Arithmetic

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Arithmetic

Arithmetic or arithmetics (from the Greek

Greek language

Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European....
word a???µ?? = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science

Science

In its broadest sense, science refers to any systematic knowledge or practice. In its more usual restricted sense, science refers to a system of acquiring knowledge based on scientific method, as well as to the organized body of knowledge gained through such research....
and business

Business

A business is a legally recognized organization designed to provide good s and/or Service to consumers. Businesses are predominant in capitalism economies, most being privately owned and formed to earn profit that will increase the wealth of its owners....
calculations. In common usage, the word refers to a branch of (or the forerunner of) mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
which records elementary properties of certain operations on number

Number

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Encyclopedia

Arithmetic or arithmetics (from the Greek

Greek language

Science

Business

Mathematics

Number

Mathematician

A mathematician is a person whose primary area of study and/or research is the field of mathematics....
s sometimes use the term (higher) arithmetic when referring to number theory

Number theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
, but this should not be confused with elementary arithmetic

Elementary arithmetic

Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division ....
.

History

The prehistory of arithmetic is limited to a very small number of small artifacts indicating a clear conception of addition and subtraction, the best-known being the Ishango bone

Ishango bone

The Ishango bone is a bone tool, dated to the Upper Paleolithic era, about 18000 to 20000 BC. It is a dark brown length of bone, the fibula of a baboon, with a sharp piece of quartz affixed to one end, perhaps for engraving or writing....
from central Africa

Democratic Republic of the Congo

The Democratic Republic of the Congo , is a country in central Africa with a small length of Atlantic coastline. It is the third largest list of African countries in order of geographical area....
, dating from somewhere between 18,000 and 20,000 BC.

It is clear that the Babylonians had solid knowledge of almost all aspects of elementary arithmetic by 1800 BC, although historians can only guess at the methods utilized to generate the arithmetical results - as shown, for instance, in the clay tablet Plimpton 322

Plimpton 322

Of the approximately half million Babylonian clay tablets excavated since the beginning of the 19th century, several thousand are of a mathematical nature....
, which appears to be a list of Pythagorean triple

Pythagorean triple

A Pythagorean triple consists of three positive integers a, b, and c, such that . Such a triple is commonly written , and a well-known example is ....
s, but with no workings to show how the list was originally produced. Likewise, the Egyptian

Ancient Egypt

Ancient Egypt was an Ancient history civilization in eastern North Africa, concentrated along the lower reaches of the Nile in what is now the modern nation of Egypt....
Rhind Mathematical Papyrus

Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scotland antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum....
(dating from c. 1650 BC, though evidently a copy of an older text from c. 1850 BC) shows evidence of addition, subtraction, multiplication, and division being used within a unit fraction

Egyptian fraction

An Egyptian fraction is the sum of distinct unit fractions, such as . That is, each Fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other....
system.

Nicomachus

Nicomachus

Nicomachus was an important mathematician in the ancient world and is best known for his works Introduction to Arithmetic and The Manual of Harmonics in Greek language....
(c. AD60 - c. AD120) summarised the philosophical Pythagorean

Pythagoreanism

Pythagoreanism is a term used for the esoteric and metaphysics beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a very inspirational source for Plato and Platonism....
approach to numbers, and their relationships to each other, in his Introduction to Arithmetic
Introduction to Arithmetic

Introduction to Arithmetic was written by Nicomachus almost two thousand years ago, and contains both philosophical prose and very basic mathematical ideas....
. At this time, basic arithmetical operations were highly complicated affairs; it was the method known as the "Method of the Indians" (Latin "Modus Indorum") that became the arithmetic that we know today. Indian arithmetic was much simpler than Greek arithmetic due to the simplicity of the Indian number system, which had a zero

0 (number)

Positional notation

A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, Geometric progression, called the radix or radix of that numeral system....
. The 7th century Syriac

Syriac Christianity

Syriac Christianity is a culturally and linguistically distinctive community within Eastern Christianity. It has its roots in the Near East, and is represented by a number of Christian denominations today, mainly in the Middle East and in Kerala, India....
bishop Severus Sebhokt mentioned this method with admiration, stating however that the Method of the Indians was beyond description. The Arabs learned this new method and called it hesab. Fibonacci

