NUMBER SYSTEMS

IntroductionNumber 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.

Binary(Base 2)

Octal(Base 8)

Decimal(Base 10)

Hexadecimal(Base 16)

0 0 0 01 1 1 110 2 2 211 3 3 3100 4 4 4101 5 5 5110 6 6 6111 7 7 71000 10 8 81001 11 9 91010 12 10 A1011 13 11 B1100 14 12 C1101 15 13 D1110 16 14 E1111 17 15 F10000

20 16 10

11111111

377 255 FF

11111010001

3721

2001 7D1

Number SystemsThe decimal number system is in universal use today, except in computers, which uses the binary number system. The decimal system requires ten different symbols, or digits, to represent all numbers; it is known as a base-10 system. The binary system requires only two digits, 0 and 1. Some cultures have used systems based on other numbers. The Babylonians, for example, used a base-60 number system.

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 (see Number Theory). 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.

Integers

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.

RATIONAL 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 Numbers

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.The irrational numbers, together with the rational numbers (numbers that can be produced by dividing one integer by another), make up the set of real numbers.

Decimal numbersDecimal numbers come from fractions: They correspond to particular fractions. Nowadays, a decimal number indicates a number written with a decimal point and followed by a number of figures, 23.45 for example

.

The idea of place values can be extended to accommodate fractions. Instead of writing 1 �(one and two-tenths), we can use a decimal point (.) to represent the same fraction as 1.2. Just as places to the left of the decimal represent units, tens, hundreds, and so on, those to the right of the decimal represent.

Multiplying decimals is similar to multiplying integers, except that the position of the decimal point must be kept in mind. First, multiply decimal numbers as if they were integers, without considering the decimal points. Then place the decimal point at the appropriate position in the product so that the number of decimal places is the same as the total number of decimal places in the numbers being multiplied. For example, in multiplying 0.3 by 0.5

In cases where the divisor is a decimal number, convert the problem to one in which the divisor is an integer; division may then proceed as in the above example. To divide 14 by 0.7, for example, convert the divisor to an integer by multiplying it by 10: (0.7)(10) = 7. Then multiply the dividend by an equal amount. We can understand this procedure more easily by considering the division rewritten as a fraction. Multiplying both numerator and denominator by the same amount will not change the value of the fraction:

Negative numbersThere is evidence that negative numbers were being used in India in the 7th century. It is important to note that Indians used zero, a necessary precondition to conceiving negative numbers. Negative numbers were called debt numbers for commercial reasons, just as today's bank statements have two columns entitled credit (for receipts) and debit (for expenditure).

The use of negative numbers in the West came much later. Italian Renaissance mathematicians, who were specialists in algebra (the science of solving equations), understood that without negative numbers they could not solve certain equations (x + 7 = 0, for example). However, they were unsure whether to call these solutions proper numbers. And even in the 17th century, French mathematician Descartes described negative numbers as false numbers. It was not until the 19th century that negative numbers were finally treated as true numbers.

The arithmetic operation of addition is basically a means of counting quickly and is indicated by the plus sign (+). We could place 4 apples and 5 more apples in a row, then count them individually from 1 to 9. Addition, however, makes it possible to count all of the apples in a single step (4 + 5 = 9).We call the end result of addition the sum. The simplest sums are usually memorized. This table shows the sums of any two numbers between zero and nine:

To find the sum of any two numbers from 0 to 9, locate one of the numbers in the vertical column on the left side of the table and the other number in the horizontal row at the top. The sum is the number in the body of the table that lies at the intersection of the column and row that have been selected. For example, 6 + 7 = 13.We can easily add long lists of numbers with more than one digit by repeatedly adding one digit at a time. For example, if the numbers 27, 32, and 49 are listed in a column so that all the units are in a line, all the tens are in a line, and so on, finding their sum is relatively simple:

1. How do we decide the group(s) to which a number belongs?We can distinguish several groups of numbers.

denotes the set of whole numbers.  =  . denotes the set of integers.

denotes the set of rational numbers. These are numbers that can be written in the form where and * (therefore they are whole

quotients). For example, and are rational numbers, but

is not a rational number. denotes the set of real numbers. This set

comprises all the numbers that we use. They can be shown on a coordinate number line:

Every real number is represented by a point and each point represents a real number. This set includes irrational numbers, i.e., real numbers that are not rational.These sets of numbers are subsets of each other as follows: N,Z,R          . This means that a whole number is also an integer; an integer is also a decimal, etc.

To recognize a number's type:

Simplify how it is written as much as possible. In the case of an irreducible quotient a/b , carry out the division. If it is finite (i.e., if the remainder is zero) then a/b is a decimal; if it is infinite, it is referred to as recurring and a/b is a rational number that is not a decimal. If the number cannot be written as a quotient of integers, then it is irrational

PythagorasConsidered the first true mathematician, Pythagoras in the 6th century bc emphasized the study of mathematics as a means to understanding all relationships in the natural world. His followers, known as Pythagoreans, were the first to teach that the Earth is a sphere revolving around the Sun. This detail showing Pythagoras surrounded by his disciples comes from a fresco known as the School of Athens (1510-1511), by Italian Renaissance painter Raphael.

Rules of calculation: The sum of two positive numbers is positive; the sum of two negative numbers is negative; the absolute value of the sum of two numbers with the same sign is the sum of the absolute values of these numbers.Examples: (+7) + (+2) = +9. As (+7) and (+2) are positive, the result is positive. We obtain 9 when we calculate 7 + 2. (-4) + (-6) = -10. As (-4) and (-6) are negative, the result is negative. We obtain 10 when we calculate 4 + 6. B. The two numbers have different signs

Rules of calculation: The sign of the sum of two numbers with different signs is the sign of the number with the greater absolute value; the absolute value of the sum of two numbers with different signs is the difference between the absolute values of the numbers (the largest minus the smallest).Particular case: The sum of two opposite numbers is equal to 0. For example, (-7) + (+7) = 0.

Examples: (+9) + (- 4) = +5. With the numbers (+9) and (-4), (+9) has the greater absolute value and so it gives its + sign to the result. We obtain 5 when we calculate 9 - 4.(+2) + (-8) = -6. In this second example, it is (-8) that has the greater absolute value and that gives its “–” sign to the result. We obtain 6 when we calculate 8 - 2