Category: maths

fuckyeahphysica: Once when lecturing in class Lord Kelvin used…


Once when lecturing in class Lord Kelvin used
the word ‘mathematician’ and then interrupting himself asked his class:
Do you know what a mathematician is?’

Stepping to his blackboard he
wrote upon it the above equation.

Then putting his finger on what he had written, he turned to
his class and said, ‘A mathematician is one to whom that is as obvious
as that twice two makes four is to you.

** Two interesting ways to arrive at the Gaussian Integral

Woah… The backlash that Lord Kelvin got after this post was just phenomenal.

There are many ways to obtain this integral (click here to know about other methods) , but here are two interesting ways to arrive at the Gaussian Integral which you may/may not have seen and may/may not be easy to follow.

Gamma Function to the rescue

If you know about factorials (5!=, you know that they make sense only for integers.

But  Gamma function
extends this to non-integers values.  This integral form allows you to
calculate factorial values such as (½)!, (¾)! and so on. 

The same can be used to evaluate the Gaussian Integral as follows:

Differentiating under the Integral sign

In this technique known as ‘Differentiating under the integral sign’, you choose an integral whose boundary values are easy integrals to evaluate.

Here I(0) and I(∞); and differentiate with respect to a parameter β instead of the variable x to obtain the result.

Have a nice Pi-Day! In memory of Stephen Hawki…

Have a nice Pi-Day! In memory of Stephen Hawking!

March 14,2018 (03/14/2018)** Einstein was born, Hawking passes…

March 14,2018 (03/14/2018)

** Einstein was born, Hawking passes away and it’s pi day…

Once when lecturing in class Lord Kelvin used the word…

Once when lecturing in class Lord Kelvin used
the word ‘mathematician’ and then interrupting himself asked his class:
Do you know what a mathematician is?’

Stepping to his blackboard he
wrote upon it the above equation.

Then putting his finger on what he had written, he turned to
his class and said, ‘A mathematician is one to whom that is as obvious
as that twice two makes four is to you.

Beautiful proofs (#4) – When Gauss was a young child…

The legend goes something like this:

Gauss’s teacher wanted to occupy his students by making them add large sets of numbers and told everyone in class to find the sum of 1+2+3+ …. + 100.

And Gauss, who was a young child (age ~ 10) quickly found the sum by just pairing up numbers:

Using this ingenious method used by Gauss allows us to write a generic formula for the sum of first n positive integers as follows:

Ask Ethan: Where Is The Line Between Mathemati…

Ask Ethan: Where Is The Line Between Mathematics And Physics?

“Where does one draw the line between abstract mathematics and physics? Is Noether’s Theorem part of the scientific corpus of knowledge, or the mathematical? What about Maldacena’s conjecture?”

We all know that one of the reasons physics is such a powerful science is because of its predictive power. If you give a physicist the equations governing your system, and the right initial conditions, they can tell you exactly how it will evolve arbitrarily far into the future, limited only by the uncertainty built into the equations themselves. But not everything is purely determined by the math of the equations. You don’t have to go to a super-advanced example to see that, either. Give me a projectile in motion, tell me its position and velocity, and the equations you get from Newton’s laws give you two possible answers for where-and-when it will hit the ground. But only one of those answers is correct! How can you tell? It requires more than just mathematics on its own; it requires you to know something about your Universe.

That’s the start of the difference between mere mathematics and physics, but the full answer goes much deeper. Curious? Find out more on this edition of Ask Ethan!



