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tendency and picks-up at challenge.
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bout, the cause to theory will be width to Cantore Mathematics presenting for
the first time arithmetic.
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Cause’ not “cause and effect” for that would end concept and atmosphere leaving
only exit as cause to proof, and that would be mathematics not arithmetic.
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Stoker, Bram’s Dracula did place the horseshoe on backwards and gallop forwards
to escape capture at his castle in the story told Wikipedia at https://simple.wikipedia.org/wiki/Vlad_III_the_Impaler
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only fact to present the equine to equus ferus as the unicorn does
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gave rise to actual: Prodigy). In such
there are “Bearded” circuses performers (https://en.wikipedia.org/wiki/Annie_Jones_(bearded_woman))
that have in our history been notoriously shown as “freak” and many have paid
to see, laugh, spectacle, and, drive dinner to conversation. This product management is well –proven and
driven many economies to product line.
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quick recovery of computation to this day.
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and subtraction and arithmetic to measure such at the current rules. To that the fact that a movie (Hollywood) has
enhanced our lot to enjoy Star Wars Chewbacca must counter the mammoth and not
the butterfly to the cotton of cocoon as a mule. To note: Men, man has never been removed from
the “bearded” syndrome as such syndrome until my generation where ‘clean shaven’
was required, this show man has been lost and the return to bearded sex via indent
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fair and Albert Einstein the return to not from will be by design for Cantore
Arithmetic for reason to mathematical.
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that Nostradamus ran to a blank page on role where Cantore mathematics will
arithmetic to show that the slow bring from and not to can calculate at Pi (https://en.wikipedia.org/wiki/Pi
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the example to Mars than arithmetic will be met.
Arithmetic
From Wikipedia, the free encyclopedia
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Arithmetic tables for children, Lausanne, 1835
Arithmetic (from the Greek ἀριθμός arithmos, 'number' and τική [τέχνη], tiké [téchne], 'art' or 'craft') is a branch of mathematics that consists of the study of numbers, especially concerning the properties of the traditional operations on them—addition, subtraction, multiplication, division, exponentiation, and extraction of roots.[1][2] Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory, and are sometimes still used to refer to a wider part of number theory.[3]
History
The prehistory of arithmetic is limited to a small number of
artifacts, which may indicate the conception of addition and
subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC, although its interpretation is disputed.[4]
The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic
operations as early as 2000 BC. These artifacts do not always reveal
the specific process used for solving problems, but the characteristics
of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, descended from tally marks used for counting. In both cases, this origin resulted in values that used a decimal base, but did not include positional notation. Complex calculations with Roman numerals required the assistance of a counting board (or the Roman abacus) to obtain the results.
Early number systems that included positional notation were not decimal, including the sexagesimal (base 60) system for Babylonian numerals, and the vigesimal (base 20) system that defined Maya numerals.
Because of this place-value concept, the ability to reuse the same
digits for different values contributed to simpler and more efficient
methods of calculation.
The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, although it originated much later than the Babylonian and Egyptian examples. Prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to numbers, and their relationships to each other, in his Introduction to Arithmetic.
Greek numerals were used by Archimedes, Diophantus and others in a positional notation
not very different from the modern notation. The ancient Greeks lacked a
symbol for zero until the Hellenistic period, and they used three
separate sets of symbols as digits:
one set for the units place, one for the tens place, and one for the
hundreds. For the thousands place, they would reuse the symbols for the
units place, and so on. Their addition algorithm was identical to the
modern method, and their multiplication algorithm was only slightly
different. Their long division algorithm was the same, and the digit-by-digit square root algorithm, popularly used as recently as the 20th century, was known to Archimedes (who may have invented it). He preferred it to Hero's method
of successive approximation because, once computed, a digit does not
change, and the square roots of perfect squares, such as 7485696,
terminate immediately as 2736. For numbers with a fractional part, such
as 546.934, they used negative powers of 60—instead of negative powers
of 10 for the fractional part 0.934.[5]
The ancient Chinese had advanced arithmetic studies dating from
the Shang Dynasty and continuing through the Tang Dynasty, from basic
numbers to advanced algebra. The ancient Chinese used a positional
notation similar to that of the Greeks. Since they also lacked a symbol
for zero,
they had one set of symbols for the units place, and a second set for
the tens place. For the hundreds place, they then reused the symbols for
the units place, and so on. Their symbols were based on the ancient counting rods.
