INTRODUCTION

A number system defines a set of values used to represent quantity. We talk about the number of people attending school, number of modules taken per students 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.

NUMBERSThe Numbers play an vital role in our life.

Counting numbers begin from 1.

The smallest number is 1.

Next number ,called Successor is obtained by adding 1 to the number.

However, there is No largest number.

Number 0 stands for Nothingness.

BRIEF INTRODUCTION ABOUT NUMBERS

A number is a mathematical object used

In counting and numbering. Numerals are often used for labels ,for ordering serial numbers , and for codes like ISBNs.In mathematics the definition of number have been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers.

BRIEF INTRODUCTION ABOUT NUMBERS

Numbers were probably first used many thousand of years ago in commerce, and initially only whole numbers and perhaps rational numbers were needed. But already in Babylonian times, practical problems of geometry began to require square roots .

Certain procedures which take one or more numbers as input and produce a number as output a number are called numerical operations.

NATURAL NUMBERS

All counting numbers which start from 1 and continuous till infinity are natural numbers.

They have only positive value numbers.

Examples: 1,2,3,4……

WHOLE NUMBERS

Whole numbers are the natural number with the number 0 added to them.

They have one number zero as neither nor negative but all numbers are positive.

Examples: 0,1,2,3,4……

POSITIVE NUMBERS

The numbers which have +sign are known as positive numbers.

They include the numbers to the right of 0.

Examples: 2,3,4……

NEGATIVE NUMBERS

The numbers which have –sign are known as negative numbers.

They include numbers to the left of 0.

Examples: -3,-2,-1……

INTEGERS

Integers are natural numbers with 0 their and negative values.

They have neutral number 0,the positive numbers with their negative counterparts.

Examples: -4,-3,-2,-1,0-1,2….

REAL NUMBERS

Real numbers comprise of any number you can think or use in everyday life.

Each real number represents A unique number of the number line.

Real numbers are the compilation of all types of numbers

All types of numbers come under real numbers. The collection of rational and irrational numbers together make up real numbers.

REAL NUMBE

RS

POSITIVE AND

NEGATIVE etc

NATURAL AND

WHOLE

INTEGERS

RATIONAL

AND IRRATION

AL

RATIONAL NUMBERS

Rational numbers are those numbers which cannot be written as integers.

They are written as an integer divided by another integer and the denominator is not 0 and both numbers do not have common factors.

Rational numbers can be called fractions.

Rational numbers have either ending or non terminating repeating decimal expansions.

There are infinitely many rational numbers between any two given rational numbers.

Examples:16,-8,1/2,3.5,1.3333..,-3/4….

IRRATIONAL NUMBERSAll numbers which cannot be written as an integer upon integer where the denominator is 0 and both integers are co-prime are irrational numbers.

They are non-terminating non-repeating decimal expansions.

The roots of prime numbers are irrational.

Examples: √17,∏……

R.DEDEKINDWhile teaching calculus for the first time at the polytechnic Dedekind came with the notion now called a Dedekind cut (German: schnitt), a new standard definition of real numbers. The idea behind a cut is that an irrational number divides the rational numbers into two classes (sets), with all the members of one class (upper) being strictly greater than the members of the other (lower) class. For example, the square root of 2 puts all the negative numbers and the numbers whose square roots are less than two into the lower class, and the positive numbers whose squares are greater than two into the upper class. Every location on the number line continuum contains either a rational or an irrational number. Thus there are no empty locations, gaps or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet “stetigkeit und Irrationale

(1831-1916)

In 1874, while on a holiday in Interlaken, Dedekind met cantor. Thus began an enduring relationship of mutual respect, and Dedekind became one of the first mathematicians to admire Cantor’s work on infinite number sets, proving a valued ally in Cantor’s battle with Kronecker, who was philosophically opposed to Cantor’s transfinite numbers.If there existed a one-to-one correspondence between two sets, Dedekind said that the two sets were “similar”. He invoked the similarity to give the first precise definition of an infinite set: a set is infinite when it is “similar to a proper part of itself,” in modern terminology, is equinumerous to one of it’s proper subsets.(This is known as Dedekind’s theorem.) Thus the set N of natural numbers can be shown to be similar to the subset of N whose members are the squares of every member of N,(N N2).

zheian”(continuity and irrational numbers); in modern terminology vollstandigkeit, completeness.

