Consider the use of tau (τ ~ 6.28) as a more natural unit for circular measures than pi (π ~ 3.14). I have a good colleague at school who counter-argues in this fashion: "But it's only a conversion by a factor of two, which should be trivial for us as mathematicians to deal with. And if our students can't handle that, then perhaps they shouldn't be in college."
Among the possible responses to this, here's a quick one (and specific to part of the curriculum that we teach): Scientific notation is a number written in the format \(a \cdot 10^b\). But imagine if instead we had defined it to be the format \(a \cdot 5^b\). The difference in the base is also only a factor of 2, but consider how much more complicated it is to convert between standard notation and this revised scientific notation.
Lesson: Consider your choice of basis carefully.
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But we only square or sometimes cube pi. We don't often use it with arbitrary exponents.
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