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Writing it equal to 0 isn't a hack, it's a common method of understanding a function. You factor polynomials by setting them equal to 0, for example. In the ca
by kej 10y ago
Writing it equal to 0 isn't a hack, it's a common method of understanding a function. You factor polynomials by setting them equal to 0, for example.
In the case of Euler's identity, what we're really asking is "what values of x make e^(ix) + 1 = 0 true?" and the answer is "every multiple of π". Using τ instead hides half of the answers.
- kazinator 10y agoYou mean "every multiple of 2π". The value -1 only comes around once per revolution of the unit circle!
- kazinator 10y agoYou mean "every multiple of 2π, shifted by π". The value -1 only comes around once per revolution of the unit circle! The solutions to e^(ix) + 1 = 0 are { π, 3π, 5π, ... } Whereas the solutions to e^(ix) - 1 = 0 are: { 0, 2π, 4π, 3π, ... } i.e. { 0, τ, 2τ, 3τ, ... } Also, note that when we set a polynomial to zero, the roots appear subtracted on the opposite side from the independent variable: (x - r0)(x - r1)...(x - rn) = 0 In the Tau-oriented Euler formula written homogeneously, there is a vague analogy to this since we're similarly subtracting that 1: e^(ix) - 1 = 0
- kej 10y agoPoor phrasing and math-before-coffee on my part. The point I was going for is that e^(ix) has real values for each multiple of π. That is mathematically interesting, and is obscured if you use τ instead.
- kazinator 10y agoYes; similarly, that a sinusoidal crosses zero twice in each period is obscured by a focus on the period itself.