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Arithmetic Definition

Arithmetic is a branch of mathematics dealing with integers or, more generally, numerical computation. Arithmetic operations include addition, congruence calculation, division, factorization, multiplication, power computation, root extraction, subtraction, logarithms, and calculations involving modulo n. 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. Arithmetic is derived from the Greek terms arithmos and tiké meaning number and art respectively. 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. Arithmetic was part of the quadrivium taught in medieval universities. A mnemonic for the spelling of arithmetic is a rat in the house may eat the ice cream.

Modular arithmetic is the arithmetic of congruences. Floating-point arithmetic is the arithmetic performed on real numbers by computers or other automated devices using a fixed number of bits. The fundamental theorem of arithmetic, also called the unique factorization theorem, states that any positive integer can be represented in exactly one way as a product of primes. The Löwenheim-Skolem theorem, which is a fundamental result in model theory, establishes the existence of nonstandard models of arithmetic.

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. 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 ours. 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 ours, and their multiplication algorithm was only very 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 doesn't 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.

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. It is a complicated question to determine exactly when the Chinese started calculating with positional representation, but it was definitely before 400 BC. The ancient Chinese were the first to meaningfully discover, understand, and apply negative numbers as explained in the Nine Chapters on the Mathematical Art (Jiuzhang Suanshu), which was written by Liu Hui.

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. This approach 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."[5] The Arabs also learned this new method and called it hesab.

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 Indoram) surpasses any known method to compute. It's a marvelous method. They do their computations using nine figures and symbol zero". 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 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.

Leibniz's Stepped Reckoner illustrated above was the first calculator that could perform all four arithmetic operations.

Arithmetic Operations

The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, 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.

Addition (+)

Addition 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 identity element for a binary operation is the number that, when combined with any number, yields the same number as the result. According to the rules of addition, adding 0 to any number yields that same number, so 0 is the additive identity. The inverse of a number with respect to a binary operation is the number that, when combined with any number, yields the identity with respect to this operation. So the inverse of a number with respect to addition (its additive inverse, or the opposite number) is the number that yields the additive identity, 0, when added to the original number; it is immediately obvious that this is the negative of the original number. For example, the additive inverse of 7 is −7, since 7 + (−7) = 0.

Addition can be interpreted geometrically as in the following instance. If we have two sticks of lengths 3 and 6, then, if we place the sticks one after the other, the length of the stick thus formed is 9, since 3 + 6 = 9.

Subtraction (-)

Subtraction 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. 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, in modern algebra the construction of this inverse operation is often discarded in favor of introducing the concept of inverse elements, as noted under Addition, and to look at subtraction 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 widespread 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 (x or ∙ 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, 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, say x, greater than 1 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. (Again, 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. So, 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 apply repeated addition, or employ tables or slide rules, thereby mapping the multiplication to addition and back. These methods are outdated and 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(÷ or /)

Division 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 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 for subtraction, in modern algebra the construction of the division is discarded in favor of constructing the inverse elements with respect to multiplication, as introduced there. That is, division is a multiplication with the dividend and the reciprocal of the divisor as factors, that is a ÷ b = a × 1/b.

Within natural numbers there is also a different, but related notion, the Euclidean division, giving two results of 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 R < Q.

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 x 32 x 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, such as 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 Indoram 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 (the ones place, tens place, hundreds place) and, with a radix point, using those same symbols to represent fractions (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 zero 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 worl