On 6/28 this year I introduced quite a few people to the circle constant tau, as I usually do. Surfing around on the web there were quite a few articles explaining why tau is better than pi, and there’s always the Tau Manifesto. But none of them cut to the chase in quite the way I wanted. There is one reason that stands out to me as by far the most important reason to start using tau as a circle constant, and that is that it makes basic trigonometry way more intuitive to learn. Here’s why.
If you think back to advanced algebra in high school you might remember having to master radian measures of angles. Instead of 30 degrees, you had to use pi/6. Instead of 90 degrees, pi/2. Your teacher may have given you a list of common angles to memorize in pi radians so that you could convert these angles easily from one to the other. Very likely, you found this confusing and frustrating. As you finally mastered it, you may have been able to visualize these fractions of pi as fractions of a half a circle. Stop and ponder that for a moment… fractions of a half of a circle?
Radians measure angles by tracing out a fraction of the circumference on the unit circle. 3.14159… radians gets you half way around the unit circle. 6.2831… radians on the other hand, gets you all the way around the circle! This is tau, equal to 2 times pi. If you use tau as your circle constant then angles measured in radians suddenly become easy. Now instead of thinking of fractions of a half of a circle, you just have to think of fractions of a circle!
I have a fond memory of one of the first times I introduced tau to a student I was tutoring. We met in the evening and she told me that she had come away from her math class that day feeling confused. They were beginning to study radians. I asked her if she wanted to know a better way of writing these radian angles even if it was going to be different than the way her teacher was writing it. She agreed so I showed her tau radians and how they related to pi radians. After a very short explanation she pointedly picked up her pencil and proceeded to cross out her entire two pages of notes from class that day, in order to re-do them using tau. Ah, the delights of elegant and intuitive notation!
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