Fibonacci

Leonardo of Pisa , also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italy mathematician, considered by some "the most talented mathematician of the Middle Ages"....
(also known as Leonardo of Pisa) introduced the "Method of the Indians" to Europe in 1202. In his book "Liber Abaci

Liber Abaci

Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. Its title has two common translations, The Book of the Abacus or The Book of Calculation....
", Fibonacci says that, compared with this new method, all other methods had been mistakes. In the Middle Ages

Middle Ages

File:Karl 1 mit papst gelasius gregor1 sacramentar v karl d kahlen.jpgThe Middle Ages of European history are a period in history which lasted for roughly a millennium, commonly dated from the fall of the Roman Empire in the 5th century to the beginning of the Early Modern Period in the 16th century, marked by the division of Western Christi...
, arithmetic was one of the seven liberal arts

Liberal arts

The term liberal arts refers to the education derived from the Classical education curriculum....
taught in universities.

Modern algorithms for arithmetic (both for hand and electronic computation) were made possible by the introduction of Arabic numerals

Arabic numerals

The 'arabic numerals', or 'Hindu numerals' are the ten digits , which?along with Decimal Number System by which a sequence was read as a number?were originally defined by Indian mathematics, later modified and transferred to North African Islamic mathematics and transmitted to Europe in the Middle Ages, whence they spread around the wo...
and decimal

Decimal

The decimal numeral system has 10 as its Base . It is the most widely used numeral system....
place notation for numbers. Arabic numeral based arithmetic was developed by the great Indian mathematicians Aryabhatta, Brahmagupta

Brahmagupta

Brahmagupta was an Indian Indian mathematics and Indian astronomy....
and Bhaskara I

Bhaskara I

Bhaskara was a 7th century Indian mathematician, who was apparently the first to write numbers in the Hindu-Arabic numeral system decimal with a circle for the 0 , and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work....
. Aryabhatta tried different place value notations and Brahmagupta added zero to the Indian number system. Brahmagupta developed modern multiplication, division, addition and subtraction based on Arabic numerals. Although it is now considered elementary, its simplicity

Simplicity

Simplicity is the property, condition, or quality of being simple or un-combined. It often denotes beauty, purity or clarity. Simple things are usually easier to explain and understand than complicated ones....
is the culmination of thousands of years of mathematical development. By contrast, the ancient mathematician Archimedes

Archimedes

Archimedes of Syracuse was a Greek mathematics, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity....
devoted an entire work, The Sand Reckoner

The Sand Reckoner

The Sand Reckoner is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe....
, to devising a notation for a certain large integer. The flourishing of algebra

Algebra

Algebra is a branch of mathematics concerning the study of structure , relation , and quantity. Together with geometry, mathematical analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics....
in the medieval Islamic world and in Renaissance

Renaissance

The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the late Middle Ages and later spreading to the rest of Europe....
Europe

Europe

Europe is, conventionally, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally divided from Asia to its east by the water divide of the Ural Mountains, the Ural , the Caspian Sea, and by the Caucasus Mountains to the southeast....
was an outgrowth of the enormous simplification of computation

Computation

Computation is a general term for any type of information processing. This includes phenomena ranging from human thinking to calculations with a more narrow meaning....
through decimal

Decimal

The decimal numeral system has 10 as its Base . It is the most widely used numeral system....
notation.

Decimal arithmetic

Decimal notation constructs all real numbers from the basic digits, the first ten non-negative integers 0,1,2,...,9. A decimal numeral consists of a sequence of these basic digits, with the "denomination" of each digit depending on its position with respect to the decimal point: 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). An essential part of this notation (and a major stumbling block in achieving it) was conceiving of zero

0 (number)

Algorism

Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and mathematical table to the digits....
comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one to its right. This notation allows the addition of arbitrary numbers by adding the digits in each place, which is accomplished with a 10 x 10 addition table. (A sum of digits which exceeds 9 must have its 10-digit carried to the next place leftward.) One can make a similar algorithm for multiplying arbitrary numbers because the set of denominations is closed under multiplication. Subtraction and division are achieved by similar, though more complicated algorithms.