Real’s Math Ask Meme

  1. What math classes have you taken?
  2. What math classes did you do best in?
  3. What math classes did you like the most?
  4. What math classes did you do worst in?
  5. Are there areas of math that you enjoy? What are they?
  6. Why do you learn math?
  7. What do you like about math?
  8. Least favorite notation you’ve ever seen?
  9. Do you have any favorite theorems?
  10. Better yet, do you have any least favorite theorems?
  11. Tell me a funny math story.
  12. Who actually invented calculus?
  13. Do you have any stories of Mathematical failure you’d like to share?
  14. Do you think you’re good at math? Do you expect more from yourself?
  15. Do other people think you’re good at math?
  16. Do you know anyone who doesn’t think they’re good at math but you look up to anyway? Do you think they are?
  17. Are there any great female Mathematicians (living or dead) you would give a shout-out to?
  18. Can you share a good math problem you’ve solved recently?
  19. How did you solve it?
  20. Can you share any problem solving tips?
  21. Have you ever taken a competitive exam?
  22. Do you have any friends on Tumblr that also do math?
  23. Will P=NP? Why or why not?
  24. Do you feel the riemann zeta function has any non-trivial zeroes off the ½ line?
  25. Who is your favorite Mathematician?
  26. Who is your least favorite Mathematician?
  27. Do you know any good math jokes?
  28. You’re at the club and Andrew Wiles proves your girl’s last theorem. WYD?
  29. You’re at the club and Grigori Perlman brushes his gorgeous locks of hair to the side and then proves your girl’s conjecture. WYD?
  30. Who is/was the most attractive Mathematician, living or dead? (And why is it Grigori Perlman?)
  31. Can you share a math pickup line?
  32. Can you share many math pickup lines?
  33. Can you keep delivering math pickup lines until my pants dissapear?
  34. Have you ever dated a Mathematician?
  35. Would you date someone who dislikes math?
  36. Would you date someone who’s better than you at math?
  37. Have you ever used math in a novel or entertaining way?
  38. Have you learned any math on your own recently?
  39. When’s the last time you computed something without a calculator?
  40. What’s the silliest Mathematical mistake you’ve ever made?
  41. Which is better named? The Chicken McNugget theorem? Or the Hairy Ball theorem?
  42. Is it really the answer to life, the universe, and everything? Was it the answer on an exam ever? If not, did you put it down anyway to be a wise-ass?
  43. Did you ever fail a math class?
  44. Is math a challenge for you?
  45. Are you a Formalist, Logicist, or Platonist?
  46. Are you close with a math professor?
  47. Just how big is a big number?
  48. Has math changed you?
  49. What’s your favorite number system? Integers? Reals? Rationals? Hyper-reals? Surreals? Complex? Natural numbers?
  50. How do you feel about Norman Wildberger?
  51. Favorite casual math book?
  52. Do you have favorite math textbooks? If so, what are they?
  53. Do you collect anything that is math-related?
  54. Do you have a shrine Terence Tao in your bedroom? If not, where is it?
  55. Where is your most favorite place to do math?
  56. Do you have a favorite sequence? Is it in the OEIS?
  57. What inspired you to do math?
  58. Do you have any favorite/cool math websites you’d like to share?
  59. Can you reccomend any online resources for math?
  60. What’s you favorite number? (Wise-ass answers allowed)
  61. Does 6 really *deserve* to be called a perfect number? What the h*ck did it ever do?
  62. Are there any non-interesting numbers?
  63. How many grains of sand are in a heap of sand?
  64. What’s something your followers don’t know that you’d be willing to share?
  65. Have you ever tried to figure out the prime factors of your phone number?
  66. If yes to 65, what are they? If no, will you let me figure them out for you? 😉
  67. Do you have any math tatoos?
  68. Do you want any math tatoos?
  69. Wanna test my theory that symmetry makes everything more fun?
  70. Do you like Mathematical paradoxes?
  71. 👀
  72. Are you a fan of algorithms? If so, which are your favorite?
  73. Can you program? What languages do you know?

I’m a physicists, but still

Dynamics of Love affairs: FYP!’s Valentine’s Day Gift Box


There is no simple love story.

If it’s simple, its not love.

If it’s love, its complicated.

How complicated can it get? Well let’s find out!

Romeo and Juliet

Let’s consider a love affair between Romeo and Juliet.

We shall use functions R(t) and J(t) to quantify their love:

  R(t) –> Romeo’s love/hate for Juliet at time t

  J(t) –> Juliet’s love/hate for Romeo at time t

These quantities are positive for love and negative for hate.


