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I was very pleased on reading this news that Government of India has decided to celebrate the upcoming year 2012 as the National Mathematical Year. This is 125th birth anniversary of math-wizard Srinivasa Ramanujan (1887-1920). He is one of the greatest mathematicians India ever produced. Well this is ‘not’ the main reason for appointing 2012 as National Mathematical Year as it is only a tribute to him. Main reason is the emptiness of mathematical awareness in Indian Students. First of all there are only a few graduating with Mathematics and second, many not choosing mathematics as a primary subject at primary levels. As mathematics is not a very earning stream, most students want to go for professional courses such as Engineering, Medicine, Business and Management. Remaining graduates who enjoy science, skip through either physical or chemical sciences. Engineering craze has developed the field of Computer Science but not so much in theoretical Computer Science, which is one of the most recommended branches in mathematics. Statistics and Combinatorics are almost ‘died’ in many of Indian Universities and Colleges. No one wants to deal with those brain cracking math-problems: neither students nor professors. Institutes where mathematics is being taught are struggling with the lack of talented lecturers. Talented mathematicians don’t want to teach here since they aren’t getting much money and ordinary lecturers can’t do much more. India is almost ‘zero’ in Mathematics and some people including critics still roar that we discovered ‘zero’, ‘pi’ and we had Ramanujan.
Mathematics is beautiful and there is no place of ugly mathematics in this world. Mathematics is originated from creativity and it develops with research papers. Research Papers aren’t only very detailed and tough to understand for general student, but also interesting. Here, I have collected the list of some excellent articles and research papers (belong mainly to Math) which I have read and are easily available online. The main source of this list is ArXiv.org and you may find several research papers on ArXiv by visiting http://arxiv.org/.
If you know any other paper/article which you find extremely interesting and that is not listed here, then please do comment mentioning the article name and URL. Papers/articles are cited as paper-title first, then http url and at last author-name.
[It is better adviced to open these links to a new tab/window for smooth reading. ]
Applied mathematics is one which is used in day-to-day life, in solving tensions (problems) or in business purposes. Let me write an example:
George had some money. He gave 14 Dollars to Matthew. Now he has 27 dollars. How much money had he?
If you are familiar with day to day calculations —you must say that George had 41 dollars, and since he had 41, gave 14 to Matthew saving 27 dollars. That’s right? Off course! This is a general(layman) approach. ‘How will we achieve it mathematically?’ —We shall restate the above problem as another statement (meaning the same):
some moneydollars. He gave 14 dollars to Matthew. Now he has 27 dollars. How much money he had?Find the value of .
This is equivalent to the problem asked above. I have just replaced ‘some money’ by ‘x dollars’. As ‘some’ senses as unknown quantity— does the same. Now all we need to get the value of x.
When solving for , we should have a plan like this:
|He gave to Matthew||14 dollars|
|Now he must have||dollars|
But problem says that he has 27 dollars left. This implies that dollars are equal to 27 dollars.
contains an alphabet x which we assumed to be unknown—can have any certain value. Statements (like ) containing unknown quantities and an equality are called Equations. The unknown quantities used in equations are called variables, usually represented by bottom letters in english alphabet (e.g.,). Top letters of alphabet (..) are usually used to represent constants (one whose value is known, but not shown).
Now let we concentrate on the problem again. We have the equation x-14=27.
Now adding 14 to both sides of the equal sign:
So, is 41. This means George had 41 dollars. And this answer is equal to the answer we found practically. Solving problems practically are not always possible, specially when complicated problems encountered —we use theory of equations. To solve equations, you need to know only four basic operations viz., Addition, Subtraction, Multiplication and Division; and also about the properties of equality sign.
We could also deal above problem as this way:
-14 transfers to another side, which makes the change in sign of the value, i.e., +14.
When we transport a number from left side to right of the equal sign, the sign of the number changes and vice-versa. As here -14 converts into +14; +18 converts into -18 in example below:
Please note, any number not having a sign before its value is deemed to be positive—e.g., 179 and +179 are the same, in theory of equations.
Before we proceed, why not take another example?
Marry had seven sheep. Marry’s uncle gifted her some more sheep. She has eighteen sheep now. How many sheep did her uncle gift?
First of all, how would you state it as an equation?
or, (just to illustrate that 7=+7)
So, Marry’s uncle gifted her 9 sheep. ///
Now tackle this problem,
Monty had some cricket balls. Graham had double number of balls as compared to Monty. Adam had also 6 cricket balls. They all collected their balls and found that total number of cricket balls was 27. How many balls had Monty and Graham?
As usual our first step to solve this problem must be to restate it as an equation. We do it like this:
Monty had (let) x balls.
Then Graham must had balls.
Adam had 6 balls.
The total sum=
But that is 27 according to our question.
