arithmetic

In 1617 Scottish mathematician John Napier published his description of what was arguably the first mechanical calculator – a set of numbered rods, usually made of bone or ivory and therefore known as Napier's bones. Using them, multiplication became merely a process of reading off the appropriate figures and making simple additions.

Branch of mathematics concerned with the study of numbers and their properties. The fundamental operations of arithmetic are addition, subtraction, multiplication, and division. Raising to powers (for example, squaring or cubing a number), the extraction of roots (for example, square roots), percentages, fractions, and ratios are developed from these operations.

Forms of simple arithmetic existed in prehistoric times. In China, Egypt, Babylon, and early civilizations generally, arithmetic was used for commercial purposes, records of taxation, and astronomy. During the Dark Ages in Europe, knowledge of arithmetic was preserved in India and later among the Arabs. European mathematics revived with the development of trade and overseas exploration. Hindu-Arabic numerals replaced Roman numerals, allowing calculations to be made on paper, instead of by the abacus.

The essential feature of this number system was the introduction of zero, which allows us to have a place–value system. The decimal numeral system employs ten numerals (0,1,2,3,4,5,6,7,8,9) and is said to operate in ‘base ten’. In a base-ten number, each position has a value ten times that of the position to its immediate right; for example, in the number 23 the numeral 3 represents three units (ones), and the numeral 2 represents two tens. The Babylonians, however, used a complex base-sixty system, residues of which are found today in the number of minutes in each hour and in angular measurement (6 x 60 degrees). The Mayas used a base-twenty system.

There have been many inventions and developments to make the manipulation of the arithmetic processes easier, such as the invention of logarithms by Scottish mathematician John Napier in 1614 and of the slide rule in the period 1620–30. Since then, many forms of ready reckoners, mechanical and electronic calculators, and computers have been invented.

Modern computers fundamentally operate in base two, using only two numerals (0,1), known as a binary system. In binary, each position has a value twice as great as the position to its immediate right, so that for example binary 111 (or 111₂) is equal to 7 in the decimal system, and binary 1111 (or 1111₂) is equal to 15. Because the main operations of subtraction, multiplication, and division can be reduced mathematically to addition, digital computers carry out calculations by adding, usually in binary numbers in which the numerals 0 and 1 can be represented by off and on pulses of electric current.

Modular or modulo arithmetic, sometimes known as residue arithmetic or clock arithmetic, can take only a specific number of digits, whatever the value. For example, in modulo 4 (mod 4) the only values any number can take are 0, 1, 2, or 3. In this system, 7 is written as 3 mod 4, and 35 is also 3 mod 4. Notice 3 is the residue, or remainder, when 7 or 35 is divided by 4. This form of arithmetic is often illustrated on a circle. It deals with events recurring in regular cycles, and is used in describing the functioning of petrol engines, electrical generators, and so on. For example, in the mod 12, the answer to a question as to what time it will be in five hours if it is now ten o'clock can be expressed 10 + 5 = 3.

Properties of numbers:

Associative law All the properties of numbers may be deduced from this law, which states that the sum of a set of numbers is the same whatever the order of addition, and that the product of a set of numbers is the same whatever the order of multiplication.

Commutative law A special case of the associative law produces commutativity where there are only two numbers in the set. For example,

a + b = b + a

ab = ba

Distributive law The distributive law for multiplication over addition states that, given a set of numbers a, b, c, ... and a multiplier m,

m(a + b + c + ...) = ma + mb + mc + ..

For example,

9 × 132 = (9 × 100) + (9 × 30) + (9 × 2)

The distributive law does not apply for addition over multiplication; for example,

7 + (3 × 5) ≠ (7 + 3) × (7 + 5)

Identities Zero is described as the identity for addition because adding zero to any number has no effect on that number.

n + 0 = 0 + n = n

One is the identity for multiplication because multiplying any number by one leaves that number unchanged.

n × 1 = 1 × n = n

Negatives Every number has a negative −n such that:

n + (−n) = 0

Inverse Every number (except 0) has an inverse 1/n such that:

n × 1/n = 1