3 Mathematical Concepts that Aren't Hard to Understand – But will Blow Your Mind Anyway
Math isn't always easy for everyone, which means that some of the more complex and interesting concepts in mathematics can't really be enjoyed by the general public with ease. These include highly complex, yet intriguing ideas, such as quantum superposition, the perturbation theory, or algebraic ideas like spectral sequences.
The point of this post, however, is to blow your mind with complex mathematical theories almost anyone can understand. So, let's take a look at three different mathematical concepts that will get you excited about math.
Dividing by zero
Dividing by zero is the quintessential mathematical concept that almost all of us were taught at school as being impossible; or, if you divide by zero, you will get an answer which is either undefined, or equivilant to infinity, depending upon who you ask – mathematician or physicist.
You can probably best understand why dividing by zero results in an undefined answer if you think about how division and multiplication are related. For example, 12 divided by 6 equals 2 because 6 times 2 is 12. So, 12 divided by 0 equals x would mean that 0 times x must equal 12.
But no value would work for x because 0 times any number is 0. So division by 0 doesn't work.
There is a way to find the answer though, and it deals with something called a limit. Take a look at the graph below of Y = 1/x and we'll explain.
The graph above is what happens when you graph the equation of y=1/x (y equals 1 divided by x). You'll notice that as X gets closer and closer to 0, the graph approaches infinity. If you were to expand this graph, you'd see that the red lines never touch the y-axis, or the position of x=0. This principle is known as a limit – the x gets infinitely close to the value of 0, but never exactly reaches it.
The principle of limits is foundational to mathematics, calculus, and physics.
While you might have been taught that dividing by zero is impossible, it's actually a highly complex idea that can be expressed in a variety of different ways, and it is also important in some physics and mathematics concepts.
In the video below, the teacher does an incredibly impressive job explaining division by zero in simple terms.
The Bailey-Borwein-Plouffe formula
The Bailey-Borwein-Plouffe Formula, commonly known as the BBP Formula, lets to skip to any digit of Pi without knowing the entire number (which is impossible).
Stepping back for a moment, Pi is the ratio of a circle's circumference to its diameter. The number produced by this ration is both infinite and a constant. The digits of Pi do not change, and they are thought to go on forever without repeating. So, what if you wanted to determine the 2340184000th digit of Pi? Previously, you would need to compute all the digits that come before using high-precision arithmetic, or a computer algorithm. However, in the mid 1990's, a remarkable new formula for was discovered by David Bailey, Peter Borwein, and Simon Plouffe (BBP). The BBP formula lets you determine any nth digit of Pi.
Here's how the formula works, for any nth number that you want to find, you split the infinite sum of the nth number in hexadecimal. This is what this looks like in formulaic terms.
If you substitute the digit of pi you want to find in for k, you'll be left with the answer in 16-bit hexadecimal.
Now, why is this useful? Well, it's useful for calculating any nth digit of Pi, of course! The video below breaks down some interesting math from the Simpsons, and it also discusses the BBP formula.
Tupper's Self-Referential formula
Tupper's Self-Referential Formula is a graphing formula that can be used by plotting software to plot almost everything.
And here's the formula:
The symbols and together denote the floor function: for a real number a, the floor a of a is the largest integer that's no bigger than a. For example, 4.2 = 4. The function mod (a,b) computes the remainder you get when you divide a by b, for example mod (8,3) = 2.
The plot works by either coloring a square on a graph or not colouing it: a square with coordinates (x,y) is colored if the inequality is true for x and y. If not the square is left blank.
If you plot the graph for many values of x and y, the outcome looks like this:
Now, while you might think that the image above is the formula written out in a weird blocky text, well, you'd be right. But the way it was written is what's interesting about this formula. That is the output of the Tupper's formula when you let N equal this number (it has 543 integers):
Yes, that's right, the formula plots a bitmap picture of itself. Hence, this is sometimes called Tupper's self-referential formula.
If you look at the squares with y coordinates between N and N+16 (and ignore all squares with y-coordinates less than N and greater than N+16, you will see the bitmap image of Tupper’s formula itself.
Now, say we wanted to change this 543-digit value of N and scroll up and down the y-axis to see what plots we get. As we scroll up and down the y-axis from minus infinity (indicated by the downwards direction) to plus infinity (indicated by the upward direction), we find that any picture that can be represented by a grid of pixels of dimensions 106x17 using two colors is somewhere in the plot of the formula for a particular value of N.
This formula was proposed by Tupper in his 2001 SIGGRAPH paper. It doesn't serve much practical purpose other than to demonstrate some of Tupper's ideas on 2-dimensional computer graphing calculations.
But at the end of the day, does that really even matter? It's a formula that graphs itself!
Hopefully, your mind is blown. If not, I'd suggest you explore harder mathematical and physics concepts like quantum physics.