Histograms

Histogram: a graphical display of data using bars of different heights.

It is similar to a Bar Chart, but a histogram groups numbers into ranges .

The height of each bar shows how many fall into each range.

And you decide what ranges to use!

DIFFERENCE BETWEEN BAR GRAPH AND HISTOGRAM:

BAR GRAPH

Bar Graphs

A Bar Graph (also called Bar Chart) is a graphical display of data using bars of different heights.

Imagine you just did a survey of your friends to find which kind of movie they liked best:

Table: Favorite Type of Movie
Comedy	Action	Romance	Drama	SciFi
4	5	6	1	4

We can show that on a bar graph like this:

                It is a really good way to show relative sizes: we can see which types of movie are most liked, and which are least liked, at a glance.

               We can use bar graphs to show the relative sizes of many things, such as what type of car people have, how many customers a shop has on different days and so on.

DATA

What is Data?

Data is a collection of facts, such as numbers, words, measurements, observations or even just descriptions of things.

Qualitative vs Quantitative

Data can be qualitative or quantitative.

• Qualitative data is descriptive information (it describes something)
• Quantitative data is numerical information (numbers)

Quantitative data can be Discrete or Continuous:

• Discrete data can only take certain values (like whole numbers)
• Continuous data can take any value (within a range)

More Examples

Qualitative:

• Your friends' favorite holiday destination
• The most common given names in your town
• How people describe the smell of a new perfume

Quantitative:

• Height (Continuous)
• Weight (Continuous)
• Petals on a flower (Discrete)
• Customers in a shop (Discrete)

Surds

When we can't simplify a number to remove a square root (or cube root etc) then it is a surd.

Example: √2 (square root of 2) can't be simplified further so it is a surd

Example: √4 (square root of 4) can be simplified (to 2), so it is not a surd!

Have a look at some more examples:

Number	Simplified	As a Decimal	Surd or not?
√2	√2	1.4142135...(etc)	Surd
√3	√3	1.7320508...(etc)	Surd
√4	2	2	Not a surd
√¼	½	0.5	Not a surd
3√11	3√11	2.2239800...(etc)	Surd
3√27	3	3	Not a surd
5√3	5√3	1.2457309...(etc)	Surd

The surds have a decimal which goes on forever without repeating, and are Irrational Numbers.

ALGEBRA - EQUATION AND FORMULA

Equations and Formulas

What is an Equation?

An equation says that two things are equal. It will have an equals sign "=" like this:

x

+

2

=

6

That equations says: what is on the left (x + 2) is equal to what is on the right (6)

So an equation is like a statement "this equals that"

(Note: this equation has the solution x=4)

What is a Formula?

A formula is a fact or rule that uses mathematical symbols.

It will usually have:

• an equals sign (=)
• two or more variables (x, y, etc) that stand in for values we don't know yet

                 WILLIAM  TIMOTHY  GOWERS

                              Sir William Timothy Gowers, FRS (born 20 November 1963) is a British mathematician. He is a Royal Society Research Professor at the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge, where he also holds the Rouse Ball chair, and is a Fellow of Trinity College, Cambridge. In 1998, he received the Fields Medal for research connecting the fields of functional analysis and combinatorics.
  
                             Gowers has written several works popularising mathematics, including
 Mathematics: A Very Short Introduction (2002), which describes modern mathematical research for the general reader. He was consulted about the 2005 film Proof, starring Gwyneth Paltrow and Anthony Hopkins. He edited The Princeton Companion to Mathematics (2008), which traces the development of various branches and concepts of modern mathematics. For his work on this book, he won the 2011 Euler Book Prize of the Mathematical Association of America.

             MICHAEL  ATIYAH  CONTRIBUTION  TO

                               MATHEMATICS

           “One of the world's greatest living mathematicians, Sir Michael Atiyah has made fundamental contributions to many areas of mathematics, but especially to topology, geometry and analysis. From his first major contribution—topological K-theory—to his more recent work on quantum field theory, Sir Michael has been influential in the development of new theoretical tools and has supplied far-reaching insights. He is a notable collaborator, with his name linked with other outstanding mathematicians through their joint research. A superb lecturer, he possesses the ability to explain sophisticated mathematics in a simple geometric way.”

