Designs such as the Japanese soroban have been used for practical calculations of up to
multi-digit numbers. Any particular abacus design supports multiple methods to perform
calculations, including the four basic operations and square and cube roots. Some of these
methods work with non-natural numbers (numbers such as 1.5 and 3⁄4).
Although calculators and computers are commonly used today instead of abacuses, abacuses
remain in everyday use in some countries. Merchants, traders, and clerks in some parts
of Eastern Europe, Russia, China, and Africa use abacuses. The abacus remains in common
use as a scoring system in non-electronic table games. Others may use an abacus due
to visual impairment that prevents the use of a calculator.[1]