The exact time where the Chinese started calculating with positional
representation is unknown, though it is known that the adoption started
before 400 BC.[6] The ancient Chinese were the first to meaningfully discover, understand, and apply negative numbers. This is explained in the Nine Chapters on the Mathematical Art (Jiuzhang Suanshu), which was written by Liu Hui dated back to 2nd century BC.
The gradual development of the Hindu–Arabic numeral system
independently devised the place-value concept and positional notation,
which combined the simpler methods for computations with a decimal base,
and the use of a digit representing 0.
This allowed the system to consistently represent both large and small
integers—an approach which eventually replaced all other systems. In the
early 6th century AD, the Indian mathematician Aryabhata incorporated an existing version of this system in his work, and experimented with different notations. In the 7th century, Brahmagupta
established the use of 0 as a separate number, and determined the
results for multiplication, division, addition and subtraction of zero
and all other numbers—except for the result of division by zero. His contemporary, the Syriac bishop Severus Sebokht
(650 AD) said, "Indians possess a method of calculation that no word
can praise enough. Their rational system of mathematics, or of their
method of calculation. I mean the system using nine symbols."[7] The Arabs also learned this new method and called it hesab.
Leibniz's
Stepped Reckoner was the first calculator that could perform all four arithmetic operations.
Although the Codex Vigilanus described an early form of Arabic numerals (omitting 0) by 976 AD, Leonardo of Pisa (Fibonacci) was primarily responsible for spreading their use throughout Europe after the publication of his book Liber Abaci in 1202. He wrote, "The method of the Indians (Latin Modus Indorum) surpasses any known method to compute. It's a marvelous method. They do their computations using nine figures and symbol zero".[8]
In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities.
The flourishing of algebra in the medieval Islamic world, and also in Renaissance Europe, was an outgrowth of the enormous simplification of computation through decimal notation.
Various types of tools have been invented and widely used to
assist in numeric calculations. Before Renaissance, they were various
types of abaci. More recent examples include slide rules, nomograms and mechanical calculators, such as Pascal's calculator. At present, they have been supplanted by electronic calculators and computers.
Arithmetic operations
The basic arithmetic operations are addition, subtraction,
multiplication and division. Although arithmetic also includes more
advanced operations, such as manipulations of percentages,[2] square roots, exponentiation, logarithmic functions, and even trigonometric functions, in the same vein as logarithms (prosthaphaeresis).
Arithmetic expressions must be evaluated according to the intended
sequence of operations. There are several methods to specify this,
either—most common, together with infix notation—explicitly using parentheses and relying on precedence rules, or using a prefix or postfix
notation, which uniquely fix the order of execution by themselves. Any
set of objects upon which all four arithmetic operations (except division by zero) can be performed, and where these four operations obey the usual laws (including distributivity), is called a field.[9]
Addition
Addition, denoted by the symbol 
, is the most basic operation of arithmetic. In its simple form, addition combines two numbers, the addends or terms, into a single number, the sum of the numbers (such as 2 + 2 = 4 or 3 + 5 = 8).
Adding finitely many numbers can be viewed as repeated simple addition; this procedure is known as summation, a term also used to denote the definition for "adding infinitely many numbers" in an infinite series. Repeated addition of the number 1 is the most basic form of counting; the result of adding 1 is usually called the successor of the original number.
Addition is commutative and associative, so the order in which finitely many terms are added does not matter.
The number 0 has the property that, when added to any number, it yields that same number; so, it is the identity element of addition, or the additive identity.
For every number x, there is a number denoted –x, called the opposite of x, such that x + (–x) = 0 and (–x) + x = 0. So, the opposite of x is the inverse of x with respect to addition, or the additive inverse of x. For example, the opposite of 7 is −7, since 7 + (−7) = 0.
Addition can also be interpreted geometrically, as in the following example.
If we have two sticks of lengths 2 and 5, then, if the sticks are aligned one after the other, the length of the combined stick becomes 7, since 2 + 5 = 7.
Subtraction
Subtraction, denoted by the symbol 
, is the inverse operation to addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend: D = M − S.
Resorting to the previously established addition, this is to say that
the difference is the number that, when added to the subtrahend, results
in the minuend: D + S = M.[1]
For positive arguments M and S holds:
- If the minuend is larger than the subtrahend, the difference D is positive.
- If the minuend is smaller than the subtrahend, the difference D is negative.
In any case, if minuend and subtrahend are equal, the difference D = 0.
Subtraction is neither commutative nor associative.