From the Abel Prize Foundation:

“The second Abel Prize has been awarded jointly to Michael Francis Atiyah and Isadore M. Singer. The Atiyah-Singer index theorem is one of the great landmarks of twentieth-century mathematics . . . . the culmination and crowning achievement of a more than one-hundred-year-old evolution of ideas, from Stokes’s theorem, which students learn in calculus classes, to sophisticated modern theories like Hodge’s theory of harmonic integrals and Hirzebruch’s signature theorem. The problem solved by the Atiyah-Singer theorem is truly ubiquitous. In the forty years since its discovery, the theorem has had innumerable applications, first in mathematics and then, beginning in the late 1970s, in theoretical physics: gauge theory, instantons, monopoles, string theory, the theory of anomalies, etc.”

               TERENCE TAO- MATHEMATICIAN

WORKS IN MATHEMATICS:

                                Terence Tao has done work in many different fields of mathematics (although as you can assume from Paul's focus on Tao's number theory work, arguably his most profound contributions have been in that field). Terence Tao specializes in partial differential equations, analytic number theory, harmonic analysis, combinations, matrix theory and ergodic theory. The actual contributions he has made are mainly in number theory, like the Green-Tao theorem, but he has made many other advances in many of the aforementioned fields, they are just not as profound as the Green-Tao theorem and some of his other working in number theory. Terence Tao is a mathematical genius by all definitions (also a genius in most other fields of academia), he has mastered most of the concepts in mathematics and has done research for many years in an attempt to better our understanding of the world through mathematics. The only way that I can truly stress the influence he has had in mathematics, without you having to understand any of the mathematics is to say that he won the Fields Medal, which is considered the "Nobel Prize of Mathematics" and is only awarded every four years.

                              C.S. SESHADRI

                       C.S. Seshadri FRS (born 29 February 1932) is an eminent Indian mathematician. He is the founder and Director-Emeritus of the Chennai Mathematical Institute, and is known for his work in algebraic geometry. The Seshadri constant is named after him.
He is a recipient of the Padma Bhushan in 2009,the third highest civilian honor in the country. Research work:

                Seshadri's main work is in algebraic geometry. His work with M S Narasimhan on unitary vector bundles and the Narasimhan–Seshadri theorem has influenced the field. His work on Geometric Invariant Theory and on Schubert varieties, in particular his introduction of standard monomial theory, is widely recognized. Seshadri's contributions include the creation of the Chennai Mathematical Institute, an institute for the study of mathematics in India.

                   NARENDRA  KARMARKAR

Karmarkar’s Famous algorithm

                           Karmarkar’s algorithm solves the various linear Programming problems in the polynomial time. These problems are mostly represented by a “n” variables and a “m” constraints. The previous used method for solving these type of problems consisted a lot of problem representation by a “x” sided solid with a “y” vertices, where all the solution was then approached by traversing it from the vertex tovertex. Karmarkar’s novel method approaches all the solution by method of cutting through the all the above solid in all its traversal.        

                      Consequently, the complex optimization problems are also solved much faster using the method of Karmarkar algorithm. A practical example of this type of efficiency is the solution to a very complex problem in the communications network optimization where all the solution time taken was reduced from the weeks to some days. His algorithm thus enables the faster business and various policy decisions. Karmarkar’s algorithm has also stimulated all the development of the several other used interior point methods, some of which are now used in the current codes for solving the linear programs.

                          HARISH  CHANDRA

CONTRIBUTION TO MATHEMATICS:

                The word mathematics is like a dangerous monster for some people. Many of the students who consider mathematics as their enemy would have thought that the subject mathematics should never have existed. But in this world of billions of population, there are many who not only love mathematics but take it up as their career.

Harish Chandra was one among such people in the olden times who was a well-known mathematician. Not only did he like the subject mathematics, but also made significant contributions to the world of mathematics which is undoubtedly praiseworthy. Harish Chandra was born on 11th of October, 1923 in Kanpur. He belonged to an upper middle class family which was a benefit to the educational career of Harish Chandra.

    Harish Chandra researched on “Semi Simple Lie Groups”. This was indeed his best research. He explained about the lie algebra where the ideal of the number were 0 and itself. Other achievements of Harish Chandra include Weyl’s character formula analogue, Plancherel measure for semi simple groups, philosophy of cusp forms, etc. he also worked at Neumann Professor in 1968 at the Institute of Advanced Study.