G.CANTOR

(I845-1918)

Georg Cantor was a German mathematician, best known as a inventor of set theory, which has became fundamental theory in mathematics. Cantor established the importance of one-to-one Correspondence between the members of two sets, defined infinite and well ordered sets and proved that the real numbers are “more numerous” than the natural numbers. In fact Cantor’s method of proof of this theorem implies an existence of an “infinity of infinities”. He defined the cardinal and the ordinal and their arithmetic. Cantor’s theory of transfinite numbers was originally regarded as so counter-intuitive and shocking that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincare and later from Hermann Weyl and L.E.J. Brouwer while Ludwig Wittgenstein raised philosophical objections .

Some Christian theologians saw Cantor’s work as a challenge to uniqueness as the absolute infinity in the nature of god. On one occasion equating the theory of transfinite numbers with pantheism-a proposition which Cantor vigorously refuted . The objections to his work were occasionally fierce: Poincare referred to Cantor’s idea as a “grave disease” infecting the discipline of mathematics, and Kronecker’s public opposition and personal attacks included describing Cantor as “a renegade” and “a corrupter of youth.” Kronecker even objected to Cantor’s proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writhing decades after Cantor’s death, it was lamented that mathematics is “ridden through and through with the pernicious idioms of set theory,” which he dismissed as “utter nonsense” that is “laughable” and “wrong”.

Cantor’s recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of his contemporaries , some have explained these episodes as a probable manifestations of a bipolar disorder.The harsh criticism has been matched by later accolades. In 1904, the Royal society awarded Cantor its Sylvester medal, the highest honor it can confer for work in mathematics. It has been suggested that Cantor believed his theory of transfinite numbers had been communicated to him by God. David Hilbert defended it from its critics by famously declaring; “No one shall expel us from the paradise that Cantor has created.”

PYTHAGORES

(569BC-479BC)

Pythagoras of Samos was a lonian Greek Philosopher ,mathematician , and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him. He was born on the island of Samos, and might have travelled widely in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC , he moved to Croton a Greek colony in southern Italy, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical theories. The society took an active role in politics of croton, but this eventually led to their downfall. The Pythagorean meeting-places were burned , and Pythagoras was forced to flee the city. He is said to have ended his days in Metapontum.

Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. He is often revered as a great mathematician, mystic and scientist, but he is best known for the Pythagorean theorem his bears his name. However, because legend obfuscation could his work even more than with the other pre-Socratic philosophers, one can give account of his teaching to a little extent, and some have questioned whether he contributed much to mathematics and natural philosophy. Many of he accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors. Whether or not his disciples believed that everything was related to mathematics and that numbers were the ultimate reality is unknown. It was said that he was the first man to call himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato, and through him, all of western philosophy.

ARCHIMEDES

(287BC-212BC)

AryabhattaAD. 476-550

The Greek genius Archimedes was the first to compute digits in the decimal expansion of ∏. He showed 3.140845<∏<3.142857. Aryabhatta , the great Indian mathematician and astronomer, found the value of ∏ correct to four decimal places (3.1416). Using high speed computers and advanced algorithms, ∏ has been computed to over 1.24 trillion decimal places!

REAL NUMBERS AND THEIR DECIMAL EXPANSIONS•Decimal expansions of Rational numbers:

1. The decimal expansion of rational of a rational number is either terminating or non- recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational.

2. The decimal expansion of an irrational number is non- terminating recurring. Moreover, a number whose decimal expansion is a non-terminating non recurring is irrational.

