When sinx, cosx and e^x goes to a party

JC Mathematics

cosx, sinx and e^x go to a party. sinx and cosx are partying away but e^x is miserable and anti social. cosx and sinx go up to e^x and say ‘what’s wrong, why don’t you integrate?’

e^x replied, It doesn’t make any difference does it?

This little joke should remind students about integrating the three terms. And a side note, they are really related, for instance in Complex Numbers.

The Birthday Paradox

JC Mathematics

This is an interesting probability problem (paradox). And no, this isn’t about the Cheryl’s Birthday Problem.

In probability theory, the birthday problem or birthday paradox[1] concerns the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday. -Wikipedia

Credits: Wikipedia
Credits: Wikipedia

The above graph shows how many people you need to approach to find someone who has the same birthday with you!

This shows how counter-intuitive probability is! And like what I always tell my students, don’t use intuition for probability but formulas.

Why is 0! = 1?

JC Mathematics

This question is probably very baffling to several students. Many students will exclaim 0!=0 to me, but this is incorrect. To understand why 0!=1, we need to first look at what n! means; n! is the number of ways to arrange n objects in a row. And we all know that n!=1 \times 2 \times 3 \ ... \times n. So shouldn’t 0!=0?

Think about this, the number of ways to arrange 1 object is 1, that is, put the object there by itself. However, the number of ways to arrange 0 object is one! Cos there is nothing to arrange so we still have one way to do it.

Give it some thought and feel free to discuss with me!

Related video by Dr James Grime

Patterns in prime numbers

Patterns in prime numbers

University Mathematics

So you still think prime numbers are random?

Take a look at this really cool project, it kinda reminds me of the sieve of Eratosthenes: El Patrón de los Números Primos: Prime Number Patterns – Jason Davies

You can examine more on divisor plot as well.

Prime numbers has mystified mathematicians for centuries, there seems to be a pattern but mathematicians aren’t able to find a method to predict the next sequence or understand what exact cause prime numbers to be the sequence it currently is.