Fun Math

9 cool facts!

First, I find it ~~odd~~ to have to evaluate $\int_{-1}^1 x e^{x^2} \cos x - \vert x \vert dx$ via brut force, rather find it to be an ~~odd~~ function & it thus becomes identically $0$ 😉

DID YOU KNOW…there are different sizes of $\infty$ ? To see this consider two worlds with DMV offices, world A and world B. In world A our $\infty$ size is countable meaning that if the person handing out tickets starts handing tickets out, he or she would need to hand tickets out till the end of time but could give each person a ticket with a number, in world B our $\infty$ size is the un-countable size so the person would not even be able to hand out tickets because there are just so darn many people, any two people you find there was someone standing between them and this gets repeated so between any two people there’s always someone standing there! there is no way to hand anything out with numbers in world B. Therefore world B has a “larger” $\infty$ size than world A.
DID YOU KNOW…there are two ways of measuring the size of a set? Either by its total number of members VS. how much space it takes up. For example, in the unit interval $[0,1]$ there are actually the same number of numbers as there are people on world B from above! which is $\infty$ ly many! However, the amount of space $[0,1]$ takes up is rather small, its just the length which is 1! Funny isn’t it?!
DID YOU KNOW… you can tell if 3 fits in any number without using a calculator?! We all know if 2,5, or 10 fit inside a number (if it’s even, ends in 0 or 5, or ends in 0, respectively). What about the number 3? Well turns out if you add the digits up and this number is divisible by 3, then the number is. For example, 108, 1,000,008, are both divisible by 3 as the sum of their digits is 9 which is divisible by 3! There is also one for 7 and 11! Click here for the 3-Rule, here for the 7-Rule, and here for the 11-Rule!!! And to see even more number tricks, click here!
DID YOU KNOW…the product of any two integers is equal to the product of their LCM (least common multiple) and GCF (greatest common factor)! Click here for a video on this!Did you know the product of any two integers is equal to the product of their LCM (least common multiple) and GCF (greatest common factor)! To see this, click here !
DID YOU KNOW…why $i^2=-1$ ? This is because in the complex plane (which looks just like your typical $xy-$ plane) the point $i$ is where $(0,1)$ is on the plane and denoted by the point $(0,i)$ and when you multiply $(0,i)(0,i)=(-1,0)$ which is just $-1$ . Here the imaginary axis plays the role of $y-$ axis from ordinary plane. To see this, click here!
DID YOU KNOW…the set of all sets is no longer a set itself but a Class of sets. I always think of it as the group of all people in the world isn’t a person this is now a group of people.
DID YOU KNOW…there are more real numbers (also denoted $\mathbb{R}$ ) between 0 and 1 than there are integers (denoted $\mathbb{Z}$ ) in the world? Crazy to think, huh?!
DID YOU KNOW…the square root of any prime number is no longer a rational number. So $\sqrt{2}$ is irrational.
DID YOU KNOW… that the reason for calculus being created as apposed to only elementary algebra is that in algebra we can only find the area of a box or perfect circle, and find an average rate of change when an object moves from point A to point B. With Calculus however one can find the area under a blob or curvy looking thing that does not have constant curvature as circles do. Furthermore, using Calculus one can find the instantaneous rate of change while moving from point A to point B.
DID YOU KNOW… that any integer in the world can be (uniquely) written in a finite number of 0’s and 1’s! Crazy so one could write the year 2021 as 1111110011, To see this, click here! !!