Last week, you might remember that I mentioned something about Euler’s Identity getting better. In fact, it makes it more natural too, as I will show you. But before we go all the way there, let’s begin with the basics.
First off, let’s start with pi. Pi is approximately equal to 3.14 and is the ratio of a circle’s circumference to its diameter. It is probably considered to be one of the most important numbers in geometry and trigonometry. You might hear pi referred to as the “circle constant.”
The true circle constant should be a number that comes out of the definition of a circle, correct? And what is the definition of a circle? According to dictionary.com, this would be:
A closed plane curve consisting of all points at a given distance from a point within it called the center.
This definition just defined two lengths in the circle:
The circumference: A closed plane curve
The radius: A given distance from a point within it called the center
In other words, the circle constant should be the ratio between the circumference and the radius, right? This number comes out to the number tau. Tau is in fact 2π, but it is called tau since it is quicker and easier to say and write it that way. There is no doubt that tau should be the circle constant.
What about all of pi’s trigonometric significance, and other properties involving circles. Pretty much all of them are made better with tau. Next week, I will show you that even things that are made more complicated with tau are still more natural or helpful.
However, I didn’t want to keep you waiting too long about Euler’s Identity. Here it is again:
e^iπ = -1
Its rewritten form that we call God’s Equation is below:
e^iπ + 1 = 0
What if we use tau? It looks like this:
e^iτ = 1
Pretty good! Tau just took the little ugly thing and fixed it. Some people are actually upset that we lost a little zero, since zero is significant too. Well, no worries!
e^iτ = 1 + 0
I actually found out something about the new version of Euler’s Identity that was really surprising, and mixes in another past cool math stuff post which can never be bad. Remember the very first post on imaginary numbers back in October when we learned about the cube roots of one? If you haven’t definitely check it out. It was a really awesome post. Just search “cube root” and it will come up.
This post taught that there are two square roots of one, three cube roots of one, four fourth roots of one, and so on. Well, turns out that these roots can be found with the new Euler’s Identity.
e^iτ/1 = 1
e^iτ/2 = -1
e^iτ/3 = -1/2 + i√3/2
e^iτ/4 = i
e^iτ/6 = 1/2 + i√3/2
e^iτ/8 = √2/2 + i√2/2
e^iτ/12 = √3/2 + i/2
When I saw that, all I was shocked that that actually worked, but it also makes tau such a beautiful number. Don’t get me wrong, e and i definitely get credit, but they don’t have such promising numbers to take their place.
Since tau has so many cool things about it, I am turning it into one of my little three week series, like I did with graphing calculators and at CTY. Even though it’s kind of late, have a happy half-tau day!