               Harish Chandra was a great mathematician of the twentieth century. His achievements and contributions were praiseworthy. He was honoured with AMS Cole Prize in 1954 for his outstanding work ‘Representations of Semi Simple Lie algebras and groups’. He was the Fellow of the Royal Society. He was given the Ramanujan Medal in 1974 for his wonderful works in mathematics. Not only in India, but Harish Chandra’s achievements were honoured outside India as well. He was given an honorary degree in the Yale University. Harish Chandra Research Institute (HRI) was started by the Government of India in the honour of his contributions in the field of mathematics.

                       C.RADHAKRISHNA RAO

CONTRIBUTIONS TO STATISTICAL THEORY AND APPLICATIONS

                                       C. Radhakrishna Rao is among the world leaders in statistical science over the last six decades. His research, scholarship, and professional services have had a profound influence on theory and applications of statistics.

Technical terms such as, Cramer-Rao inequality, Rao-Blackwellization, Rao’s Score Test, Fisher-Rao and Rao Theorems on second order efficiency of an estimator, Rao
metric and distance, Analysis of Dispersion (MANOVA) and Canonical Variate analysis and G-inverse of matrices appear in all standard books on statistics. Cramer-Rao Bound and Rao-Blackwellization are the most frequently quoted key words in statistical and engineering literature. Special uses of Cramer-Rao Bound under the technical term, Quantum Cramer- Rao Bound have appeared in Quantum Physics. Rao-Blackwellization has found applications in adaptive sampling, particle filtering in high-dimensional state spaces, dynamic Bayesian networks etc. These results have led to contributions of strategic significance to signal detection, tracking of non-friendly planes and recognition of objects by shape.

Rao has made some significant contributions to combinatorial mathematics for use in design of experiments, the most important of which is Orthogonal arrays (OA).The basic paper on the subject appeared in Proc. Edinburgh Math. Soc. (the referee of the paper reported that it is a fresh and original piece of work). The Japanese Quality Control Expert, G.Taguchi made extensive use of OA’s (described by Forbes Magazine as “new mantra” for industries), in industrial experimentation.
Rao defined a generalized inverse (g-inverse) of a matrix (singular or rectangular) and demonstrated its usefulness in the study of linear models and singular multivariate normal distributions.

He is the author of 14 books and about 350 research papers. Three of his books have been translated into several European and Chinese and Japanese languages

                     SRINIVASA  RAMANUJAN

Contribution to Mathematics

            His chief contribution in mathematics lies mainly in analysis, game theory and infinite series. He made in depth analysis in order to solve various mathematical problems by bringing to light new and novel ideas that gave impetus to progress of game theory. Such was his mathematical genius that he discovered his own theorems. It was because of his keen insight and natural intelligence that he came up with infinite series for π

                     

This series made up the basis of certain algorithms that are used today. One such remarkable instance is when he solved the bivariate problem of his roommate at spur of moment with a novel answer that solved the whole class of problems through continued fraction. Besides that he also led to draw some formerly unknown identities such as by linking coefficients of and providing identities for hyperbolic secant.

      D.R.KAPRELAR DISCOVERIES

Kaprekar constant

In 1949, Kaprekar discovered an interesting property of the number 6174, which was subsequently named the Kaprekar constant. He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical. Thus, starting with 1234, we have:

4321 − 1234 = 3087, then

8730 − 0378 = 8352, and

8532 − 2358 = 6174.

Repeating from this point onward leaves the same number (7641 − 1467 = 6174). In general, when the operation converges it does so in at most seven iterations.

A similar constant for 3 digits is 495. However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for other digit lengths or bases other than 10, the Kaprekar's routine algorithm described above may in general terminate in multiple different constants or repeated cycles, depending on the starting value.

Kaprekar number

Another class of numbers Kaprekar described are the Kaprekar numbers. A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number (e.g. 45, since 45²=2025, and 20+25=45, also 9, 55, 99 etc.) However, note the restriction that the two numbers are positive; for example, 100 is not a Kaprekar number even though 100²=10000, and 100+00 = 100. This operation, of taking the rightmost digits of a square, and adding it to the integer formed by the leftmost digits, is known as the Kaprekar operation.

