What are irrationals?

irrational numbers

Irrational numbers


 A number that can not be written in the form p/q, where p and q are integers and q is non zero. So we can say a number which is not rational is irrational. Thus, √2, √5, π, etc. are such examples of irrational numbers.


In the view of decimal expansion


As we  know the decimal expansion of √2 =.  1.4142135623730950488016887242096..., π = 3.14159265358979323846264338327950...., e = 2.71828183, etc., which are irrational numbers.

In above examples we have seen that decimal expansion is non terminating as well as non repeating. Now, let us summarize our results in the following form:

The decimal expansion of a irrational number is neither terminating nor non terminating recurring (non repeating). Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational.

         The collection of irrational numbers is denoted by Q' (Compliment of Q). 

The Greek genius Archimedes(287 BC - 212 BC) was the first to compute digits in the decimal expansion of π. He showed 3.140845 < π < 3.142857. Aryabhatta (476 AD - 550 AD), the great Indian mathematician and astronomer, found the value of π correct up to four decimal places. Using high speed computers and advanced algorithms, π has been computed to over 1.24 trillion decimal places.


Properties of irrational numbers:

  • No fraction is irrational number.
  • No natural number is irrational number.
  • The integer 0 is not irrational number.
  • All integers are not irrational numbers.
  • Here we can't say what is next irrational number, i.e. we can't say successor or predecessor of a irrational number as there are infinitely many irrational numbers, even uncountable.
  • Irrationals have a denseness property, i.e. the closure of a collection of irrational numbers is closed.
  • Between any two given irrational number, there are infinitely many rationals as well as irrationals.
  • Square-root of every every positive integer which is not perfect square is irrational number.
  • There is nothing common between rationals and irrationals.

PETER LUDWIG MEJDELL SYLOW

                                          
sylow
sylow


Ludwig Sylow was born on December 12, 1832, in Christiania (now Oslo), Norway. While a student at Christiania University, Sylow won a gold medal for competitive problem solving. In 1855, he became a high school teacher, and despite the long hours required by this teaching duties, Sylow found time to study the papers of Abel.  During the school year 1862-1863, Sylow received a temporary appointment at Christiana University and gave lectures on Galois's theory and permutation groups. Among his students that year was the great mathematician Sophus Lie, after who's Lie algebras and Lie groups are named. From 1873 to 1881, Sylow, with some help from Lie, prepared a new edition of Abel's work. In 1902, Sylow and Ellington Hoslt published Abel's correspondence.

    Sylow's great discovery, Sylow's Theorem, came in 1872. Upon learning of Sylow's result, C. Jorden called it "one of the essential points in the theory of permutations." The result took on greater importance when the theory of abstract groups flowered on the late 19th century and early 20th century.

   In 1869, Sylow's was offered a professorship at Christiania University, but turned it down. Upon Sylow's retirement at the age 6t from high school teaching, Lie mounted a successful campaign to establish a chair for Sylow's at Christiania University. Sylow held this position until his death on September 7, 1918.
 

WHAT ARE RATIONALS & PROPERTIES ?


rational numbers


Rational numbers: A number that can be written in the form p/q, where p and q are integers and q is non zero. Thus, 2/3, 5/9, 31/57, -73/1134 are such examples of rational numbers.

In the view of decimal expansion

As we know that any fraction can be converted into decimal number upon dividing the numerator by its denominator. For example: 1/2 = 0.5, 1/3 = 0.33333....., 1/4 = 0.25, 14/11= 1.272727272....,  etc.

In above examples we have seen either the decimal expansion is terminating or repeating but what about the decimal expansion which are neither terminating nor repeating. About the later case we'll discuss later on. Now, let us summarise our results in the following form:

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

               The collection of rational numbers is denoted by Q.

Properties of rational numbers:

  • All fractions are rational numbers.
  • All natural numbers are rational numbers but all rational numbers are not natural numbers.
  • The integer 0 is a rational number.
  • All integers are rational numbers.
  • Here we can't say what is next rational number, i.e. we can't say successor or predecessor of a rational number as there are infinitely many rational numbers, even countably infinite.
  • Rationals have a denseness property, i.e. the closure of a collection of rational numbers is closed.
  • Between any two given rational numbers, we can always find another rational number and hence infinitely many rationals between any two given rationals.

ALL ABOUT FRACTIONS

Fractions represent parts of a whole.

fractions
fractions



For example, if an apple is divided into three equal parts, each part is called one third and is denoted  by 1/3. We call 1/3 a fraction. Similalry, 1/4 , 2/5 , 3/7 , and 5/4 are also fractions.
      
In any fraction a/b, the dividend a is called the numerator and dividend b is called the denominator. For example, in the fraction 5/9, the numerator is 5 and the denominator is 9.

Classification of Fractions
           
Common, simple or vulgar fraction
The numerator and the denominator are integers in a common fraction, also known as a simple fraction or vulgar fraction.
Example 3/5 , 7/9 , 11/7 , 3/2 are some simple fractions.

