NUMBER SYSTEM

A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a

consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for

other numbers in different bases.

Ideally a number system will…..

Represent a useful set of numbers (e.g. all integers, or rational numbers)

Give every number represented a unique representation (or at least a standard representation)

Reflect the algebraic and arithmetic structure of the numbers. For example, the usual decimal representation of whole

numbers gives every whole number a unique representation as a finite sequence of digits. However, when decimal representation is used for the rational or real numbers, such numbers in general have an infinite number of representations, for example 2.31 can also be written as 2.310, 2.3100000, 2.309999999…, etc., all of which have the same meaning except for some scientific and other contexts where greater precision is implied by a larger number of figures shown.

Contribution of Indians

The most commonly used system of numerals is known as Arabic numerals or HinduTwo Indian mathematicians are credited with developing them. Aryabhata of Kusumapura developed the place-value notation in the 5th century and a century later Brahmagupta introduced the symbol for zero.[1] The numeral system and the zero concept, developed by the Hindus in India slowly spread to other surrounding countries due to their commercial and military activities with India. The Arabs adopted it and modified them. Even today, the Arabs called the numerals they use 'Rakam Al-Hind' or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread it to the western world due to their trade links with them.

Positional Notation positional system, also known as place-value

notation. Again working in base 10, ten different digits 0, ..., 9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1. Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu-Arabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system.

Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems need a large number of different symbols for the different powers of 10; a positional system needs only ten different symbols (assuming that it uses base 10).

The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the arithmetic numerals 0,1,2,3,4,5,6,7,8,9 and the geometricnumerals 1,10,100,1000,10000... respectively. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. The sign-value system does not need arithmetic numerals because they are made by repetition (except for the Ionic system), and the positional system does not need geometric numerals because they are made by position. However, the spoken language uses both arithmetic and geometric numerals.

Representation of rational no. onon number line

p/q where p and q are integers and q ! 0 are knownas rational numbers. The collection of numbers of the formp/q , where q > 0 is denoted by Q.

Rational numbers include natural numbers, whole numbers, integers and all negative and positive fractions. Here we can visualize how the girl collected all the rational numbers in a bag.

Rational numbers can also be represented on the number line and here we can see a picture of a girl walking on the number line.To express rational numbers appropriately on the number line, divide each unit length into as many number of equal parts as the denominator of the rational number and then mark the given number on the number line.

Successive magnification

The process of visualization of representation of numbers on the number line through a magnifying glass is known as the process of successive magnification.

Exponents of Real NumbersThis module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret. Objectives of this module: understand exponential notation, be able to read exponential notation, understand how to use exponential notation with the order of operations.

RationalizationIn elementary algebra, root rationalisation is a process by which surds in the denominator of an irrational fraction are eliminated.These surds may be monomials or binomials involving square roots, in simple examples. There are wide extensions to the technique.

Rationalizing FactorIt is difficult to deal with the expression having square root in the denominator. This raises a need of removing square root from the denominator. It can be done by rationlising the denominator.

Key Concepts 1. Numbers 1, 2, 3…….∞, which are used for counting are

called Natural numbers and are denoted by N. 2. 0 when included with the natural numbers form a new

set of numbers called Whole number denoted by W 3. -1,-2,-3……………..-∞ are the negative of natural

numbers. 4. The negative of natural numbers, 0 and the natural

number together constitutes integers denoted by Z. 5. The numbers which can be represented in the form of

p/q where q 0 ≠ and p and q are integers are called Rational

numbers. Rational numbers are denoted by Q. If p and q are co prime then

the rational number is in its simplest form. 6. Irrational numbers are the numbers which are non-

terminating and non-repeating. 7. Rational and irrational numbers together constitute

Real numbers and it is denoted by R. 8. Equivalent rational numbers (or fractions) have same

(equal) values when written in the simplest form. 9. Terminating fractions are the fractions which leaves

remainder 0 on division. 10. Recurring fractions are the fractions which never

leave a remainder 0 on division. 11. There are infinitely many rational numbers between

any two rational numbers. 12. If Prime factors of the denominator are 2 or 5 or both

only. Then the number is terminating else repeating/recurring. 13. Two numbers p & q are said to be co-prime if,

numbers p & q have no common factors other than 1.

16. Real numbers satisfy the commutative, associate and distributive

law of addition and multiplication. 17. Commutative law of addition: If a and b are two real

numbers then, a + b = b + a 18. Commutative law of multiplication: If a and b are two

real numbers then, a. b = b. a 19. Associative law of addition: If a, b and c are real

numbers then, a + (b + c) = (a + b) + c 20. Associative law of multiplication: If a, b and c are

real numbers then, a. (b. c) = (a. b). c 21. Distributive of multiplication with respect to

addition: If a, b and c are real numbers then, a. (b+ c) = a. b + a. c 22. Removing the radical sign from the denominator is

called rationalisation of denominator. 23. The multiplication factor used for rationalizing the

denominator is called the rationalizing factor. 24. The exponent is the number of times the base is

multiplied by itself. 25. In the exponential representation m a , a is called the base and m is called the exponent or power. 26. If a number is to the left of the number on the

number line, it is less than the other number. If it is to the right, then it is

greater than the number. 27. There is one to one correspondence between the set

of real numbers and the set of point on the number line.

The origins and history of number system.

We call them Arabic Numerals, but our numbers actually find their origins in the history of the Hindus of India. They have changed greatly over the centuries, passing first to the Arabs of the Middle East and finally to Europe in the Middle Ages, and are now the most commonly used numbers throughout the world.

ThanksName - Prajjwal KushwahaClass - flyers-1-bRoll no -28

Special thanks to Mr. pradeep Kumar lodha