Here multiplication sign converts into division sign, when transferred.
Since , we can say that Monty had 7 balls (instead of x balls) and Graham had 14 (instead of ).
Types of Equations
They are many types of algebraic equations (we suffix ‘Algebraic’ because it includes variables which are part of algebra) depending on their properties. In common we classify them into two main parts:
1. Equations with one variable (univariable algebraic equations, or just Univariables)
2. Equations with more than one variables (multivariable algebraic equations, or just Multivariables)
Equations consisting of only one variable are called univariable equations.
All of the equations we solved above are univariables since they contain only one variable (x). Other examples are:
(e is a constant).
Univariables are further divided into many categories depending upon the degree of the variable. Some most common are:
- Linear Univariables: Equations having the maximum power (degree) of the variable 1.
is a general example of linear equations in one variable, where a, b and c are arbitrary constants.
- Quadratic Equations: Also known as Square Equations, are ones in which the maximum power of the variable is 2.
is a general example of quadratic equations, where a,b,c are constants.
- Cubic Equations: Equations of third degree (maximum power=3) are called Cubic.
A cubic equation is of type ; where a,b,c,d are constants.
- Quartic Equations: Equations of fourth degree are Quartic.
A quartic equation is of type .
Similarly, equation of an n-th degree can be defined if the variable of the equation has maximum power n.
Some equations have more than one variables, as etc. Such equations are termed as Multivariable Equations. Depending on the number of variables present in the equations, multivariable equations can be classified as:
1. Bi-variable Equations - Equations having exactly two variables are called bi-variables.
; ; , where k is constant; etc are equations with two variables.
Bivariable equations can also be divided into many categories, as same as univariables were.
A.Linear Bivariable Equations: Power of a variable or sum of powers of product of two variables does not exceed 1.
For example: is a linear but is not.
B. Second Order Bivariable Equations: Power of a variable or sum of powers of product of two variables does not exceed 2.
For example: , are of second order.
Similarly you can easily define n-th order Bivariable equations.
2. Tri-variable Equations: Equations having exactly three variables are called tri-variable equations.
; ; , where k is constant; etc are trivariables. (Further classification of Trivariables are not necessary, but I hope that you can divide them into more categories as we did above.)
Similary, you can easily define any n-variable equation as an equation in which the number of variables is n.
Out of these equations, we shall discuss only Linear Univariable Equations here (actually we are discussing them). ////
We have already discussed them above, for particular example. Here we’ll discuss them for general cases.
As told earlier, a general example of linear univariable equation is .
We can adjust it by transfering constants to one side and keeping variable to other.
this is the required solution.
Example: Solve .
About “A Trip To Mathematics”:
A Trip to Mathematics is an indefinitely long series, aimed on generally interested readers and other undergraduate students. This series will deal Basic Mathematics as well as Advanced Mathematics in very interactive manners. Each post of this series is kept small that reader be able to grasp concepts. Critics and suggestions are invited in form of comments.
What is Logic?
If mathematics is regarded as a language, then logic is its grammar.
In other words, logical precision has the same importance in mathematics as grammatical accuracy in a language. As linguistic grammar has sentences, statements— logic has them too. Let we discuss about Sentence &Statements, then we shall proceed to further logic .
Sentence & Statements
A sentence is a collection of some words, those together having some sense.
- Math is a tough subject.
- English is not a tough subject.
- Math and English both are tough subjects.
- Either Math or English is tough subject.
- If Math is a tough subject, then English is also a tough subject.
- Math is a tough subject, if and only if English is a tough subject.
Just have a quick look on above collections of words. Those are sentences, since they have some meaning too. First sentence is called Prime Sentence, i.e., sentence which either contains no connectives or, by choice, is regarded as “indivisible”. The five words
- if …. then
- if and only if
or their combinations are called ‘connectives‘. The sentences (all but first) are called composite sentences, i.e., a declarative sentence (statement) in which one or more connectives appear. Remember that there is no difference between a sentence and statement in general logic. In this series, sentences and statements would have the same meaning.
not: A sentence which is modified by the word “not” is called the negation of the original sentence.
For example: “English is not a tough subject” is the negation of “English is a tough subject“. Also, “3 is not a prime” is the negation of “3 is a prime“. Always note that negation doesn’t really mean the converse of a sentence. For example, you can not write “English is a simple subject” as the negation of “English is a tough subject“.
In mathematical writings, symbols are often used for conciseness. The negation of sentences/statements is expressed by putting a slash (/) over that symbol which incorporates the principal verb in the statement.
For example: The statement (read ‘x is equal to y’) is negated as (read ‘x is not equal to y‘). Similarly, (read ‘x does not belong to set A‘) is the negation of (read ‘x belongs to set A‘).
