math

A complex-looking simple problem

| Cosmoshivani

I would like to share the solution of a very simple problem. I tried solving it during an exam. It looked really complex (to me) at first but then I finally did it!

Problem: There is a square of side r. There are two circles drawn with radius equal to the side of the square. What is the probability of a particle to be found in the shaded (yellow) region?

Usually when we solve the probability problems (not that I am an expert), the formula that we use is, (number of desired items e.g. red balls) divided by (total number of items e.g. red and yellow balls). Similarly, here we can use, (area of shaded region) divided by (total area). To calculate it, we need to figure out the area of the yellow region.

It can be seen, that there are two circles here. And the area of circle is, \pi r^2, but the circle inside the square is just \frac{1}{4}^{th}, so area of first circle inside the square is, \frac{1}{4} \pi r^2. The area of square itself will be r^2. Now if we subtract the area of the circle from the square, we get the area of one purple region. r^2 - \frac{1}{4} \pi r^2 = r^2(1 - \frac{\pi}{4} ). The area of second purple region will also be the same, and hence the total area is, 2 [r^2(1 - \frac{\pi}{4})]. In the end what we need to do is, subtract this from the area of square to get the area of yellow region. Which is going to be, r^2 - 2 [r^2(1 - \frac{\pi}{4})] = -r^2 + \frac{\pi}{2}r^2.

Finally, we can get the probability of the particle to be found in the yellow region, which is, \frac{[-r^2 + \frac{\pi}{2}r^2]}{r^2} = \frac{\pi}{2} - 1.

Math reading challenge 2020

| Cosmoshivani

So, I found this article by Evelyn lamb in her blog Roots of Unity at the end of 2019. It had some prompts to help you find math related books. I wasn’t able to complete all of them but I did try some.

I read three books for the prompt, “A work of fiction in which a main character is a mathematician“. The very first book I read was, “The devotion of suspect X“. It was beautiful. Set in Japan, it’s about a mathematician Ishigami, who is a High school teacher. One day, when he was trying to take his life, someone rings the doorbell. His new neighbors, Yasuko and her daughter Misato. Without knowing, they save his life. And from that point, he does everything to protect this family next door. It’s mysterious, and very emotional at times. Next was, “The Housekeeper and the Professor“. Due to a terrible accident in 1970, a mathematician is now only able to remember last 80 minutes of his life. Like a hard disk which has only 80 minutes of memory limit; anything new is recorded on top of that. So whenever he wakes up in morning, he doesn’t remember a single thing that happened yesterday. Many housekeepers are assigned to take care of him but no one lasts. But it’s about to change. The tenth housekeeper, who is the youngest in the company takes the job and then starts a story of friendship between the housekeeper, her son and the professor. They go on many trips and he tells them interesting things like the importance of square root, amicable numbers, perfect numbers, triangle numbers etc. But the good days come to an end. His 80 minute memory tape is also broken and he cannot retain even a minute of new memories. Sadly, in the end, he is transferred to a new care facility. Then I read “Uncle Petros and Goldbach’s Conjecture“. Its story revolves around a mathematician who is obsessed with the Goldbach’s conjecture (Every even number greater than 2 can be written as a sum of prime numbers). And how he puts in so much effort to discourage his nephew from pursuing higher mathematics, because he doesn’t want him to get mad and obsessed like himself. It also has a very tragic ending.

For the next prompt, “A graphic novel about math or mathematicians“, I read “Logicomix: An Epic Search for Truth“. It had really awesome script and beautiful drawings. I read about Russel’s paradox for the first time in it, which is introduced by Bertie Russell himself. It goes something like this, if a set consists of all the sets that do not contain themselves, then does that set contain itself as an element? If it does then it does not follow the property that it should not contain itself; If it doesn’t contain itself then it does follow the property and thus it belongs to the set but then again if it does it shouldn’t and if it doesn’t then it should.

For the most part, the story follows the life of Bertrand Russell. There are many events of his life from childhood to a university student; how he started questioning everything that others took for granted. It also contains many dialogues and fictional meetings between famous mathematicians, which makes it even more interesting.

logicomix
Logicomix

So basically, all I read was mathematical fiction. But I did learn some concepts in number theory and logic here and there.