mathematics on S Anand

Correlating subjects

Sun, 12 Feb 2012 04:08:02 +0000

A question from Dorai get me thinking: does being good at maths help in programming?

I don’t have a personal view. But since Reportbee has data on the Class 12 examination results for the last three years, we thought we could do a bit of analysis.

Here’s the correlation of the scores of various subjects with Computer Science.

Correlation	Subject
0.79	CHEMISTRY
0.79	PHYSICS
0.75	ENGLISH
0.75	MATHEMATICS
0.72	LANGUAGE
0.67	BIOLOGY
0.66	ECONOMICS
0.66	COMMERCE
0.65	ACCOUNTANCY
0.56	HISTORY
0.52	GEOGRAPHY

It almost breaks neatly into four groups.

Physics & Chemistry, both of which have a correlation of 0.79, and clearly are the most correlated with Computer Science
Maths, English & Language, which have a correlation of 0.72 – 0.75
Biology, Economics, Commerce and Accountancy, which hover at around 0.66
History & Geography, which are 0.52 – 0.56

The results in 2010 are almost exactly the same.

Correlation	Subject
0.78	PHYSICS
0.78	CHEMISTRY
0.75	ENGLISH
0.75	MATHEMATICS
0.73	LANGUAGE
0.67	ACCOUNTANCY
0.65	ECONOMICS
0.65	COMMERCE
0.64	BIOLOGY
0.60	GEOGRAPHY
0.55	HISTORY

I’m not sure what it is that leads to this kind of correlation. In fact, the full correlation between every pair of subjects (for 2011) is below:

What inferences would you draw from this?

And what do you think is the reason for this?

Comments

Arun Ravindran 12 Feb 2012 5:53 am: My inference is purely anecdotal but might be helpful in explaining this data. I had chosen Computer Science for my +2 in 1998. It was taught in C++ and mostly involved memorising operations on Data Structures such as Lists, Stacks and Queues. The C++ standard library had to be memorized including I/O, string and file functions. The exams were basically a test of memory rather than attacking a new problem space mathematically. Guess which are the other subjects which involve memorising a huge set of symbolic facts? – Chemistry and to a certain extend, Physics. I believe the data is more revealing of our Computer Science pedagogical and evaluation methods than the subject itself.
S Anand 12 Feb 2012 6:46 am: Good point. It’s quite debatable whether marks in computer science are indicative of programming ability.
Anamika 12 Feb 2012 10:21 am: Netruvarai neram poga villaiyae, unadhu arugae neram podhavillaiyae…!
Sathya 16 Feb 2012 12:56 pm: In my experience, I’ve come across rock star programmers who have sound grasp of mathematics. But i dont have statistics to prove them. As we all know, programming (esp functional) is heavily influenced by mathematics. Coming to inferences, I would take the dataset with a pinch of salt. Is it diverse enough to be statistically significant ? I agree with most of the comments on relating “mugging up programs” to being good at Chemistry in particular.
Sathyaraj 12 Feb 2012 7:44 pm: My 2 cents based on experience. People who take the computer science group generally have to always take one language and English subject apart from taking Physics, Chemistry, Maths. They would not be able to take classes in accountancy, economics, history, etc. Some of the students at the top of the class would have realized that in order to differentiate in terms of coming first in class/school, one would need to excel in language and english. It is given that one needs to get excellent scores in physics, chemistry and maths to get an overall good percentage. Also the fact, that scoring a big total would enable them to get admission in colleges in like BITS pilani which looks at overall score. Students who take the biology or accountancy group have no such pressure for them to excel in English and language subjects. Hope it makes sense. Also, if you can do a correlation between computer science students and the language that they take, I believe you will find that the majority of students would have chosen french rather than Tamil or English.
Shankar V 13 Feb 2012 6:23 am: Anand Correlation is a misleading statistic if the dependent and independent variables are not “really” related that way. There could be compounding effects within the independent variables leading to wrong correlation coefficients. There are ways to detect and cleanse compounding - design of experiments (in statistics) deals in depth on this. It is quite possible that there could be some relationship in how the computer science and math exams are scheduled and the scores in these subjects. If they are too close to each other, the student may have had lesser time to prepare for the math exam. For example, if Physics and Math were scheduled one after the other, and Computer Science test is after the Math test, the student may have prepared better for Physics, not have had enough time for Math and then recovered to do well in Computer Science. This can be a pattern in the entire class as it is normal for kids to focus more on Physics (the dreaded subject!) v/s Math. The statistic may not reveal ability or natural alignment of the subjects.
Evgeni 11 Dec 2012 3:37 am: I’d rearrange subjects such that they’re more clustered together by correlation, like in a TreeMap view. That way it’s easier to see the relationships.
amanjot kaur 10 Sep 2015 12:28 pm: i need correlation of commerce with language

Absolutely convergent series

Thu, 31 Aug 2006 12:00:00 +0000

I’ve seen many proofs that 1=2. Here’s a classic.

