Name-Harsh

Class - 9th dRoll no - 34Math's

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A number system defines a set of values used to represent a quantity. We talk about the number of people attending school, number of modules taken per student etc. Quantifying items and values in relation to each other is helpful for us to make sense of our environment. The study of numbers is not only related to computers. We apply numbers everyday, and knowing how numbers work, will give us an insight of how computers manipulate and store numbers.

Number System

Number System• IntroductionDenotation of Symbols in Number System Real numbers - R Rational Numbers

- Q Integers - Z Whole Numbers -

W Natural Numbers -

N

Real NumbersIn 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

the integer −5 and the fraction 4/3, and all

the irrational numbers such as √2 (1.41421356… the square

root of two, an irrational algebraic number)

and π (3.14159265…, a transcendental number).

Real Numbers

IrrationalRational

It is divided into two parts :-

Rational And

Irrational

Rational numbersIn mathematics,

a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. Since q may be equal to 1, every integer is a rational number.

IntegersWhole Numbers

Natural Numbers

Rational Numbers are divided into three main

parts :-

Rational

Integers Whole Natural

1. INTEGERSAn integer is

a number that can be written without a fractional or decimal

component. For example, 21, 4, and −2048 are integers; 9.75, 5½, and √2 are

not integers.

The set of integers is a subset of the real numbers, and consists of the natural

numbers (0, 1, 2, 3, ...) and the negatives of the non-zero natural numbers (−1, −2)

2. Whole numbersWhole number is collection of

positive numbers and zero. Whole number also called as integer. The whole number is represented as {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ….}. The set of

whole numbers may be finite or infinite. The finite defines the numbers in the set are

countable. Infinite set means the numbers are

uncountable. . Zero is neither a fraction nor a decimal, so

zero is an whole number.

3. Natural Numbers

In mathematics, the natural numbers are those used

for counting and ordering . Properties of the natural

numbers related to divisibility, such as the

distribution of prime numbers, are studied in number theory. The

natural numbers had their origins in the words used to count things, beginning with

the number 1.

Irrational

numbersIn mathematics, an irrational number is

any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is

non-zero. Pythagoras property :

In a right angled triangle sum of square of two sides is always equal to square of third side.

(HYPOTENUSE)

Pythagoras (582- 500bc), Greek

philosopher and mathematician, whose doctrines

strongly influenced Plato.

Born on the island of Sámos, Pythagoras was instructed in the teachings of the early Ionian

philosophers Thales, Anaximander, and Anaximenes. Pythagoras is said to have been driven from Sámos by his disgust for the tyranny of Polycrates. About

530 bc Pythagoras settled in Crotona, a Greek colony in southern Italy, where he founded a movement with

religious, political, and philosophical aims, known as Pythagoreanism. The philosophy of Pythagoras is known

only through the work of his disciples.

Pythagoras

Decimal Expansion

 

 

IN CASE (I) THE REMAINDER NEVER BECOMES ZERO AND REPEATS AFTER A CERTAIN STAGE FORCING

THE DECIMAL EXPANSION TO GO FOR EVER. THESE TYPE OF DECIMAL EXPANSIONS ARE KNOWN AS

NON-TERMINATING REPEATING DECIMAL EXPANSION.

IN CASE (II) THE REMAINDER BECOMES ZERO AFTER A CERTAIN STAGE. THIS TYPE OF DECIMAL

EXPANSION IS COMMONLY KNOWN AS TERMINATING DECIMAL EXPANSION.

IN CASE (III) THE REMINDER NEVER BECOMES ZERO AND NEVER REPEATS. THESE TYPE OF DECIMAL

EXPANSION ARE CALLED NON-TERMINATING NON-REPEATING DECIMAL EXPANSION.

Note:-

  

Archimedes He was a Greek

mathematician. He was the first to compute the digits

in the decimal expansion of π (pi). He showed that -

3.140845 < π < 3.142857

Archimedes (287-212 BC), preeminent Greek

mathematician and inventor, who wrote important works on plane and solid geometry, arithmetic, and mechanics.

Archimedes was born in Syracuse, Sicily, and educated in

Alexandria, Egypt. In pure mathematics he anticipated many of the discoveries of modern science, such as the integral calculus, through his studies of the areas and volumes of curved solid figures and the areas of plane

figures. He also proved that the volume of a sphere is two-thirds the volume of a cylinder that circumscribes the

sphere. In mechanics, Archimedes defined the principle of the lever and is credited with inventing the compound

pulley. During his stay in Egypt he invented the hydraulic screw for raising water from a lower to a higher level. He is

best known for discovering the law of hydrostatics, often called Archimedes' principle, which states that a body

immersed in fluid loses weight equal to the weight of the amount of fluid it displaces. This discovery is said to have

been made as Archimedes stepped into his bath and perceived the displaced water overflowing.

Archimedes

ARYABHATTA Aryabhatta ( 476 – 550 A.D) ,

the great mathematician and astronomer , found the value of π correct to four decimal places (3.1456). Using high speed computers and advance algorithms , π has been computed to over 1.24 trillion decimal places.

The most commonly used system of numerals is known as Arabic numerals or Hindu Two Indian mathematicians are credited with developing them

• 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.

ARYABHATTA • Aryabhatta of Kusumapura

 developed the place-value notation in the 5th century and a century later Brahmagupta introduced the symbol for zero.

R. Dedekind

R. Dedekind :Julius Wilhelm Richard

Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), algebraic number theory and the foundations of the real numbers.

In 1870s two German mathematicians; Cantor and Dedekind, showed that corresponding to every real number, there is a point on the number line, and corresponding to every point on the number line, there exists a unique real number.

G. Cantor

Georg Cantor , (born March 3, 1845, St. Petersburg, Russia died Jan. 6, 1918, Halle, Ger.), German mathematician who founded set theory and introduced the mathematically meaningful concept of transfinite numbers, indefinitely large but distinct from one another .

G. Cantor

BabyloniansBabylonians, were famous for their astrological observations and calculations, and used a sexagesimal (base-60) numbering system. In addition to using base sixty, the Babylonians also made use of six and ten as sub-bases. The Babylonians sexagesimal system which first appeared around 1900 to 1800 BC, is also credited with being the first known place-value of a particular digit depends on both the digit itself and its position within the number . This as an extremely important development, because – prior to place-value system – people were obliged to use different symbol to represent different power of a base.

Euclid : Euclid was an ancient

mathematician from Alexandria, who is best known for his major work, Elements. He told about the division lemma, according to which, A prime number that divides a product of two integers must divide one of the two integer.  He was active in Alexandria during the reign of Ptolemy I (323–283 BC).

Euclid :

His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor.

Euclid :

Euclid :

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