Statements are sometimes represented by symbols like p, q, r, s etc. With this notation there is a symbol, or ¬ (read as ‘not’) for negation. For example if ‘p’ stands for the statement “Terence Tao is a professor” then [or ¬p] is read as ‘not p’ and states for “Terence Tao is not a professor.” Sometimes ~p is also used for the negation of p.
and: The word “and” is used to join two sentences to form a composite sentence which is called the conjunction of the two sentences. For example, the sentence “I am writing, and my sister is reading” is the conjunction of the two sentences: “I am writing” and “My sister is reading“. In ordinary language (English), words like “but, while” are used as approximate synonyms for “and“, however in math, we shall ignore possible differences in shades of meaning which might accompany the use of one in the place of the other. This allows us to write “I am writing but my sister is reading” having the same mathematical meaning as above.
The standard notation for conjunction is , read as ‘and‘. If p and q are statements then their conjunction is denoted by and is read as ‘p and q’.
or: A sentence formed by connecting two sentences with the word “or” is called the disjunction of the two sentences. For example, “Justin Bieber is a celebrity, or Sachin Tendulkar is a footballer.” is a disjunction of “Justin Bieber is a celebrity” and “Sachin Tendulkar is a footballer“.
Sometimes we put the word ‘either‘ before the first statement to make the disjunction sound nice, but it is not necessary to do so, so far as a logician is concerned. The symbolic notation for disjunction is read ‘or’. If p and q are two statements, their disjunction is represented by and read as p or q.
if….then: From two sentences we may construct one of the from “If . . . . . then . . .“; which is called a conditional sentence. The sentence immediately following IF is the antecedent, and the sentence immediately following THEN is the consequent. For example, “If 5 <6, 6<7, then 5<7” is a conditional sentence whith “5<6, 6<7” as antecedent and “5<7” as consequent. If p and q are antecedent and consequent sentences respectively, then the conditional sentence can be written as:
“If p then q”.
This can be mathematically represented as and is read as “p implies q” and the statement sometimes is also called implication statement. Several other ways are available to paraphrase implication statements including:
- If p then q
- p implies q
- q follows from p
- q is a logical consequence of p
- p (is true) only if q (is true)
- p is a sufficient condition for q
- q is a necessary condition for p
If and Only If: The phrase “if and only if” (abbreviated as ‘iff‘) is used to obtain a bi-conditional sentence. For example, “A triangle is called a right angled triangle, if and only if one of its angles is 90°.” This sentence can be understood in either ways: “A triangle is called a right angled triangle if one of its angles is 90°” and “One of angles of a triangle is 90° if the triangle is right angled triangle.” This means that first prime sentence implies second prime sentence and second prime sentence implies first one. (This is why ‘iff’ is sometimes called double-implication.)
Another example is “A glass is half filled iff that glass is half empty.“
If p and q are two statements, then we regard the biconditional statement as “p if and only if q” or “p iff q” and mathematically represent by ” “. represents double implication and read as ‘if and only if’.
In the statement , the implication is called direct implication and the implication is called the converse implication of the statement.
Other terms in logic:
Stronger and Weaker Statements: A statement p is stronger than a statement q (or that q is weaker than p) if the implication statement is true.
Strictly Stronger and Strictly Weaker Statements: The word ‘stronger‘(or weaker) does not necessarily mean ‘strictly stronger‘ (or strictly weaker).
For example, every statement is stronger than itself, since . The apparent paradox here is purely linguistical. If we want to avoid it, we should replace the word stronger by the phrase ‘stronger than‘ or ‘possibly as strong as‘.
If is true but its converse is false (), then we say that p is strictly stronger than q (or that q is strictly weaker than p). For example it is easy to say that a given quadrilateral is a rhombus that to say it is a parallelogram. Another understandable example is that ” If a blog is hosted on WordPress.com, it is powered with WordPress software.” is true but ” If a blog is powered with WordPress software , it is hosted on WordPress.com” is not true.
What exactly is the difference between a mathematician, a physicist and a layman?
Let us suppose they all start measuring the angles of hundreds of triangles of various shapes, find the sum in each case and keep a record.
Suppose the layman finds that with one or two exceptions the sum in each case comes out to be 180 degrees. He will ignore the exceptions and state ‘The sum of the three angles in a triangle is 180 degrees.’
A physicist will be more cautious in dealing the exceptional cases. He will examine then more carefully. If he finds that the sum in them some where 179 degrees to 181 degrees, say, then if will attribute the deviation to experimental errors. He will state a law – ‘The sum of the three angles of any triangle is 180 degrees.’ He will then watch happily as the rest of the world puts his law to test and finds that it holds good in thousands of different cases, until somebody comes up with a triangle in which the law fails miserably. The physicist now has to withdraw his law altogether or else to replace it by some other law which holds good in all the cases tried. Even this new law may have to be modified at a later date. And this will continue without end.