Complex fraction
A fraction in which the numerator or the denominator or the both of them also contain fractions, is called a complex fractions.
For example: (4/5)/(7/9), (3/5)/(8/7), etc.

Proper fraction
A vulgar fraction in which the numerator is less than the denominator.
For example: 3/5, 4/9, etc.

Improper fraction
A vulgar fraction in which the numerator is greater than the denominator.
For example: 4/3, 8/5, etc.

Mixed fraction
An integer together with a proper fraction, is called a mixed fraction.
From example: 67/5, etc.

         A mixed fraction can easily be converted into an improper fraction as follows.

 Mixed fraction = [ (integral part) × (denominator of the proper fraction) + (numerator of the proper fraction)] / denominator.
OR
    Mixed fraction = quotient in (numerator ÷ denominator) + remainder/ denominator

Equivalent or equal fractions
If the numerator and denominator of a fraction are multiplied or divided by the same non-zero number, then we get an equivalent or equal fraction.
For example: 2/5 = 6/15, 7/9 = 42/54, etc.

This brings us to conclusion that the value of a fraction remains same of both its numerator and denominator are multiplied or divided by the same number.

• Two fractions a/b and c/d   are equivalent fractions if ad = BC.

Simplest form of fraction: A fraction is said to be in the simplest form of if its numerator and denominator have no factor in common except 1.
For example: 2/3, 7/9, 11/51, etc.

Like and unlike fractions: Two or more fractions are called like fractions of they have the same denominator. Fractions that do not have the same denominator are called unlike fractions.
For example: 2/9, 5/9, 7/9 are like fractions where as 2/5, 3/7,6/11 are unlike fractions.

JOSEPH LOUIS LAGRANGE

                                    
                                     
lagrange
Lagrange
          JOSEPH LOUIS LAGRANGE was born on Italy of Franchise ancestry on January 25, 1736. He became captivated by mathematics at an early age when he read on essay on Halley on Newton's calculus. At the age of 19, he became professor of mathematics at the Royal Artillery School in Turin. Lagrange made significant contributions to many branches of mathematics and physics, among them the theory of numbers, the theory of equations, ordinary and partial differential equations, the calculus of variations, analytic geometry, fluid dynamics, and celestial mechanics. His methods for solving third and fourth degree polynomial equations by radicals laid the groundwork for the group-theoretic approach to solving polynomials taken by Galois. Lagrange was a very careful writer with a clear and elegant style.
                               
lagrange stamp
Lagrange stamp




                "Lagrange is the Lofty Pyramid of the Mathematical Sciences."
                                                                          NAPOLEON BONAPARTE 

At the age of 40, Lagrange was appointed Head of Berlin Academy, succeeding Euler
 In offering this appointment, Frederick The Great proclaimed that the  "greatest king in Europe" ought to have the "greatest mathematician in Europe" at his court. In 1787, Lagrange was invited to Paris by Louis XVI and became a good friend of the king and his wife, Marie Antoinette. In 1793, Lagrange headed a commission, which included Laplace and Lavoisier, to devise a new system of weights and measures. Out of this came the metric system. Late in his life he made a count by Napoleon. Lagrange died on April 10, 1813.

WHAT ARE INTEGERS AND PROPERTIES ?


integers
Integers


As we are now we'll known with natural numbers & for each natural number a (say), there is a number -a such that a+(-a) = 0. This, 1+(-1)= 0, 2+(-2)= 0, ......
The numbers ....,-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, ... are called integers. 1, 2, 3, 4, ... are called position integers, while -1, -2, -3, -4, -5, ... are called negative integers. The integer 0(zero) is neither positive nore negative.

The set of integers is denoted by Z or I. It is clear that all the natural numbers and all the whole numbers are integers but all the integers are neither natural nor whole numbers.

Some properties of integers:

  • Every integer is greater than every Negative integers
  • Zero is greater than every negative integer and is less than every positive integer.
  • Every negative integer is smaller than every positive integer.
  • Given two positive integers, the one with the bigger absolute value is greater.
  • Given two negative integers, the one with smaller absolute value is greater.

Absolute value: The absolute value of an integer is the whole number obtained by disregarding its sign.  The absolute value of an integer a is denoted by |a|
 Thus |-51| = (absolute value of -51) = 51,
          |0|= (absolute value of 0) = 0,a|.
          |-2| = (absolute value of -2) = 2,
          |43| = (absolute value of 43) =43.

WHAT ARE WHOLE NUMBERS ?

 
whole numbers
Whole numbers


The natural numbers along with 0 (zero) form a collection of whole numbers.
 Whole numbers = 0, 1, 2, 3, 4, 5, 6, .....
A whole number is either zero or a natural number.

Properties of whole numbers:
  • The first and smallest whole number is 0 (zero).
  • The last and the greatest whole number can not be obtained, like natural numbers, whole number are also infinite.
  • The difference between two consecutive whole numbers is 1 (one).
  • By adding 1 to any whole number, its next whole number is obtained.
  • All natural numbers are whole numbers, but all whole numbers are not natural numbers.

