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A bit more algebra

By working with polynomials we can justify these definitions purely algebraically without doing any differentiation, which I hand-waved away as “a bit more algebra” in the post.

For example, from the angle sum identity

sin(ɑ + β) = sin ɑ cos β + cos ɑ sin β

we know that

sin(a + bε) = sin a cos bε + cos a sin bε

and then from the Taylor series

sin x = x - (x³ / 3!) + (x⁵ / 5!) - … and
cos x = 1 - (x² / 2!) + (x⁴ / 4!) - …

we can see that

sin bε = bε - ((bε)³ / 3!) + ((bε)⁵ / 5!) - … = bε - 0 + 0 - … = bε and
cos bε = 1 - ((bε)² / 2!) + ((bε)⁴ / 4!) - … = 1 - 0 + 0 - … = 1

just because ε² = 0, and so

sin(a + bε) = sin a + bε cos a

which is what we wanted. I think this is appealing because it depends only on knowing that ε² = 0, in the same way that addition & multiplication of dual numbers does, rather than any analytic or geometric intuition about infinitesimals.

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