For that reason, the construction of this inverse operation in modern
algebra is often discarded in favor of introducing the concept of
inverse elements (as sketched under § Addition), where subtraction is regarded as adding the additive inverse of the subtrahend to the minuend, that is, a − b = a + (−b). The immediate price of discarding the binary operation of subtraction is the introduction of the (trivial) unary operation, delivering the additive inverse for any given number, and losing the immediate access to the notion of difference, which is potentially misleading when negative arguments are involved.
For any representation of numbers, there are methods for
calculating results, some of which are particularly advantageous in
exploiting procedures, existing for one operation, by small alterations
also for others. For example, digital computers can reuse existing
adding-circuitry and save additional circuits for implementing a
subtraction, by employing the method of two's complement for representing the additive inverses, which is extremely easy to implement in hardware (negation). The trade-off is the halving of the number range for a fixed word length.
A formerly wide spread method to achieve a correct change amount, knowing the due and given amounts, is the counting up method, which does not explicitly generate the value of the difference. Suppose an amount P is given in order to pay the required amount Q, with P greater than Q. Rather than explicitly performing the subtraction P − Q = C and counting out that amount C in change, money is counted out starting with the successor of Q, and continuing in the steps of the currency, until P is reached. Although the amount counted out must equal the result of the subtraction P − Q, the subtraction was never really done and the value of P − Q is not supplied by this method.
Multiplication
Multiplication, denoted by the symbols 
or 
, is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the product. The two original numbers are called the multiplier and the multiplicand, mostly both are simply called factors.
Multiplication may be viewed as a scaling operation. If the
numbers are imagined as lying in a line, multiplication by a number
greater than 1, say x, is the same as stretching everything away from 0 uniformly, in such a way that the number 1 itself is stretched to where x
was. Similarly, multiplying by a number less than 1 can be imagined as
squeezing towards 0, in such a way that 1 goes to the multiplicand.
Another view on multiplication of integer numbers (extendable to
rationals but not very accessible for real numbers) is by considering it
as repeated addition. For example. 3 × 4 corresponds to either adding 3 times a 4, or 4 times a 3, giving the same result. There are different opinions on the advantageousness of these paradigmata in math education.
Multiplication is commutative and associative; further, it is distributive over addition and subtraction. The multiplicative identity is 1, since multiplying any number by 1 yields that same number. The multiplicative inverse for any number except 0 is the reciprocal of this number, because multiplying the reciprocal of any number by the number itself yields the multiplicative identity 1. 0 is the only number without a multiplicative inverse, and the result of multiplying any number and 0 is again 0. One says that 0 is not contained in the multiplicative group of the numbers.
The product of a and b is written as a × b or a·b. When a or b are expressions not written simply with digits, it is also written by simple juxtaposition: ab.
In computer programming languages and software packages (in which one
can only use characters normally found on a keyboard), it is often
written with an asterisk: a * b.
Algorithms implementing the operation of multiplication for
various representations of numbers are by far more costly and laborious
than those for addition. Those accessible for manual computation either
rely on breaking down the factors to single place values and applying
repeated addition, or on employing tables or slide rules,
thereby mapping multiplication to addition and vice versa. These
methods are outdated and are gradually replaced by mobile devices.
Computers utilize diverse sophisticated and highly optimized algorithms,
to implement multiplication and division for the various number formats
supported in their system.
Division
Division, denoted by the symbols 
or 
, is essentially the inverse operation to multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero
is undefined. For distinct positive numbers, if the dividend is larger
than the divisor, the quotient is greater than 1, otherwise it is less
than or equal to 1 (a similar rule applies for negative numbers). The
quotient multiplied by the divisor always yields the dividend.
Division is neither commutative nor associative. So as explained in § Subtraction,
the construction of the division in modern algebra is discarded in
favor of constructing the inverse elements with respect to
multiplication, as introduced in § Multiplication. Hence division is the multiplication of the dividend with the reciprocal of the divisor as factors, that is, a ÷ b = a × 1/b.
Within the natural numbers, there is also a different but related notion called Euclidean division, which outputs two numbers after "dividing" a natural N (numerator) by a natural D (denominator): first a natural Q (quotient), and second a natural R (remainder) such that N = D×Q + R and 0 ≤ R < Q.
In some contexts, including computer programming and advanced
arithmetic, division is extended with another output for the remainder.
This is often treated as a separate operation, the Modulo operation, denoted by the symbol 
or the word 
, though sometimes a second output for one "divmod" operation.[10] In either case, Modular arithmetic
has a variety of use cases. Different implementations of division
(floored, truncated, Euclidean, etc.) correspond with different
implementations of modulus.
