# MY DIGITAL NOTEBOOK

A Personal Blog On Mathematical Sciences and Technology

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# Tag Archives: Math

## Welcome 2012 – The National Mathematical Year in India

Wednesday, December 28th, 2011 08:23 / 18 Comments

Srinivasa Ramanujan (Photo credit: Wikipedia)

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.

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## Interesting and Must Read Papers and Articles in Mathematics

Tuesday, December 27th, 2011 09:10 / 2 Comments

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. ]

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## A Trip to Mathematics: Part-I Logic

Tuesday, October 25th, 2011 13:34 / 5 Comments

# 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.

For example:

1. Math is a tough subject.
2. English is not a tough subject.
3. Math and English both are tough subjects.
4. Either Math or English is tough subject.
5. If Math is a tough subject, then English is also a tough subject.
6. Math is a tough subject, 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., . The five words

• not
• and
• or
• if …. then
• if and only if

or their combinations are called ‘connectives‘. The sentences (all but first) are called composite sentences, i.e., a ) 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.

# Connectives

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 $x=y$ (read ‘x is equal to y’) is negated as $x \ne y$ (read ‘x is not equal to y‘). Similarly, $x \notin A$ (read ‘x does not belong to set A‘) is the negation of $x \in A$ (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, $\not$ or ¬ (read as ‘not’) for negation. For example if ‘p’ stands for the statement “Terence Tao is a professor” then $\not p$ [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 $\wedge$, read as ‘and‘. If p and q are statements then their conjunction is denoted by $p \wedge q$ 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 $\vee$ read ‘or’. If p and q are two statements, their disjunction is represented by $p \vee q$ 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 $p \Rightarrow q$ 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:

1. If p then q
2. p implies q
3. q follows from p
4. q is a logical consequence of p
5. p (is true) only if q (is true)
6. p is a sufficient condition for q
7. 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 ” $p\iff q$ “. $\iff$ represents double implication and read as ‘if and only if’.

In the statement $p \iff q$, the implication $p \Rightarrow q$ is called direct implication and the implication $q \Rightarrow p$ 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 $p \Rightarrow q$ 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 $p \Rightarrow p$. 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 $p \Rightarrow q$ is true but its converse is false ($q \not \Rightarrow p$), 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.

# Logical Approach

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.

# Inductive Approach

In inductive approach, we make a few observations and generalise. Exceptions are generally not counted in inductive approach.

# Deductive 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:   $p_1$         Every man is mortal.
$p_2$                              Ram is a man.
———————————————————————————-
Conclusion:                   $q$ Ram is mortal.

Symbolically, let us denote the premises of an argument by $p_1, p_2, \ldots , p_n$ and its conclusion by $q$. Then the argument is the statement $(p_1 \wedge p_2 \wedge \ldots \wedge p_n) \Rightarrow q$. If this implication is true, the argument is valid otherwise it is invalid.

To be continued……

###### Suggested Readings:

• Basic logic – connectives – NOT (gowers.wordpress.com)
• Welcome to the Cambridge Mathematical Tripos (gowers.wordpress.com)
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## 381654729 : An Interesting Number Happened To Me Today

Saturday, September 17th, 2011 18:21 / 5 Comments

Image via Wikipedia

You might be thinking why am I writing about an individual number? Actually, in previous year annual exams, my registration number was 381654729. Which is just an ‘ordinary’ 9-digit long number. I never cared about it- and forgot it after exam results were announced. But today morning, when I opened “Mathematics Today” magazine’s October 2010, page 8; I was brilliantly shocked. 381654729 is a nine digit number with each of the digits from 1 to 9 appearing once. The whole number is divisible by 9. If you remove the right-most digit, the remaining eight-digit number is divisible by 8. Again removing the next-right-most digit leaves a seven-digit number that is divisible by 7. Similarly, removing next-rightmost digit leaves a six-digit number that is divisible by 6. This property continues all the way down to one digit.
Further research on this number provided a term for this number as Poly-divisible Number.
And I also noticed that a similar problem has been asked in U S A Mathematical Talent Search  competition. See the first question in the doc below:

To view this document in appropriate size click on View tab of the doc.

After this beautiful incident, I would like to quote a statement here:

Mathematical Wonders happen with Mathematicians.

Numbers always chase me.

###### Related articles

• Two Interesting Math Problems (wpgaurav.wordpress.com)
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## Social Networks for Math Majors

Tuesday, September 13th, 2011 11:44 / 9 Comments

Math or Mathematics is not as difficult as it is thought to be. Mathematical Patterns, Structures, Geometry and its use in everyday life make it beautiful. ‘Math majors’ term generally include Math students, Math professors and researchers or Mathematicians. Internet has always been a tonic for learners and whole internet is supposed to be a social network, in which one shares and others read, one asks & others answer. There are thousands of social networks (and growing) where you enjoy your days, share fun etc. However there are only a few social (mathematical) networks which are completely focused on math and related sciences. But these are brilliantly good enough to demonstrate the wisdom of mathematicians. I have tried to list my favorite social networking websites on mathematics. Please have a read and give feedback in form of comments
Click On Images To Visit Corresponding Websites.

# Math.Stack Exchange

Mathematics StackExchange Website

Mathematics Stack exchange is a website dedicated to all types of mathematical discussions. You can ask questions, give answers, comment on questions and vote for it. Registration is very easy and takes seconds. Depending on your work, you are given ‘reputations’. Depending upon some special works, you are also given some privileges.

This is a free, community driven Q&A for people studying math at any level and professionals in related fields. It is a part of the Stack Exchange network of Q&A websites, and it was created through the open democratic process defined at Stack Exchange Area 51. (more…)

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