The (not-so-subtle) error in the above proof is that we’re cancelling (a-b) on both sides, when (a-b) equals zero. That is, we’re dividing by zero on both sides. That completely invalidates the equality.

Another proof uses the fact that the square root of a number can be both positive or negative.

(Proving -1=1 is the same as proving 1=2. Once you have one wrong proof, you can prove every other falsehood.)

The flaw here is that the square root of 1 is 1 and -1. So right after the square root symbol appears, every equation should have a plus-or-minus symbol on both sides.

The most convincing proof uses absolutely convergent series as the key idea. Here’s how the proof goes.

Most people initially think that the flaw is in the re-arrangement of the series. That’s not true! The re-arrangement works just fine, and you can prove that every term is correct to infinity.

The flaw is subtler.

When an infinite series is summed, it can be summed in any order. But the total may vary depending on the order you sum it up! You are guaranteed that the total is the same only if the series is absolutely convergent. That is, if the sum of the absolute values of each number is finite. (See the Wikipedia article on the Riemann series theorem.)

For the log 2 series, it’s not absolutely convergent. The series diverges, as shown below:

So, by re-arranging the series for log 2, we’ve invalidated the equality anyway.

This fact once saved an entire class. We had a problem in our first year physics course to which the answer was the series above. (It had to do with calculating the electromagnetic potential created by an array of charges.) Since the series is not absolutely convergent, and every possible answer was correct, the whole class got marks for this question, as long as they attempted it.

Comments

Arun 1 Sep 2006 1:18 pm: hi da how much u analyze and write amazing i read ur blog continuously for the past four years keep blogging interesting
S Anand 1 Sep 2006 5:28 pm: Thanks! It’s tough to manage, but nowadays I have a lot more time than I used to.
saurabh 2 Apr 2007 7:28 am: very subtle indeed. Keep up…
Alan 27 Apr 2015 10:29 pm: How can it be proves that “dividing by zero on both sides…. invalidates the equality.”? Isn’t the correct answer to 0/0 = we don’t know?
Lolguy 4 Dec 2014 9:52 pm: The number 2 is wrong. It should read like: sqrt(1) sqrt(1) = sqrt(-1) sqrt(-1); sqrt(1) = sqrt(1); (since sqrt(-1) x sqrt(-1) = sqrt(-1 x-1) = sqrt(1)) 1 = 1
Fabricio 21 Dec 2014 1:48 pm:

The number 2 is wrong. It should read like: […]Lolguy,The second proof is wrong, yes; though not for the reason you stated (which is also wrong reasoning).The error in the proof is the third line, sqrt(1/-1) = sqrt(-1/1) , which does not make sense, formally; even though it looks suggestively meaningful.The implication from the second to the third line would be equivalent to saying that,Carl is the same person as Carl, [ Carl = Carl ]Therefore, [any] son of Carl is the same person as [any] son of Carl. [ son_of(Carl) = son_of(Carl) ]This will only be always true if Carl has only one son; evidently, if he has more than one, any two given sons of Carl need not necessarily be the same person.A analogue situation happens when defining the square root of a number:In context of the set of real numbers, it is possible (and actually done so, in many places) to define the square root of a number x as ’the positive number y, such as y^2=x’ ; (The generalization for n-th roots, n pair, coming easily from that, in an analogue manner.)This solves the issue for square roots (or any other roots) under the real numbers: defined in this way, they determine a function (a relation between sets A and B, such that to each element in A one, and *only* one element in B is associated) and can be used as an operation that can be applied to both sides of an equation and still preserve its truth (in the same way it is valid to state that x = y implies that x^2 = y^2 , or that x = y implies that log (x) = log (y) ).Defined in this way, sqrt(x) constitutes a function, and does, as all functions do, possess the property of being a univocal relation: for each x, there is only one f(x), or in this particular case, only one sqrt(x).When it comes to the set of complex numbers, though, the plot thickens, because for any given non-zero complex number z, there will be n different complex numbers z_i (i = 1, … , n) such that (z_i)^n = z .It is said, usually: ’each complex number z has n complex nth-roots.’ (Thus, 2 square roots, 3 cubic roots, and so on.)Now this way of defining z_i as being an n-th root of z does not have the property of being univocal, and therefore does not constitute a function and cannot be used as an equality-preserving operation applied to both sides of an equation.This is why -1 = -1 does *not* imply that sqrt(-1) = sqrt(-1) ;After all, i^2 = -1 and, therefore, it is a complex square root of -1 (qualifying to be on the left side of the second equation),;At the same time as (-i)^2 = -1 and, thus, is also a complex square root of -1 (and could be on the right side of the second equation).Evidently, -i = i is false, as they are distinct complex numbers.It is worthy noting:The definition of a complex root could be restricted to make it a function.It could be that *the* complex n-th root of z is defined to be the number w with the smallest argument (the angle theta in the complex plane representation of w) such that w^n = z .This, however, is not done anywhere. Rather, the concept is defined by saying that w_i is a complex n-th root of z if w_i^n = z . And that implicates that there is more than one n-th root (for n>1). The complex root thus defined is sometimes said to be a ‘mutivalued/plurivocal function,’ in textbooks.(This is usually carried on to extend the concept of function, and does not mean that the complex root is a function proper. It is a case of ‘abuse of language.’ See https://en.wikipedia.org/wiki/Abuse_of_notation )Thus, when you write sqrt(-1) and look down at it, in the paper, no matter how suggestive it is of being ‘a number,’ it is not.It is but an abuse of language by which you are representing different numbers with one single graphical symbol. And therefore it cannot be used in equations in that same manner that you use ‘x’ , normally.While x represents, in the context of the equation, only one number; sqrt(-1) represents more than one number and therefore, sqrt(-1) = sqrt(-1) is wrong.Abuse of notation can be used effectively to produce correct results, if communications, or reasoning, take place between two parties that are well informed and conscious of the limitations of the notationally abused language. What is gained, many times, is a more compact way of expressing ideas well known to both parties.For instance, it can be economical to write and think 1/0 = infinity, or, lim (1/x) as x->0 = infinity, even though, strictly, both of these ’equation-like’ arrays of symbols have no meaning.If both parties are well trained in the definition and properties of limits, the shorthand notation can be used to perform calculations correctly, saving a great deal of paper and time needed to write formally correct statements regarding the (non-existent) limit of 1/x as x approaches 0.The danger here, is uninformed third-parties reading those and due to lack of training, drawing false conclusions.