Fundamental theorem of arithmetic
The fundamental theorem of arithmetic states that any integer greater
than 1 has a unique prime factorization (a representation of a number
as the product of prime factors), excluding the order of the factors.
For example, 252 only has one prime factorization:
- 252 = 22 × 32 × 71
Euclid's Elements first introduced this theorem, and gave a partial proof (which is called Euclid's lemma). The fundamental theorem of arithmetic was first proven by Carl Friedrich Gauss.
The fundamental theorem of arithmetic is one of the reasons why 1 is not considered a prime number. Other reasons include the sieve of Eratosthenes,
and the definition of a prime number itself (a natural number greater
than 1 that cannot be formed by multiplying two smaller natural
numbers.).
Decimal arithmetic
Decimal representation refers exclusively, in common use, to the written numeral system employing arabic numerals as the digits for a radix 10 ("decimal") positional notation; however, any numeral system based on powers of 10, e.g., Greek, Cyrillic, Roman, or Chinese numerals may conceptually be described as "decimal notation" or "decimal representation".
Modern methods for four fundamental operations (addition, subtraction, multiplication and division) were first devised by Brahmagupta
of India. This was known during medieval Europe as "Modus Indorum" or
Method of the Indians. Positional notation (also known as "place-value
notation") refers to the representation or encoding of numbers using the same symbol for the different orders of magnitude (e.g., the "ones place", "tens place", "hundreds place") and, with a radix point, using those same symbols to represent fractions (e.g., the "tenths place", "hundredths place"). For example, 507.36 denotes 5 hundreds (102), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10−1) plus 6 hundredths (10−2).
The concept of 0
as a number comparable to the other basic digits is essential to this
notation, as is the concept of 0's use as a placeholder, and as is the
definition of multiplication and addition with 0. The use of 0 as a
placeholder and, therefore, the use of a positional notation is first
attested to in the Jain text from India entitled the Lokavibhâga, dated 458 AD and it was only in the early 13th century that these concepts, transmitted via the scholarship of the Arabic world, were introduced into Europe by Fibonacci[11] using the Hindu–Arabic numeral system.
Algorism
comprises all of the rules for performing arithmetic computations using
this type of written numeral. For example, addition produces the sum of
two arbitrary numbers. The result is calculated by the repeated
addition of single digits from each number that occupies the same
position, proceeding from right to left. An addition table with ten rows
and ten columns displays all possible values for each sum. If an
individual sum exceeds the value 9, the result is represented with two
digits. The rightmost digit is the value for the current position, and
the result for the subsequent addition of the digits to the left
increases by the value of the second (leftmost) digit, which is always
one (if not zero). This adjustment is termed a carry of the value 1.
The process for multiplying two arbitrary numbers is similar to
the process for addition. A multiplication table with ten rows and ten
columns lists the results for each pair of digits. If an individual
product of a pair of digits exceeds 9, the carry adjustment
increases the result of any subsequent multiplication from digits to the
left by a value equal to the second (leftmost) digit, which is any
value from 1 to 8 (9 × 9 = 81). Additional steps define the final result.
Similar techniques exist for subtraction and division.
The creation of a correct process for multiplication relies on
the relationship between values of adjacent digits. The value for any
single digit in a numeral depends on its position. Also, each position
to the left represents a value ten times larger than the position to the
right. In mathematical terms, the exponent for the radix
(base) of 10 increases by 1 (to the left) or decreases by 1 (to the
right). Therefore, the value for any arbitrary digit is multiplied by a
value of the form 10n with integer n. The list of values corresponding to all possible positions for a single digit is written as {..., 102, 10, 1, 10−1, 10−2, ...}.
Repeated multiplication of any value in this list by 10 produces
another value in the list. In mathematical terminology, this
characteristic is defined as closure, and the previous list is described as closed under multiplication.
It is the basis for correctly finding the results of multiplication
using the previous technique. This outcome is one example of the uses of
number theory.
Compound unit arithmetic
Compound[12] unit arithmetic is the application of arithmetic operations to mixed radix
quantities such as feet and inches; gallons and pints; pounds,
shillings and pence; and so on. Before decimal-based systems of money
and units of measure, compound unit arithmetic was widely used in
commerce and industry.
