Introduction to Arithmetic- The Number System

Number Systems

Number Systems, in mathematics, various notational systems that have been or are being used to represent the abstract quantities called numbers. A number system is defined by the base it uses, the base being the number of different symbols required by the system to represent any of the infinite series of numbers. Thus, the decimal system in universal use today (except for computer application) requires ten different symbols, or digits, to represent numbers and is therefore a base-10 system.

Throughout history, many different number systems have been used; in fact, any whole number greater than 1 can be used as a base. Some cultures have used systems based on the numbers 3, 4, or 5. The Babylonians used the sexagesimal system, based on the number 60, and the Romans used (for some purposes) the duodecimal system, based on the number 12. The Mayas used the vigesimal system, based on the number 20. The binary system, based on the number 2, was used by some tribes and, together with the system based on 8, is used today in computer systems.


Interger
Integer, any number that is a natural number (the counting numbers 1, 2, 3, 4, ...), a negative of a natural number (-1, -2, -3, -4, ...), or zero. A large proportion of mathematics has been devoted to integers because of their immediate application to real situations.

Any integer greater than 1 that is divisible only by itself and 1 is called a prime number. Every integer has a unique set of prime factors, that is, a list of prime numbers that when multiplied together produce the integer concerned. For example, the prime factors of the integer 42 are 2, 3, and 7.

Base (mathematics), the number of different single-digit symbols used in a particular number system. In the usual counting system of numbers, the decimal system (with symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9), the base is 10. In the binary number system, which has only the symbols 1 and 0, the base is 2. A base is also a number that, when raised to a particular power (that is, when multiplied by itself a particular number of times, as in 10² = 10 x 10 = 100), has a logarithm equal to the power. For example, the logarithm of 100 to the base 10 is 2. In geometry, the term base is used to denote the line or area on which a polygon or solid stands.

Roman Numeral

The system of number symbols created by the Romans had the merit of expressing all numbers from 1 to 1,000,000 with a total of seven symbols: I for 1, V for 5, X for 10, L for 50, C for 100, D for 500, and M for 1000. Roman numerals are read from left to right. The symbols representing the largest quantities are placed at the left; immediately to the right of those are the symbols representing the next largest quantities, and so on. The symbols are usually added together. For example, LX = 60, and MMCIII = 2103. When a numeral is smaller than the numeral to the right, however, the numeral on the left should be subtracted from the numeral on the right. For instance, XIV = 14 and IX = 9.  represents 1,000,000—a small bar placed over the numeral multiplies the numeral by 1000. Thus, theoretically, it is possible, by using an infinite number of bars, to express the numbers from 1 to infinity. In practice, however, one bar is usually used; two are rarely used, and more than two are almost never used. Roman numerals are still used today, more than 2000 years after their introduction. The Roman system's one drawback, however, is that it is not suitable for rapid written calculations.

Arabic Numeral
The common system of number notation in use in most parts of the world today is the Arabic system. This system was first developed by the Hindus and was in use in India in the 3rd century bc. At that time the numerals 1, 4, and 6 were written in substantially the same form used today. The Hindu numeral system was probably introduced into the Arab world about the 7th or 8th century ad. The first recorded use of the system in Europe was in ad 976.

The important innovation in the Arabic system was the use of positional notation, in which individual number symbols assume different values according to their position in the written numeral. Positional notation is made possible by the use of a symbol for zero. The symbol 0 makes it possible to differentiate between 11, 101, and 1001 without the use of additional symbols, and all numbers can be expressed in terms of ten symbols, the numerals from 1 to 9 plus 0. Positional notation also greatly simplifies all forms of written numerical calculation.

Rational Number
Rational Numbers, class of numbers that are the result of dividing one integer by another. Integers are the negative and positive whole numbers (… -3, -2, -1, 0, 1, 2, 3, …). The numbers ’, 5 (5/1), and –1.4 (-7/5) are therefore rational numbers because they are quotients (results of division) of two integers. Rational numbers are a subset of the real numbers, which also include the set of irrational numbers. Irrational numbers are numbers such as pi (p), the square root of two (Ã), and the mathematical constant e that are not the quotient of any two integers.

All rational numbers can be written as decimal numbers. The decimals may have a definite termination point (such as 5 or 3.427) or they may endlessly repeat in a pattern (such as 1.8888… or 2.18181…).

Irrational Number.

Irrational Numbers, class of numbers that cannot be produced by dividing any integer by another integer. Integers comprise the positive whole numbers, negative whole numbers, and zero: …-3, -2, -1, 0, 1, 2, 3…. Examples of irrational numbers include the square root of two (Ã, 1.41421356…), pi (p, 3.14159265…), and the mathematical constant e (2.71828182…). When expressed as decimals these numbers can never be fully written out as they have an infinite number of decimal places which never fall into a repeating pattern.

Real Number
Real Numbers, class of numbers comprising all positive and negative numbers, together with zero. Real numbers include the rational numbers. Rational numbers comprise all numbers that are equal to the quotient (result of dividing one number by another) of two integers, which are the positive and negative whole numbers: 1, 2, 3…, -1, -2, -3…, and 0. Thus the numbers Š, 7 (7/1), and –1.2 (-6/5) are rational numbers. In addition to rational numbers, real numbers include irrational numbers. Irrational numbers are numbers such as the square root of two (Ã), pi (p), and e that are not the quotient of any two integers. The real numbers are a subset of the complex numbers, which also include the set of imaginary numbers—numbers that are a multiple of i, where i is the square root of –1—as well as numbers that are a combination of real and imaginary numbers, such as 2 + 3i.
Real numbers can all be written as decimal numbers. The decimals may have a definite termination point (such as 5 or 3.427), endlessly repeat in a pattern (such as 2.12121…), or continue forever with no pattern (3.14159265…).

The idea of real numbers arose when ancient Greek mathematicians encountered difficulties with using only rational numbers. They discovered, for example, that à is not rational. The numbers p and e are often encountered in geometry and physics (p occurs in the equations for the area and perimeter of a circle, for instance). The recognition of these important 'irrational' numbers resulted in the creation of the set of real numbers.

Imaginary Number
Imaginary Numbers, numbers formed by multiplying a real number times i, where i is the square root of minus 1. The square root of any negative number can be expressed using i. The square root of -16, for example, is equal to i times the square root of 16 (iÂ):
= ÁÂ = iÂ

Certain equations, such as x² = -1, have no real number solutions. There simply is no real number that can be substituted for x in order to fulfill this equation because the square of any real number, positive or negative, is always positive. During the 16th century mathematicians invented the concept of a whole new set of numbers, the imaginary numbers, to deal with such equations.
Any sum of a real number and an imaginary number, such as 3.2 + 2i, is a complex number. More generally, complex numbers are all of the form a + bi, where a and b are real numbers but bi (the product of a real number and i) is an imaginary number. Complex numbers include all real numbers because a + bi = a when b is equal to zero. Complex numbers also include all imaginary numbers because a + bi = bi (an imaginary number) when a is equal to zero.

Complementary Number, in number theory, the number obtained by subtracting a number from its base. For example, the complement of 7 in numbers to base 10 is 3. Complementary numbers are necessary in computing, as the only mathematical operation of which digital computers (including pocket calculators) are directly capable is addition. Two numbers can be subtracted by adding one number to the complement of the other. Two numbers can be divided by using successive subtraction (which, using complements, becomes successive addition). Multiplication can be performed by using successive addition.

The four main operations of arithmetic can thus all be reduced to various types of addition and be performed within the capability of a digital computer, using the binary number system.

- Encarta