Why Tau is Better Than Pi
First of all, this post was inspired by ViHart, if you don't know who she is, she's got some pretty interesting mathematical videos on youtube, so check that out.
Ok. Go back to when you were in elementary school and you had to memorize all these equations with pi, and you sort of got it, but you didn't really understand. Well, that's not your problem. It's not the teacher's problem either. It's pi's problem. The problem with pi and pi day is just like the problem with Christopher Columbus and Columbus Day. Sure, he was a real man that did some stuff, but he DIDN'T discover America, he DIDN'T discover that the world was round, and, let's face it, he was kind of a bad person.
Radians are a very convenient way of measuring angles. Two pi radians in a circle, pi radians a half circle. Now say I'm baking I pie, and I invite my friends over to have some. "How much pi do you want?" Your friend wants a tenth of the pi, so he says "one tenth of a pi". But since my friend and I are both extremely mathematical, I'm greatly confused whether he/she means one tenth of a radian, in which case would be one twentieth of a pi, or just one tenth of a pi. And yes, someone with common sense unlike me would clearly say the one tenth of a pi is correct, BUT WE'RE TRYING TO KEEP THIS SIMPLE RIGHT NOW.
One way we could resolve this is by just defining pi as 2pi but that makes everything even more confusing and every time someone used the word pi they would have to specify which pi they meant. So we use another greek letter, tau. Tau is approximately 6.2831853072, or 2pi.
Okay, here. You memorize the sine of pi radians and stuff like that so its easier to graph the sine wave (or use your calculator) but it doesn't have to be that way. Tau will make trigonometry much prettier. Let's try to draw a sin wave with tau units. At x=0, the y is clearly 0 too. When x = tau/4, y is 1 because we've gone a quarter of a circle over (unit circles, trigonometry). When x = tau/2, y is 0 again because we went 180 degrees around and at x= 3/4 tau, y is -1 because we've gone 90 degrees counterclockwise and yeah.
Why do we use pi? A circle is defined by its radius, there really is no correlation between the diameter and the circle besides the fact that the diameter is twice the radius. So why do we use pi when that is actually the ratio between the circumference and the diameter and not just use tau which is the ratio between the circumference and the radius and make everything easier? Because we were taught that way.
And you might say, well, doesn't this ruin Euler's Theorem because then e to the power of i multiplied by tau over 2 would be -1 which makes everything worse? Well, no. Euler's Theorem is derived from Euler's identity which is that (by the way, Euler is pronounced oiler) e to the power of i multiplied by theta is equal to the sum of cosine theta and i multiplied by sine theta. I know this is confusing, but here it is:
ei x = cos( x ) + i sin( x )
If we use tau for theta, we get the wonderful equation that e to the power of i multiplied by tau is 1.
And here's a special math puzzle:
Let
T = the infinite sum 1+2+4+8....
Let's see what 2T would be.
2T = 2+4+8+16...
As you can see, 2T looks exactly the same as T except with the one lopped off, since the series are infinite.
Thus, T = 2T+1.
The only solution for T is -1. Could a infinite series turn into a negative number? What's wrong with my reasoning?
Comment down below.
Hope you enjoyed this post,
Owen
Ok. Go back to when you were in elementary school and you had to memorize all these equations with pi, and you sort of got it, but you didn't really understand. Well, that's not your problem. It's not the teacher's problem either. It's pi's problem. The problem with pi and pi day is just like the problem with Christopher Columbus and Columbus Day. Sure, he was a real man that did some stuff, but he DIDN'T discover America, he DIDN'T discover that the world was round, and, let's face it, he was kind of a bad person.
Radians are a very convenient way of measuring angles. Two pi radians in a circle, pi radians a half circle. Now say I'm baking I pie, and I invite my friends over to have some. "How much pi do you want?" Your friend wants a tenth of the pi, so he says "one tenth of a pi". But since my friend and I are both extremely mathematical, I'm greatly confused whether he/she means one tenth of a radian, in which case would be one twentieth of a pi, or just one tenth of a pi. And yes, someone with common sense unlike me would clearly say the one tenth of a pi is correct, BUT WE'RE TRYING TO KEEP THIS SIMPLE RIGHT NOW.
One way we could resolve this is by just defining pi as 2pi but that makes everything even more confusing and every time someone used the word pi they would have to specify which pi they meant. So we use another greek letter, tau. Tau is approximately 6.2831853072, or 2pi.
Okay, here. You memorize the sine of pi radians and stuff like that so its easier to graph the sine wave (or use your calculator) but it doesn't have to be that way. Tau will make trigonometry much prettier. Let's try to draw a sin wave with tau units. At x=0, the y is clearly 0 too. When x = tau/4, y is 1 because we've gone a quarter of a circle over (unit circles, trigonometry). When x = tau/2, y is 0 again because we went 180 degrees around and at x= 3/4 tau, y is -1 because we've gone 90 degrees counterclockwise and yeah.
Why do we use pi? A circle is defined by its radius, there really is no correlation between the diameter and the circle besides the fact that the diameter is twice the radius. So why do we use pi when that is actually the ratio between the circumference and the diameter and not just use tau which is the ratio between the circumference and the radius and make everything easier? Because we were taught that way.
And you might say, well, doesn't this ruin Euler's Theorem because then e to the power of i multiplied by tau over 2 would be -1 which makes everything worse? Well, no. Euler's Theorem is derived from Euler's identity which is that (by the way, Euler is pronounced oiler) e to the power of i multiplied by theta is equal to the sum of cosine theta and i multiplied by sine theta. I know this is confusing, but here it is:
ei x = cos( x ) + i sin( x )
And here's a special math puzzle:
Let
T = the infinite sum 1+2+4+8....
Let's see what 2T would be.
2T = 2+4+8+16...
As you can see, 2T looks exactly the same as T except with the one lopped off, since the series are infinite.
Thus, T = 2T+1.
The only solution for T is -1. Could a infinite series turn into a negative number? What's wrong with my reasoning?
Comment down below.
Hope you enjoyed this post,
Owen
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