The simplest linear relationship that one can construct to describe their love affair is:


These set of primitive equations*** can be used to predict how their relationship is going to evolve with time.

Now if you had taken a course in Linear Algebra then you might know how to solve these system of linear differential equations and obtain the solution. Here’s a quick review:


INTERLUDE: What type of a person is Romeo ?

Remember that we said that Romeo’s love for Juliet evolves with the equation:


Depending on the signs of the coefficients a and b , Romeo can exhibit one of four romantic styles:

1. Eager beaver: a > 0, b > 0 (Romeo is encouraged by his own
feelings as well as Juliet’s.)


2. Narcissistic nerd: a > 0, b < 0 (Romeo wants more of what
he feels but retreats from Juliet’s feelings.)


3. Cautious (or secure) lover: a < 0, b > 0 (Romeo retreats from
his own feelings but is encouraged by Juliet’s.) –>Most people


4. Hermit: a < 0, b < 0 (Romeo retreats from his own feelings
as well as Juliet’s.)


And based on the signs of the coefficients c and d Juliet can also exhibit one of four romantic styles as well.

If we know these coefficients for Romeo and Juliet, we can look compute the trace (a+d) and determinant (ad-bc), and look up the Poincaré diagram to predict the future of the Romeo-Juliet relationship:


Example: A never-ending tale of love and hate

Let me illustrate with a special case where a = 0 , b = 1, c = (-1), d = 0:


Trace = a + d = 0

Determinant = ad – bc = 1 

Looking up the Poincaré diagram we get a centered orbit:


And we can verify this by using solving this system of linear equation numerically using applications such as pplane:


              Plot of Romeo’s love for Juliet v/s Juliet’s love for Romeo

What this plot implies is that the couple is stuck for eternity in an orbit of love and hate. Wherever the initial point of starting might be on the plane, they always end up orbiting round the origin.

They spend three quadrants of their cycle in a turmoil of love and hate but spend one entire quadrant (R > 0 & J > 0) in true and unconditional love.

Happy Valentine’s day everybody!

* This post was Inspired by Strogatz’s Book on Nonlinear Dynamics and Sprott’s paper titled ‘Dynamics of Love affairs’.

** Previous giftboxes: 2016 , 2017

*** This is the simplest system that one can analyze

People are awesome & Math is beautyThe white circles albeit…

People are awesome & Math is beauty

The white circles albeit traveling in a straight line across the circle exhibit a more collective circular behavior.

Here’s a much more real world scenario which follows similar guidelines:


If you notice, the motion of all the workers individually are also periodic in nature, but each of their motion is slightly out of phase leading to this beautiful symmetric behavior that constitutes this gif.

Truly mesmerizing!

The mathematical sciences exhibit order, symmetry and limitations; and these are the greatest forms of the beautiful

– Aristotle

This should have kept you up all night! If you read the previous…

This should have kept you up all night!

If you read the previous post on the Fourier Series, then you might have noticed that this animation was kind of lying to you.

It surely does seem to resemble a square wave but notice that the peaks in red : They are overshooting  and undershooting the maximum and minimum amplitudes.

What on earth is happening here? This goes by the name ‘Gibbs Phenomenon’.

We do not have enough terms

Remember that in Fourier Series you are trying to construct a square wave (which has sharp edges) with smooth and continuous sine and cosine waves.


Fourier series promises us to reconstruct the waveform perfectly ONLY if we provide it with the entire spectrum of frequencies.

But practically we can only work in a finite range of frequencies and when working in a finite domain this overshoot is unavoidable and does not die out.


And if you are an engineer working with a system whose maximum output must not exceed the limit, this can be quite frustrating.

Is there a way out of this ?

In order to get much smoother Fourier series, methods such as Fejér summation or Riesz summation, or sigma-approximation are employed.

Here’s the Fejér summation in action:


                                     Without Fejér summation                              


                                        With Fejér summation

Have a good one!

** Read more about the consequences of Gibbs phenomenon here