Even understanding it all, it is not uncommon, from time to time, for one to come back to those false implications/proofs using sqrt(-1) and scratch one’s head, while going ‘damn… I know this is false, but, why, again?’ Notational intuition is a powerful force within your mathematical brain. (And it shows in the history of mathematics, in episodes where bouts of prolific development in theory come right after some better notation gets introduced.) You might wanna memorize a formalist mantra designed to safeguard you against this particular situation: ‘Sqrt(-1) is not a specific number. It cannot be hurled around inside equations just like x.’ Repeat ad nauseam until all thought is drowned. :p Put this formalist mantra in your personal mantra library along with ‘Thou shalt not divide by zero.’ and also, ‘The square root of a^2 is modulus of a, and *not* just a.’ (which helps you avoid the common and tempting mistake of assuming that a^2 = b^2 implies that a = b ) Cheers, Fabricio
Ignis 5 Nov 2023 3:25 am: X≠Y, but 0=0 so X=Y 🤪

Knowing when to stop

Wed, 21 Jun 2006 12:00:00 +0000

Mathematics, marriage and finding somewhere to eat has a simple solution to all these problems. Whether you’re hiring someone, or picking a partner, or finding a house – or any problem that requires you to pick the best among N choices – here’s the rule.

Scan the first 37% of choices. Then pick the first one that’s better than anything you’ve seen so far.

The Mathematical Structure of Terrorism

Wed, 24 May 2006 12:00:00 +0000

The Mathematical Structure of Terrorism. The frequency distribution of terrorist attacks – be it in Iraq, Columbia, Afghanistan or anywhere in the world – is the same. The frequency distribution of the size of terrorist groups is the same as well.

Math will rock your world

Sat, 14 Jan 2006 12:00:00 +0000

It’s a good time to be a mathematician.

Special numbers

Tue, 08 Mar 2005 12:00:00 +0000

Every number is special. Here’s what’s special about the first 10,000 numbers (almost). For instance, “8281 is the only 4-digit square whose two 2-digit pairs are consecutive.” And “5851 is the only prime so that it, its square, and its cube all have the same sum of digits.”

Math comics

Tue, 06 May 2003 12:00:00 +0000

Math comics. Including some Calvin & Hobbes.

Poincare conjecture may have been solved

Wed, 16 Apr 2003 12:00:00 +0000

The Poincare conjecture may have been solved. via Joseph

Paradox of randomness

Wed, 13 Nov 2002 12:00:00 +0000

Another interesting piece related to complexity: a speech on the paradox of randomness by Gregory Chaitin. via missing matter

Encyclopaedia of integer sequences

Tue, 16 Apr 2002 12:00:00 +0000

Encyclopaedia of integer sequences. Funny, it couldn’t find “1,2,3,4,5,6…” though.

Mathworld

Thu, 08 Nov 2001 12:00:00 +0000

Mathworld is back.

Unsolved problems in mathematics

Fri, 01 Sep 2000 12:00:00 +0000

In 1900, David Hilbert outlined 23 unsolved problems in mathematics. Many of these have been solved today, with the notable exception of the Riemann Hypothesis. Today, if we solve any of these, we get $1 million.

Clifford Pickover

Thu, 13 Jul 2000 12:00:00 +0000

Pickover’s site is always a pleasure to visit. His puzzles are great, but you really go there for a blend of art, computers, mathematics and science.