Comment by sunshowers

8 days ago

I'm not a professional, but to me it's clear that whether i and -i are "the same" or "different" is actually quite important.

10 comments

sunshowers

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impendia  8 days ago

I'm a professional mathematician and professor.

This is a very interesting question, and a great motivator for Galois theory, kind of like a Zen koan. (e.g. "What is the sound of one hand clapping?")

But the question is inherently imprecise. As soon as you make a precise question out of it, that question can be answered trivially.

  • HelloNurse  8 days ago

    Generally, the nth roots of 1 form a cyclic group (with complex multiplication, i.e. rotation by multiples of 2pi/n).

    One of the roots is 1, choosing either adjacent one as a privileged group generator means choosing whether to draw the same complex plane clockwise or counterclockwise.

grumbelbart  8 days ago

They would never be the same. It's just that everything still works the same if you switch out every i with -i (and thus every -i with i).

  • alexey-salmin  8 days ago

    There are ways to build C that result in:

    1) Exactly one C

    2) Exactly two isomorphic Cs

    3) Infinitely many isomorphic Cs

    It's not really the question of whether i and -i are the same or not. It's the question of whether this question arises at all and in which form.

    • zozbot234  8 days ago

      The question is meaningless because isomorphic structures should be considered identical. A=A. Unless you happen to be studying the isomorphisms themselves in some broader context, in which case how the structures are identical matters. (For example, the fact that in any expression you can freely switch i with -i is a meaningful claim about how you might work with the complex numbers.)

      2 replies →

boisterousness  6 days ago

They're different. Multiplication by i gives a quarter turn counterclockwise. -i gives a quarter turn clockwise.

Opposite quarter turns cancel: (-i)(i) = (-1)(i^2) = +1

Quarter turn twice counterclockwise gives a half turn: (i)(i) = -1

Quarter turn twice clockwise also gives a half turn: (-i)(-i) = -1

  • nyeah  6 days ago

    Sure. Either that or the reverse. "They're not the same" in the sense that they can't both be clockwise. "They are the same" in the sense that we could make either one clockwise.