Decimal expansion of irrational numbers: root 2: 1.4142135623730∏950488016887242096… ∏:3.14159265358979323846264338327950…

TO PROVE SQUARE ROOT 3 IS AN IRRATIONAL NUMBERSolution:To prove that this statement is true, let us assume that square root 3 is rational so that we may writeSquare root 3 = a/bHere a and b = any two integers. We must then show that no two such integers can be found.Squaring both side3 = a2/b23 b2 = a2If b is odd then b2 is odd. Similarly, if b is even then b2, a2, and a are even. Since any choice of even values of a and b leads to a ratio a/b that can be reduced by canceling a common factor of 2.Suppose a2 is odd than then b is odd that is a = 2m=1 and b =2n=1.Putting the value of a and b in above equation 3 (4n2+4n+1) = 4m2+4m+16n2+6n+1 = 2(2m+m)The LHS of the above expression is odd and RHS is even. That is contradiction.That is square root 3 is irrational.

TO PROVE SQUARE ROOT 2 IS AN IRRATIONAL NUMBERSolution:Suppose square root 2 is rational. That means it can be written as a ratio of two integers p and q root 2 = p/qWhere we may assume that p and q have no common factors. (If there are any factors we cancel them in numerator and denominator.) Squaring on 1.both the sides gives root 2 = p2/q2 (2)Which implies p2= 2q2 (3)Thus, p2 is even. The only way this can be is that p itself is even. But then p2 is actually divisible by 4.Hence q2 and therefore q must be even. So p and q are both even which is a contradiction to our assumption that they have no common factors. The square root 2 cannot be rational!

REPRESENTING IRRATIONAL NUMBERS ON NUMBER LINE Steps:1. Draw a unit square OABC with each side 1 unit in length on

number line. Make sure that vertex O coincides with zero. Then see by the Pythagoras theorem that OC=

√12+12= √2.2. We have seen that OC= √2. Using a compass with the center O

and radius OC, draw an arc intersecting the number line at point P. Then P corresponds to 2 on the number line.

3. Construct BD of unit length perpendicular to OB. Then using the Pythagoras theorem, we see that OD = √(√2)2+12=√3. Using a compass, with center O and radius OD, draw an arc which intersects the number line at point Q . Then Q corresponds to √3

4. In the same way, we can locate n for any positive integer n, after n-1 has been located.

In geometry the square root spirals a spiral composed of continuous right angle triangles. It was first constructed by Theodorus of Cyrene.

This spiral is started with an isosceles right triangle, with each leg having unit length. Another right triangle is formed with one leg being the hypotenuse of the prior triangle (√2)and other leg having length of 1; the length of hypotenuse of this second triangle is √3. The process then repeats till √17. The spiral stops at √17. The reason is commonly believed to be that the √17 hypotenuse belongs to the last triangle that does not overlap the figure. The spiral represents many of the creations of nature. For ex. shells

SQUARE ROOT SPIRAL

SUCCESSIVE MAGNIFICATIONThe process of Visualizationof representation of numbers on number line, through a magnifying glass, is known as successive magnification.

OPERATIONS ON REAL NUMBERS Rational numbers satisfy the commutative, associative and distributive laws for addition and multiplication. Moreover, if we add, subtract, multiply or divide two rationals, we still gets rational number. Irrational numbers also satisfy the commutative, associative and distributive laws for addition and multiplication. If we add, subtract, multiply or divide two irrationals, the result may be rational or irrational. The sum of difference of rational number and irrational number is irrational. The product of quotient of a non-zero rational number with a irrational number is irrational.

IDENTIEIES RELATING TO SQUARE ROOTSI. √ab = √a √b VII. (√a)2 = aII. √a/b = √a/√bIII. (√a+√b) (a-b) = a-bIV. (a+√b) (a-√b) = a2-bV. (√a+√b) (√c+√d) = √ac + √ad + √bc + √bdVI. (√a+√b) = a+2√ab+b

LAWS OF EXPONENTS FOR REAL NUMBERSI. am x an = am+n

II. (am)n = a mn

III. am/an= am-n,m >nIV. am bm = (ab)m

V. (a)0= 1 We can extend the laws of exponents, even

when the base is a positive real number and exponents are rational numbers.

Thank you