Arithmetic operations

The traditional arithmetic operations are addition

Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
, subtraction

Subtraction

Subtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with....
, multiplication

Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
and division

Division (mathematics)

In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.Specifically, if c times b equals a, written:...
, although more advanced operations (such as manipulations of percentage

Percentage

In mathematics, a percentage is a way of expressing a number as a fraction of 100 . It is often denoted using the percent sign, "%". For example, 45% is equal to 45 / 100, or 0.45....
s, square root

Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
, exponentiation

Exponentiation

Exponentiation is a mathematics operation , written 'an', involving two numbers, the base a and the exponent n....
, and logarithmic functions

Logarithm

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
) are also sometimes included in this subject. Arithmetic is performed according to an order of operations

Order of operations

In algebra and computer programming, when a number or expression is both preceded and followed by an operator such as minus or multiplication, a rule is needed to specify which operator should be applied first; this rule is known as a precedence rule, or more informally order of operation....
. Any set of objects upon which all four operations of arithmetic can be performed (except division by zero

Division by zero

In mathematics, a division is called a division by zero if the divisor is 0 . Such a division can be formally expressed as a/0 where a is the dividend....
), and wherein these four operations obey the usual laws, is called a field.

Addition (+)

Addition is the basic operation

Operator

In mathematics, an operator is a function which operates on another function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the functions which ar...
of arithmetic. In its simplest form, addition combines two number

Number

A number is a mathematical object used in counting and measurement. A notational symbol which represents a number is called a Numeral system, but in common usage the word number is used for both the abstract object and the symbol, as well as for the numeral for the number....
s, the addends or terms
Term (mathematics)

The word term is from the Latin terminus which literally means "boundary line, limit", from the Proto-Indo-European root "peg, post, boundary"....
, into a single number, the sum of the numbers.

Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation

Summation

Summation is the addition of a set of numbers; the result is their sum or total. An interim or present total of a summation process is termed the running total....
and includes ways to add infinitely many numbers in an infinite series

Series (mathematics)

In mathematics, given an infinite set sequence of numbers , a series is informally the result of adding all those terms together: . These can be written more compactly using the summation symbol ?....
; repeated addition of the number one

1 (number)

1 is a number, number names, and the name of the glyph representing that number.It represents a single entity, the unit of counting or measurement....
is the most basic form of counting

Counting

Counting is the mathematics action of repeatedly adding one, usually to find out how many objects there are or to set aside a desired number of objects , or for well-ordered objects, to find the ordinal number of a particular object, or to find the object with a particular ordinal number....
.

Addition is commutative and associative so the order in which the terms are added does not matter. The identity element

Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
of addition (the additive identity

Additive identity

In mathematics the additive identity of a Set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x....
) is 0, that is, adding zero to any number will yield that same number. Also, the inverse element

Inverse element

In mathematics, the idea of inverse element generalises the concepts of additive inverse, in relation to addition, and Multiplicative inverse, in relation to multiplication....
of addition (the additive inverse

Additive inverse

In mathematics, the additive inverse, or opposite, of a number n is the number that, when addition to n, yields 0 .The additive inverse of F is denoted −F....
) is the opposite of any number, that is, adding the opposite of any number to the number itself will yield the additive identity, 0. For example, the opposite of 7 is (-7), so 7 + (-7) = 0. Addition can be given geometrically as follows.

If a and b are the lengths of two sticks, then if we place the sticks one after the other, the length of the stick thus formed will be a+b

Subtraction (−)

Subtraction is essentially the opposite of addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend. If the minuend is larger than the subtrahend, the difference will be positive; if the minuend is smaller than the subtrahend, the difference will be negative; and if they are equal, the difference will be zero

0 (number)

0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures....
.

Subtraction is neither commutative nor associative. For that reason, it is often helpful to look at subtraction as addition of the minuend and the opposite of the subtrahend, that is a − b = a + (−b). When written as a sum, all the properties of addition hold.

Multiplication (×, ·, or *)

Multiplication 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, sometimes both simply called factors.

Multiplication is best viewed as a scaling operation. If the real numbers are imagined as lying in a line, multiplication by a number, say , greater than 1 is the same as stretching everything away from zero uniformly, in such a way that the number 1 itself is stretched to where was. Similarly, multiplying by a number less than 1 can be imagined as squeezing towards zero. (Again, in such a way that 1 goes to the multiplicand.)