A mathematician will be the fussiest of all. If there is even a single exception, he will refrain from saying anything. Even when millions of triangles are tried without a single exception, he will not state it as a theorem that the sum of the three angles in ‘any’ triangle is 180 degrees. The reason is that there are infinitely many different types of triangles. To generalise from a million to infinity is as baseless to a mathematician as to generalise from one to a million. He will at the most make a conjecture and say that there is a ‘strong evidence’ suggesting that the conjecture is true.
The approach taken by the layman or the physicist is known as the inductive approach whereas the mathematician’s approach is called the deductive approach.
In inductive approach, we make a few observations and generalise. Exceptions are generally not counted in inductive approach.
In this approach, we deduce from something which is already proven.
Axioms or Postulates
Sometimes, when deducting theorems or conclusion from another theorems, we reach at a stage where a certain statement cannot be proved from any ‘other’ proved statement and must be taken for granted to be true, then such a statement is called an axiom or a postulate.
Each branch of mathematics has its own populates or axioms. For example, the most fundamental axiom of geometry is that infinitely many lines can be drawn passing through a single point. The whole beautiful structure of geometry is based on five or six such axioms and every theorem in geometry can be ultimately Deducted from these axioms.
Argument, Premises and Conclusion
An argument is really speaking nothing more than an implication statement. Its hypothesis consists of the conjunction of several statements, called premises. In giving an argument, its premises are first listed (in any order), then connecting all, a conclusion is given. Example of an argument:
Premises: Every man is mortal.
Ram is a man.
Conclusion: Ram is mortal.
Symbolically, let us denote the premises of an argument by and its conclusion by . Then the argument is the statement . If this implication is true, the argument is valid otherwise it is invalid.
To be continued……
- Basic logic – connectives – NOT (gowers.wordpress.com)
- Welcome to the Cambridge Mathematical Tripos (gowers.wordpress.com)
National Programm on Technology Enhanced Learning (NPTEL) provides E-learning through online Web and Video courses in Engineering, Science and humanities streams. The mission of NPTEL is to enhance the quality of Engineering education in the country by providing free online courseware. All videos of NPTEL include the lectures of Indian professors in IITs and IISc and they can be found either on NPTEL homepage or at their YouTube page. On Youtube, the Video Courses are organised as PLAYLISTS under the following Categories:
- Core Sciences
- Civil Engineering
- Computer Science and Engineering
- Electrical Engineering
- Electronics and Communication Engineering
- Mechanical Engineering
And on their website, these are arranged in a order of Subjects.
Bellow are two Course Videos, as a demo, one on Semiconductors and other on Artificial Intelligence. It would be better to watch them, before you go for whole.
1. Simple Repitition Method
The classical method, very popular as in committing poems to memory by reading them over and over.
2. Full Concentration Method
Concentrate on the topic content while learning. Do not allow your mind to wander. Focus on names and numbers. Try deliberately to remember. Your approach should not be casual. Review soon after you learn, lest memory should fade away.
3. Visual Encoding Method
Translate information into visual formats like pictures, charts, diagrams, tables and graph.
4. Logical Organisation
Matter that is logically organised is retained better than the disjointed floating bits of information. Infuse meaning into what ever you learn. No “Nonsense Syllables”.
5. Mnemonics Method
Few good people try this method to learn some complicated series. There is no harm in using memory crutches like it, after grasping the spirit of the lesson. VIBGYOR is the most famous mnemonic that helps up to list the seven rainbow colors in their appropriate order. Once, I had remembered the name of the planets in the order of their distances from the sum by the acrostics “My very enlightened mother just served us noodles-in plates”, as it tells me the names of nine planets ( off course now they are eight, but I learned it earlier than the removal of pluto), in the order: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto.
6. Mock Teaching Method
If you find a particular portion of a chapter difficult to digest, imagine that you are teaching to a student sitting before you. Explain virtually the content of the lessen. The idea will get hammered into your mind.
7. Rhyming Method
Scholars say that verses are better than prose. Change any definition into verse and try to rhym them. Remember, I discuss just verses, and not poetry.
8. Chunking Method
Very popular method. You can divide a big list into a number of bits or chunks. For example, to remember a mobile number of ten digits, people usually split it to three or more chunks.
This involves the principal of association. First make a list of ten or twenty convenient pegs or key words that you can easily recall in the right sequence. For example, in alphabetical order, ant, butterfly, cat, dog, elephant, fox, and so on (Living beings are arranged). Cartoons also help in pegging.
This is the best method for that student, who want to learn very complicated topics. When you blog about any topic, you use all the nine methods listed above in your post. I got blogging, superior method to other nine.
- What is an easy way to remember the planets (wiki.answers.com)