NAME - Punit S. Sharma

CLASS - 9th

DIVISION - ‘B’

ROLL NO. - 10

SUBJECT - Mathematics

SCHOOL - Alchemy School

Topic :- Real Number Presentation.

What is Real Number ?

Real Numbers are just the number on the number line. It is the easiest way to think of them. Basically, if you put the number in the number line, then it is a real number. Also, you have to be add, subtract, multiply, divide that number in a way is consistent with the number line. They include many types of number.

PRESENTATION ON REAL NUMBERS.

In Mathematics, a real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers , such as integer -5 & the fraction 4/3 and all the irrational numbers such as root 2 (1.41421356……) and irrational algebraic numbers.

The Latex Real Number Symbol.

The real numbers can be characterized by the important mathematical property of

“COMPLETENESS” meaning that every non-empty set that has an upper bound has a smallest such bound , a property not possessed by the rational numbers. The discovery of a suitably rigorous definition of the real

numbers. Indeed, the realization that a better definition was needed was one of the important developments of 19th century mathematics. The real numbers are uncountable i.e while both the set of all

natural no.s and the set of all real no.s are infinite sets there can be no one to one function from the real

numbers to the natural numbers.

Simple fractions have been used by the Egyptians around 1000 BC. The concept of irrationality was implicity accepted by early Indian mathematician

since Manava, who were aware that the square root of certain numbers such as 2 & 61 could not be exactly determined. Around 500 BC, the Greek

mathematicians led by Pythagoras realized the need for irrational no.s, in particular the irrationality of the

square root of 2. The middle ages brought the acceptance of zero, negative, integral and fractional

numbers, first by Indian and Chinese mathematicians.

History of the Real Numbers. History of the Real Numbers.

Then by Arabic mathematicians, who were also the first to treat irrational no.s as algebraic objects, which was made possible by the development of the algebra. Arabic mathematician merged the concepts of “number” and “magnitude” into a more general idea of real numbers. The Egyptians mathematician “Abu Kamil Shuja Ibn Aslam” (C. 850-930) was the first to accept irrational numbers as solution to quadratic equations or as co-efficients in an equation, often in the form of square roots, cube roots and fourth roots. In the 16th century, Simon Stevin created the basics for modern decimal notation , and insisted that there is no difference between rational & irrational in this regard.

Types of Real Numbers.

In the 17th century Descartes introduced the term “REAL” to describe roots of the polynomial, distinguishing them from ‘imaginary’ ones.

In the 18th and 19th century there was much work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that cannot be rational number.

-:Properties of Real Number :- Real numbers can be ordered (this is not true , for

instance of imaginary numbers) They can be added, subtracted, multiplied and

divided by non-zero numbers in an ordered way. Real numbers are used to measure continuous

quantities. They may be expressed by decimal

representations that have an infinite sequence of digits to the right of the decimal point, these are

often represented in the same form as 324.823122147….

The ellipsis (three dots) indicate that there would still be more digits to

come. More formally, real numbers have the

two basic properties of being an ordered field, and having the Least

Upper Bound Property. The first says that the real number

comprise a field with addition and multiplication as well as division by

non-zero numbers which can be totally ordered on a number line in a

way compatible with addition and multiplication.

Basic Properties :-

Property Table of Real Numbers.

The second says that if a non-empty set of real numbers has an upper bound, then it has a real Least Upper

Bound.

The second condition distinguishes the real numbers from the rational numbers.

For example :- the set of rational numbers whose square is less than 2 is a set with an upper bound but no (rational) Least Upper Bound hence, the rational

numbers do not satisfy the Least Upper Bound property.

Continue Basic Properties…

Rational Numbers :- in other words all integers, fractions and decimals (including repeating decimals)

Example :- 2, 3, -2, ½, -3/4, .34

Irrational Numbers :- the number including non-terminating & non-recurring and terminating & recurring.

Example :- Root 3 & -5 yes, irrational numbers can be ordered and put on a number line.

Types of Real Numbers with Example:-

Continue Types of Real Numbers & Examples..

1) Natural Numbers :- {1, 2, 3, 4,……}

2) Whole Numbers :- {0, 1, 2, 3, 4,….}

3) Integers :- {…., -3, -2, -1, 0, 1, 2, 3,….}

4) Rational Numbers :- All the numbers that can be written as p/q, where a and b are integers, and b is to 0.

5) Irrational Numbers :- Numbers such as 2, 7, -

6) Real Numbers :- The union of the sets of rational numbers and irrational numbers.

The Real Number System :-

BY :- Punit

Sharma.