Strike A Pause (Some Beautiful Mathematics)

Alright, here's a fun math fact that I think can be appreciated by any major: --> ANY real number (including irrational numbers) can be approximated to ANY desired degree of accuracy using rational numbers with finite decimal representations. Those who enjoy (or want to start enjoying 😂) proving such statements rigorously can engage with the following theorem (here _ denotes subscript): Theorem: Assume x >= 0. Then for every integer n >= 1 there is a finite decimal r_n = a_0.a_1a_2....a_n such that r_n <= x < r_n + 1/10^n [To clarify what is meant by r_n = a_0.a_1a_2....a_n, here's an example: For the number r_n = 8.1472 we have a_0 = 8, a_1 = 1, a_2 = 4, a_3 = 7 and a_4 = 2]

Basic calculus: Continuity: Let A ⊂ R and f : A → R be continuous. Then for each x0 ∈ A and for given ε > 0, there exists a δ(ε, x0) > 0 such that x ∈ A and | x − x0 |< δ imply | f(x) − f(x0) |< ε. (Observe that δ depends, in general, on ε as well as the point x0). A little more advanced: Uniform Continuity (Definition): A function f : A → R, where A ⊂ R is said to be uniformly continuous on A if given ε > 0, there exists δ > 0 such that whenever x, y ∈ A and | x − y |< δ, we have | f(x) − f(y) |< ε. Question 1: What is the difference between continuity and uniform continuity? Which is the stronger condition? Why? Once you've clearly resolved the above and looked at a few examples to see the difference clearly (the internet is a great resource if you know where to look!): Question 2 (Self Study): What is Lipschitz Continuity? Is it a stronger or weaker condition that continuity and uniform continuity? After you've resolved the above at the level of defin...

Did you know that: 1) Although the more commonly used geometric representation of complex numbers is the complex plane (developed by Gauss (1799) and Argand (1806)), one can also use the sphere (Riemann (19th century)) to obtain what is known as a stereographic projection (see figures below). Here points, P, of the sphere are projected from the North Pole, N, onto the tangent plane at the South Pole as follows: The line passing through N and P intersects the tangent plane at the point P'. This P' represents the sphere point P in the plane. Observe that with the exception of the North Pole itself, each point of the sphere corresponds to exactly one point in the plane. (Wait...there's coolness coming...keep reading...) 2. While we need two symbols '+infinity' and '-infinity' when we extend the real line R to infinity (the extended real line is called R*), we need only one symbol 'infinity' when extending the complex plane C to C*. This results form the...

Alright, here's some fun (and charming!) mathematics for a peaceful Sunday afternoon: Using the infinite series representation of e^x = 1 + x + x^2/2! + x^3/3! + ... + x^n/n! + ..., prove that e must be irrational. (Hint: Go after 1/e. Trap this minion and he shall lead you to the Don!) Note: No, e is not irrational by definition! (Yes, of course, if you are stuck after trying for a while, by all means consult references (or reliable sites on the internet). I'm simply putting up math problems that I find charming 😊. These problems have made me appreciate how beautiful mathematics can be if you are a bit patient.)

Alright, let's start having some maths fun with this blog 😉! This first post ought to be bread and butter for math and computer science majors, but I think it's something pretty cool for the intellectually curious anyway! Consider this : Every integer n > 1 can be represented as a product of prime factors in only one way, apart from the order of the factors. That's crisp! Isn't it? The above is known as the Unique Factorization Theorem. (It's also called the Fundamental Theorem of Arithmetic - that's how important this result is considered.) Let me know using an appropriate reaction emoticon in the comments fields below if you can prove this theorem. If not right away, then take your time - no hurries 😉!