The reason for writing such a strange blog is the sudden realisation that sometimes we don't give enough importance to the things we have around us. I don't want to get too philosophical but all this started when I was looking for a book which I treasured a lot in my schooldays. My father is a great admirer of Mathematics and used to get me books on Mathematics and mathematical puzzles. I used to like them. One of them which was my favourite at that time was by Ya. I. Perelman. It was titled "Mathematics can be Fun". There was another book by him: "Physics can be Fun" which also I liked. Now I don't know where they are. I just searched Google for the book. Got an Amazon link. "Mathematics can be Fun" is out of print.
I spent a lot of time with the book. I specially remember and liked the discussion on Diophantine equations and pythagorean triples. I was fascinated by the fact that, for three integers a,b and c, whenever a² + b² = c², there exists integers m and n such that
a = m² - n²,
b = 2mn,
c = m² + n².
The fact that one can logically argue to get these forms really fascinated me in my high school. I also liked the section on maxima and minima done with just some basic knowledge in high school algebra.
Anyway, I really liked the book and it's really sad that Mir had to stop publishing.
14 comments:
In fact its sadder ... the Mir is now out of buisness.
I remeber seeing a book 'Manoranjan Bhautiki'(amongst many other excellent ones by Mir). Does that translate into 'Physics can be fun'?
They also used to have a series called 'Little Mathematics' Library', some of which my school library (in 10th, that is when we actually had something of a library) had. That is by far the best (motivational) introductions to various 'advanced' topics I can recall. It was not crisply mathematical though - infact it didn't had had any proofs.
Yes, I too remember. This "Manoranjan Bhautiki" is in deed the same book as "Physics can be fun". There were two volumes of this book. The author was same - Ya. I. Perelman. I remeber learnig Fermat's law in optics from that book... those lovely days!
Sad that we do not get those books. Near my home, a van used to come twice/thrice a year to sell these books. They also had other Russian books for kids translated into Hindi, with many pictures and paintings. What a wonderful image of the world I had that time... Alas! I was so wrong!
@Vaibhav
Yeah! I also had those books in the Little Mathematics Library and there were some books like "Something called Nothing" and so on. They were really good books.
@Amit
Bahut nikle mere armaan lekin phir bhi kam nikle
This may be naive but if such an m and n do exist, what is the point? Why is their existence important?
p.s. great start! you have just got yourself one loyal reader.
@Samudrika
Thank you. I just changed the look a bit too... Now for your question:
The existence of m and n lets you find out all the solutions. Whatever you put in for m and n, it will you give a solution of a^2 + b^2 = c^2. For example, if you take m to be 2 and n to be 1, you get a = 3, b = 4 and c = 5. And so on.
It is easy to see that (m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2. So the above recipe will give solutions, no doubt. What is striking is that it gives ALL the solutions.
@samudrika,vivek
Another key point is that, not merely such m, n exist - but that they are integers.
And as Vivek said, any integers for m,n yield a solution too. Thus, for example, we can immediately answer if an integer can satisfy for a - it can iff there are m, n such that a=m^2 - n^2 for some integers m and n :)
Thanks, Vaibhav! That's also an important point!
isn't it an identity? I mean if I could find a=m^2 + n^2, b and c values are fixed...so are we sure, that we can ALWAYS find m and n's?
I think find our m and n's from b would be the easiest...they will be ALL factors of b/2....a problem which shanta can easily solve.
Vivek, both these books are available with me, in both hindi and angrezi bhasha....If you have someone staying in UP near Kanpur, every year there is a grand book fair, in which many MIR books can be found...upto the exhaust their stock, of course
Well finding m and n might not be all that difficult. You know c = m^2 + n^2. Now question is which of a and b is 2mn? We know 2mn + m^2 + n^2 is a perfect square, while (m^2 - n^2) + (m^2 + n^2) = 2m^2 is not. So we know which is 2mn and which is m^2 - n^2. Suppose b = 2mn. Then b+c = m^2+n^2+2mn = (m+n)^2. Thus you get the value for m+n. You already know the value for mn. Now you can solve for m and n in thousands of ways!
Yup Vivek, it is really sad that Mir has stopped publishing. What makes it worse for me is that I didn't know about it till I went to HRI for my Ph.D.. Once I had gone to the Ald University and there I spotted a van of books (I think they come there every year) and that's when I came to know about Mir publication. Too let for me to buy these great books. I should perhaps try to borrow these books from you and hope that you would forget about them. :-).
Hmm.. i had a topological childhood.
Smashing doughnuts into zero dimensional manifolds kinda things.
It was my favorite book, many years ago, back in Siberia.
Well, here is some good news for you - http://www.archive.org/details/physicsforentert035428mbp
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Another fan of Y. Perelman.
I have the book Mathematics can be fun by Y Perleman if anyone needs it I can scan it
Contact me at termvader@hotmail.com
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