AUGUSTIN LOUIS CAUCHY


cauchy
cauchy
AUGUSTIN LOUIS CAUCHY was born on August 21, 1789, in Paris, the oldest of six children. By the time he was 11, both Laplace and Lagrange had recognised Cauchy's extraordinary talent for mathematics. In school he won prizes for Greek, Latin and the humanities. At the age of 21, he was given a comemission in Napoleon's army as a civil engineer. For the next few years, Cauchy attend to his engineering duties while carrying out brilliant mathematical research on the side.

You saw that little young man? Well! He will supplant all of us in so far as we are mathematicians.     
                                                                      SPOKEN BY LAGRANGE TO LAPLACE 
                                                                      ABOUT THE 11-YEAR-OLD CAUCHY

cauchy stamp
Cauchy stamp


       In 1815, at the age of 26, Cauchy was made Professor of Mathematics at the École polytechnique and was recognised as the leading mathematics in France. Cauchy and his contemporary Gauss were the last men to know the whole of mathematics as known at their time, and both made important contributions to nearly branch, both pure and applied, as well as to physics and astronomy.

     Cauchy introduced a new level of rigor into mathematical analysis. We owe our contemporary notions of limit and continuity to him. He gave the first proof of the Fundamental Theorem of Calculus. Cauchy was the founder of complex function theory and a pioneer in the theory of permutation groups and determinants. His total written output of mathematics fills 24 large volumes and is second only to that of Euler. He wrote over 500 research papers after the age of 50. Cauchy died at the age of 67 on May 23, 1857.
  



What are natural numbers ?



natural numbers
Natural numbers



Natural numbers are those numbers by which we can count things in nature like 4 pens, 5 girls, 8 trees, etc.
These are the numbers used for counting purpose.
  So, Natural numbers = 1, 2, 3, 4, 5, 6, 7, .....

Properties of natural numbers:

  • The first and smallest natural number is 1(one).
  • The last and greatest natural number can not be obtained, infact there are infinite numbers, like counting of stars in nature.
  • The difference between any two consecutive natural numbers is obtained.
  • The fractions like: 15/43, -34/51, etc. are not natural numbers.
  • The decimal numbers like: 4.4, 2.36, 8.94,etc. are not natural numbers.
  • 0 (zero) is not a natural numbers.
  • No natural number is negative i.e. none of -2, -53, -712, etc. is a natural number.
  • If n is any natural number, then 2n is even natural number and (2n-1) is an odd natural number.

NUMBERS IN INDIAN AND INTERNATIONAL SYSTEM

INTRODUCING UNIT, NUMBER, NUMERAL AND NUMERATION


number system

NUMBERS IN INDIAN AND INTERNATIONAL SYSTEM


In mathematics unit means a single thing.
For example, a pen, a book, a girl, etc.
A unit is the first and lowest natural number as a standard of measurement.
The number written before the name of a unit indicates how many times that unit is taken.
For example, 6kg means; the number 6 is written before the unit kg.
Unit kg (kilogram) is taken 6 times.

NUMERAL AND NUMERATION


A numeral is a symbol representing a given number and numeration represents that number in words.

 NumberNumeral Numeration
      2     2      two
      36     36    thirty-six 
       0     0      zero

HINDU-ARABIC (INDIAN) SYSTEM OF NUMERATION

The Indian system of numeration is in fact the decimal system that is in use all over the world. This system was developed by the ancient Hindu-Mathematicians in India and was carried to West by the Arabs. For this reason, it is called the Hindu-Arabic system of numeration.

In India system, which is also known as denary system, the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are used to write a numeral (number). Each of these ten symbols is called a digit.

In Indian system, also known as Hindu Arabic System and in International sysytem of numeration, to read and write large quantities with ease, the group's are made with certain periods as shown below.

 Indian system Figures International system Figures
 Unit 1 Ones 1
 Tens 10 Tens 10
 1 Hundred 100 1 Hundred 100
 1 Thousand 1000 1 Thousand 1000
 10 Thousands 10000 10 Thousands 10000 
 1 Lakh 100000 100 Thousands 100000
 10 Lakhs 1000000 1 Million 1000000
 1 Crore 10000000 10 Millions 10000000
 10 Crores 100000000 100 Millions 100000000
 1 Arab 1000000000 1 Billion 1000000000
 10 Arabs 10000000000 10 Billions 10000000000
 1 Kharab 100000000000 100 Billions 100000000000
 10 Kharabs 1000000000000 1 Trillion 1000000000000
 1 Neel 10000000000000 10 Trillions 10000000000000
 10 Neels 100000000000000 100 Trillions 100000000000000
 1 Padam 1000000000000000 1 Quadrillion 1000000000000000
 10 Padam 10000000000000000 10 Quadrillions 10000000000000000
 1 Shankh 100000000000000000 100 Quadrillions 100000000000000000
 10 Shankh 1000000000000000000 1 Quintillion 1000000000000000000
 Maha-Shankh 10000000000000000000 10 Quintillions 10000000000000000000




  
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