Basic arithmetic operations
The
techniques used in compound unit arithmetic were developed over many
centuries and are well documented in many textbooks in many different
languages.[13][14][15][16]
In addition to the basic arithmetic functions encountered in decimal
arithmetic, compound unit arithmetic employs three more functions:
- Reduction,
in which a compound quantity is reduced to a single quantity—for
example, conversion of a distance expressed in yards, feet and inches to
one expressed in inches.[17]
- Expansion, the inverse function
to reduction, is the conversion of a quantity that is expressed as a
single unit of measure to a compound unit, such as expanding 24 oz to 1 lb 8 oz.
- Normalization is the conversion of a set of compound units to a standard form—for example, rewriting "1 ft 13 in" as "2 ft 1 in".
Knowledge of the relationship between the various units of measure,
their multiples and their submultiples forms an essential part of
compound unit arithmetic.
Principles of compound unit arithmetic
There are two basic approaches to compound unit arithmetic:
- Reduction–expansion method where all the compound unit
variables are reduced to single unit variables, the calculation
performed and the result expanded back to compound units. This approach
is suited for automated calculations. A typical example is the handling
of time by Microsoft Excel where all time intervals are processed internally as days and decimal fractions of a day.
- On-going normalization method in which each unit is treated
separately and the problem is continuously normalized as the solution
develops. This approach, which is widely described in classical texts,
is best suited for manual calculations. An example of the ongoing
normalization method as applied to addition is shown below.
The
addition operation is carried out from right to left; in this case,
pence are processed first, then shillings followed by pounds. The
numbers below the "answer line" are intermediate results.
The total in the pence column is 25. Since there are 12 pennies
in a shilling, 25 is divided by 12 to give 2 with a remainder of 1. The
value "1" is then written to the answer row and the value "2" carried
forward to the shillings column. This operation is repeated using the
values in the shillings column, with the additional step of adding the
value that was carried forward from the pennies column. The intermediate
total is divided by 20 as there are 20 shillings in a pound. The pound
column is then processed, but as pounds are the largest unit that is
being considered, no values are carried forward from the pounds column.
For the sake of simplicity, the example chosen did not have farthings.
Operations in practice
A scale calibrated in imperial units with an associated cost display.
During the 19th and 20th centuries various aids were developed to aid
the manipulation of compound units, particularly in commercial
applications. The most common aids were mechanical tills which were
adapted in countries such as the United Kingdom to accommodate pounds,
shillings, pennies and farthings, and ready reckoners,
which are books aimed at traders that catalogued the results of various
routine calculations such as the percentages or multiples of various
sums of money. One typical booklet[18] that ran to 150 pages tabulated multiples "from one to ten thousand at the various prices from one farthing to one pound".
The cumbersome nature of compound unit arithmetic has been recognized for many years—in 1586, the Flemish mathematician Simon Stevin published a small pamphlet called De Thiende ("the tenth")[19]
in which he declared the universal introduction of decimal coinage,
measures, and weights to be merely a question of time. In the modern
era, many conversion programs, such as that included in the Microsoft
Windows 7 operating system calculator, display compound units in a
reduced decimal format rather than using an expanded format (e.g.
"2.5 ft" is displayed rather than "2 ft 6 in").
Number theory
Until the 19th century, number theory was a synonym of "arithmetic". The addressed problems were directly related to the basic operations and concerned primality, divisibility, and the solution of equations in integers, such as Fermat's Last Theorem.
It appeared that most of these problems, although very elementary to
state, are very difficult and may not be solved without very deep
mathematics involving concepts and methods from many other branches of
mathematics. This led to new branches of number theory such as analytic number theory, algebraic number theory, Diophantine geometry and arithmetic algebraic geometry. Wiles' proof of Fermat's Last Theorem
is a typical example of the necessity of sophisticated methods, which
go far beyond the classical methods of arithmetic, for solving problems
that can be stated in elementary arithmetic.
Arithmetic in education
Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers, integers, fractions, and decimals (using the decimal place-value system). This study is sometimes known as algorism.
The difficulty and unmotivated appearance of these algorithms has
long led educators to question this curriculum, advocating the early
teaching of more central and intuitive mathematical ideas. One notable
movement in this direction was the New Math
of the 1960s and 1970s, which attempted to teach arithmetic in the
spirit of axiomatic development from set theory, an echo of the
prevailing trend in higher mathematics.[20]
Also, arithmetic was used by Islamic Scholars in order to teach application of the rulings related to Zakat and Irth. This was done in a book entitled The Best of Arithmetic by Abd-al-Fattah-al-Dumyati.[21]
The book begins with the foundations of mathematics and proceeds to its application in the later chapters.
See also
Related topics
Notes
"Arithmetic". Encyclopedia Britannica. Retrieved 2020-08-25.