Multiplication is commutative and associative; further it is distributive

Distributivity

In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra....
over addition and subtraction. The multiplicative identity is 1, that is, multiplying any number by 1 will yield that same number. Also, the multiplicative inverse

Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1....
is the reciprocal of any number, that is, multiplying the reciprocal of any number by the number itself will yield the multiplicative identity.

Division (÷ or /)

Division is essentially the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero

Division by zero

In mathematics, a division is called a division by zero if the divisor is 0 . Such a division can be formally expressed as a/0 where a is the dividend....
is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient will be greater than one, otherwise it will be less than one (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.

Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal

Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1....
of the divisor, that is a ÷ b = a × ¹/_b. When written as a product, it will obey all the properties of multiplication.

Examples

Multiplication table

×	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
2	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34	36	38	40	42	44	46	48	50
3	3	6	9	12	15	18	21	24	27	30	33	36	39	42	45	48	51	54	57	60	63	66	69	72	75
4	4	8	12	16	20	24	28	32	36	40	44	48	52	56	60	64	68	72	76	80	84	88	92	96	100
5	5	10	15	20	25	30	35	40	45	50	55	60	65	70	75	80	85	90	95	100	105	110	115	120	125
6	6	12	18	24	30	36	42	48	54	60	66	72	78	84	90	96	102	108	114	120	126	132	138	144	150
7	7	14	21	28	35	42	49	56	63	70	77	84	91	98	105	112	119	126	133	140	147	154	161	168	175
8	8	16	24	32	40	48	56	64	72	80	88	96	104	112	120	128	136	144	152	160	168	176	184	192	200
9	9	18	27	36	45	54	63	72	81	90	99	108	117	126	135	144	153	162	171	180	189	198	207	216	225
10	10	20	30	40	50	60	70	80	90	100	110	120	130	140	150	160	170	180	190	200	210	220	230	240	250
11	11	22	33	44	55	66	77	88	99	110	121	132	143	154	165	176	187	198	209	220	231	242	253	264	275
12	12	24	36	48	60	72	84	96	108	120	132	144	156	168	180	192	204	216	228	240	252	264	276	288	300
13	13	26	39	52	65	78	91	104	117	130	143	156	169	182	195	208	221	234	247	260	273	286	299	312	325
14	14	28	42	56	70	84	98	112	126	140	154	168	182	196	210	224	238	252	266	280	294	308	322	336	350
15	15	30	45	60	75	90	105	120	135	150	165	180	195	210	225	240	255	270	285	300	315	330	345	360	375
16	16	32	48	64	80	96	112	128	144	160	176	192	208	224	240	256	272	288	304	320	336	352	368	384	400
17	17	34	51	68	85	102	119	136	153	170	187	204	221	238	255	272	289	306	323	340	357	374	391	408	425
18	18	36	54	72	90	108	126	144	162	180	198	216	234	252	270	288	306	324	342	360	378	396	414	432	450
19	19	38	57	76	95	114	133	152	171	190	209	228	247	266	285	304	323	342	361	380	399	418	437	456	475
20	20	40	60	80	100	120	140	160	180	200	220	240	260	280	300	320	340	360	380	400	420	440	460	480	500
21	21	42	63	84	105	126	147	168	189	210	231	252	273	294	315	336	357	378	399	420	441	462	483	504	525
22	22	44	66	88	110	132	154	176	198	220	242	264	286	308	330	352	374	396	418	440	462	484	506	528	550
23	23	46	69	92	115	138	161	184	207	230	253	276	299	322	345	368	391	414	437	460	483	506	529	552	575
24	24	48	72	96	120	144	168	192	216	240	264	288	312	336	360	384	408	432	456	480	504	528	552	576	600
25	25	50	75	100	125	150	175	200	225	250	275	300	325	350	375	400	425	450	475	500	525	550	575	600	625

Number theory

The term arithmetic is also used to refer to number theory

Number theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
. This includes the properties of integers related to primality

Prime number

In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC....
, divisibility, and the solution of equations by integers, as well as modern research which is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic

Fundamental theorem of arithmetic

In number theory and algebraic number theory, the Fundamental Theorem of Arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers....
and arithmetic function

Arithmetic function

In number theory, an arithmetic function or arithmetical function is a function defined on the set of natural numbers that takes real or complex values....
s. A Course in Arithmetic by Serre

Jean-Pierre Serre

Jean-Pierre Serre is a French mathematician in the fields of algebraic geometry, number theory and topology. He has received numerous awards and honors for his mathematical research and exposition, including the Fields Medal in 1954 and the Abel Prize in 2003....
reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry. Number theory is also referred to as 'the higher arithmetic', as in the title of H. Davenport's

Harold Davenport

Harold Davenport was an English mathematician, known for his extensive work in number theory....
book on the subject.