Some examples of Kaprekar numbers in base 10, besides the numbers 9, 99, 999, …, are (sequence A006886 in the OEIS):

Number	Square	Decomposition
703	703² = 494209	494+209 = 703
2728	2728² = 7441984	744+1984 = 2728
5292	5292² = 28005264	28+005264 = 5292

       P.C. MAHALANOBIS CONTRIBUTIONS

Mahalanobis Distance

                         Mahalanobis Distance is one of the most widely used metric to find how much a point diverges from a distribution, based on measurements in multiple dimensions. It is widely used in the field of cluster analysis and classification. It was first proposed by Mahalanobis in 1930 in context of his study on racial likeness. From a chance meeting with Nelson Annandale, then the director of the Zoological Survey of India, at the 1920 Nagpur session of the Indian Science Congress led to Annandale asking him to analyse anthropometric measurements of Anglo-Indians in Calcutta. Mahalanobis had been influenced by the anthropometric studies published in the journal Biometrika and he chose to ask the questions on what factors influence the formation of European and Indian marriages. He wanted to examine if the Indian side came from any specific castes. He used the data collected by Annandale and the caste-specific measurements made by Herbert Risley to come up with the conclusion that the sample represented a mix of Europeans mainly with people from Bengal and Punjab but not with those from the Northwest Frontier Provinces or from Chhota Nagpur. He also concluded that the intermixture more frequently involved the higher castes than the lower ones. This analysis was described by his first scientific paper in 1922. During the course of these studies he found a way of comparing and grouping populations using a multivariate distance measure. This measure, denoted "D²" and now eponymously named Mahalanobis distance, is independent of measurement scale. Mahalanobis also took an interest in physical anthropology and in the accurate measurement of skull measurements for which he developed an instrument that he called the "profiloscope".

        INTERESTING  FACTS  OF  MATHEMATICS-2

1.   The ancient Egyptians did not use fractions.

2.   If you add up all the roulette numbers, you will get a mystical number of 666.

3.   Quadratic equations appeared in India 15 centuries ago.

4.   Euclid left many works on mathematics and geometry, which we still use. Interestingly, no information about Euclid himself was found.

      5. Rene Descartes introduced the concepts of real and imaginary numbers.

       6.Interestingly, the great emperor Napoleon Bonaparte left some mathematical works after his death.

        7.The Indian scholar Budhayana, who lived in the 6th century, is considered the first to use the Pi number.

        8.Johannes Widmann first recorded the classic signs of addition and subtraction. It happened about 500 years ago.

       INTERESTING FUN FACTS OF MATHEMATICS

1. You can cut a cake into 8 pieces by using only 3 cuts.

You just need to make two cuts in a vertical plane and one in a horizontal plane. 

 2. The numbers on opposite sides of a dice always add up to seven

 On a dice the numbers 1,2 and 3 all share a vertex.  If these three numbers run clockwise round this vertex then the dice is called left-handed and if the three numbers run anti-clockwise round the vertex, then it is a right-handed dice.   Chinese dice are normally left-handed and Western dice are normally right-handed.

3. William Shanks calculated pi to 707 decimal places but made a mistake on the 528^th digit.

Amateur mathematicia William Shanks (1812-1882) spent a good part of his life calculating mathematical constants by hand.  Shanks never found out about his mistake as it wasn’t revealed until after his death.

4. The word googol was made up by a 9-year old boy

In the 1930s an American Mathematician named Edward Kasner asked his nine-year old nephew Milton Sirotta to make up a word for him to use. Milton made up the world ‘googol’ which Edward Kasner later used to describe the number  .  The search engine Google was later named after the ‘googol’ meaning that Milton Sirotta had unwittingly helped name one of the world’s most famous companies.

5. The word “hundred” comes from the old Norse term, “hundrath”, which actually means 120 and not 100. In a room of 23 people there’s a 50% chance that two people have the same birthday.

                    DIOPHANTUS

                Diophantus was an Alexandrian Greek mathematician, born somewhere between 200 and 214 BC. Alexandria was the center of Greek culture and knowledge and Diophantus belonged to the ‘Silver Age’ of Alexandria. His life story is not known in detail however we do have some dates acquired from a mathematical puzzle known as ‘Diophantus’s Riddle’. 

Contribution to Mathematics

           ‘Arithmetika’ a major work of Diophantus, is considered to be the most noticeable and influential work done on algebra in Greek history. His style was very different; he never used general methods in working out a problem. A method used for one problem could not be used to solve even another very similar problem. Diophantus wrote many books but unfortunately only a few lasted. He did a lot of work in algebra, solving equations in terms of integers. Some of his equations resulted in more than one answer possibility. There are now called ‘Diophantine’ or ‘Indeterminate’. It was none other than Diophantus who started the use of a symbol to specify the unidentified quantities in his equations. His style of algebra is known as the ‘syncopated’ style of algebraic writing, in which he represented polynomials as one unknown.