References
- Cunnington, Susan, The Story of Arithmetic: A Short History of Its Origin and Development, Swan Sonnenschein, London, 1904
- Dickson, Leonard Eugene, History of the Theory of Numbers (3 volumes), reprints: Carnegie Institute of Washington, Washington, 1932; Chelsea, New York, 1952, 1966
- Euler, Leonhard, Elements of Algebra, Tarquin Press, 2007
- Fine, Henry Burchard (1858–1928), The Number System of Algebra Treated Theoretically and Historically, Leach, Shewell & Sanborn, Boston, 1891
- Karpinski, Louis Charles (1878–1956), The History of Arithmetic, Rand McNally, Chicago, 1925; reprint: Russell & Russell, New York, 1965
- Ore, Øystein, Number Theory and Its History, McGraw–Hill, New York, 1948
- Weil, André, Number Theory: An Approach through History, Birkhauser, Boston, 1984; reviewed: Mathematical Reviews 85c:01004
External links
 |
Look up arithmetic in Wiktionary, the free dictionary. |
 |
Wikimedia Commons has media related to Arithmetic. |
"Definition of Arithmetic". www.mathsisfun.com. Retrieved 2020-08-25.
Davenport, Harold, The Higher Arithmetic: An Introduction to the Theory of Numbers (7th ed.), Cambridge University Press, Cambridge, 1999, ISBN 0-521-63446-6.
Rudman, Peter Strom (2007). How Mathematics Happened: The First 50,000 Years. Prometheus Books. p. 64. ISBN 978-1-59102-477-4.
The Works of Archimedes, Chapter IV, Arithmetic in Archimedes, edited by T.L. Heath, Dover Publications Inc, New York, 2002.
Joseph Needham, Science and Civilization in China, Vol. 3, p. 9, Cambridge University Press, 1959.
Reference: Revue de l'Orient Chretien by François Nau pp. 327–338. (1929)
Reference: Sigler, L., "Fibonacci's Liber Abaci", Springer, 2003.
Tapson, Frank (1996). The Oxford Mathematics Study Dictionary. Oxford University Press. ISBN 0-19-914551-2.
"Python divmod() Function". W3Schools. Refsnes Data. Retrieved 2021-03-13.
Leonardo Pisano – p. 3: "Contributions to number theory" Archived 2008-06-17 at the Wayback Machine. Encyclopædia Britannica Online, 2006. Retrieved 18 September 2006.
Walkingame, Francis (1860). "The Tutor's Companion; or, Complete Practical Arithmetic" (PDF). Webb, Millington & Co. pp. 24–39. Archived from the original (PDF) on 2015-05-04.
Palaiseau, JFG (October 1816). Métrologie
universelle, ancienne et moderne: ou rapport des poids et mesures des
empires, royaumes, duchés et principautés des quatre parties du monde [Universal,
ancient and modern metrology: or report of weights and measurements of
empires, kingdoms, duchies and principalities of all parts of the world] (in French). Bordeaux. Retrieved October 30, 2011.
Jacob de Gelder (1824). Allereerste Gronden der Cijferkunst [Introduction to Numeracy] (in Dutch). 's-Gravenhage and Amsterdam: de Gebroeders van Cleef. pp. 163–176. Archived from the original on October 5, 2015. Retrieved March 2, 2011.
Malaisé, Ferdinand (1842). Theoretisch-Praktischer
Unterricht im Rechnen für die niederen Classen der Regimentsschulen der
Königl. Bayer. Infantrie und Cavalerie [Theoretical and practical instruction in arithmetic for the lower classes of the Royal Bavarian Infantry and Cavalry School] (in German). Munich. Archived from the original on 25 September 2012. Retrieved 20 March 2012.
Encyclopædia Britannica, I, Edinburgh, 1772, Arithmetick
Walkingame, Francis (1860). "The Tutor's Companion; or, Complete Practical Arithmetic" (PDF). Webb, Millington & Co. pp. 43–50. Archived from the original (PDF) on 2015-05-04.
Thomson, J (1824). The
Ready Reckoner in miniature containing accurate table from one to the
thousand at the various prices from one farthing to one pound. Montreal. ISBN 9780665947063. Archived from the original on 28 July 2013. Retrieved 25 March 2012.
O'Connor, John J.; Robertson, Edmund F. (January 2004), "Arithmetic", MacTutor History of Mathematics archive, University of St Andrews
Mathematically Correct: Glossary of Terms
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Coordinates: 16.5544°S 68.6741°W