Arithmetic in education

Primary education

A primary school is an institution where children receive the first stage of compulsory education known as Primary education. Primary school is the preferred term in the United Kingdom and many Commonwealth of Nations, and in most publications of the United Nations Educational, Scientific, and Cultural Organization ....
in mathematics often places a strong focus on algorithms for the arithmetic of natural number

Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
s, integer

Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
s, rational number

Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
s (vulgar fractions), and real number

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s (using the decimal

Decimal

The decimal numeral system has 10 as its Base . It is the most widely used numeral system....
place-value system). This study is sometimes known as algorism

Algorism

Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and mathematical table to the digits....
.

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

New math

New Math was a brief, dramatic change in the way mathematics was taught in United States grade schools, and to a lesser extent in European countries, during the 1960s....
of the 1960s and '70s, which attempted to teach arithmetic in the spirit of axiomatic development from set theory, an echo of the prevailing trend in higher mathematics.

Since the introduction of the electronic calculator

Calculator

A calculator is a device for performing mathematical calculations, distinguished from a computer by having a limited problem solving ability and an interface optimized for interactive calculation rather than programming....
, which can perform the algorithms far more efficiently than humans, an influential school of educators has argued that mechanical mastery of the standard arithmetic algorithms is no longer necessary. In their view, the first years of school mathematics could be more profitably spent on understanding higher-level ideas about what numbers are used for and relationships among number, quantity, measurement, and so on. However, most research mathematicians still consider mastery of the manual algorithms to be a necessary foundation for the study of algebra and computer science. This controversy was central to the "Math Wars

Math wars

Math wars is the debate over modern mathematics education, textbooks and curricula in the United States that was triggered by the publication in 1989 of the Curriculum and Evaluation Standards for School Mathematics by the National Council of Teachers of Mathematics and subsequent development and widespread adoption of a new generation o...
" over California's primary school curriculum in the 1990s, and continues today.

Many mathematics texts for K–12 instruction were developed, funded by grants from the United States National Science Foundation

National Science Foundation

The National Science Foundation is a United States government agency that supports fundamental research and education in all the non-medical fields of science and engineering....
based on standards created by the NCTM and given high ratings by United States Department of Education, though condemned by many mathematicians. Some widely adopted texts such as TERC

TERC

TERC may refer to:*Telomerase RNA component, a human gene.*The developers of the Investigations in Numbers, Data, and Space mathematics curriculum....
were based on the spirit of research papers which found that instruction of basic arithmetic was harmful to mathematical understanding. Rather than teaching any traditional method of arithmetic, teachers are instructed to instead guide students to invent their own (some critics claim inefficient) methods, instead using such techniques as skip counting

Skip counting

Skip counting is a mathematics technique taught as a kind of multiplication in standards-based mathematics textbooks such as TERC. In older textbooks, this technique is called counting by twos ....
, and the heavy use of manipulatives, scissors and paste, and even singing rather than multiplication tables or long division. Although such texts were designed to be a complete curricula, in the face of intense protest and criticism, many districts have chosen to circumvent the intent of such radical approaches by supplementing with traditional texts. Other districts have since adopted traditional mathematics

Traditional mathematics

Traditional mathematics is a term used to describe the predominant methods of Mathematics education in the United States in the early-to-mid 20th century....
texts and discarded such reform-based approaches as misguided failures.

Footnotes

External links

The New Student's Reference Work/Arithmetic (historical)
an early western work on arithmetic at

Arithmetic

History

Decimal arithmetic

Arithmetic operations

Addition (+)

Subtraction (−)

Multiplication (×, ·, or *)

Division (÷ or /)

Examples

Multiplication table

Number theory

Arithmetic in education

See also

Lists

Related topics

Footnotes

External links