Opposite Directions: Addition vs. Multiplication

[forcefield]

Instead of i² = -1, is it useful to think that (-i)i = 1, i.e. opposites cancel each others, kind of doing an addition instead of the multiplication ? If I replace i with "direction" it kind of makes sense. (I like to think that this is not just math but also physics.)

 

[bhobba]

The most useful way to look on i is rotation in the complex plane by 90%. -i is rotation through -90% - so lo and behold multiply them together and you get rotation through 0% - ie 1.

Thanks
Bill

 

[HallsofIvy]

I'm not sure what you mean by "useful" here. Certainly either of the two statements implies the other. I do think you are making a mistake saying "I like to think that this is not just math but also physics." Mathematics is NOT physics until you add some 'physical' information. "Direction" itself is a geometric concept, not physics. There is, of course, geometry involved in complex numbers. We can think of real numbers as giving a number line while complex numbers give us the complex plane with "real" and "imaginary" axes perpendicular to one another.

 

[arildno]

I must strongly discourage you from this type of thinking:
" opposites cancel each others"

Why?
Because your words "opposites" and "cancel each other" have totally different meanings, but in the vagueness of your language seem to be the same.

Argument:
1. How can -i and i be called "opposites"?
Only in the sense if you ADD them, you get the result 0

2. How can (-i)*i=1 be said to "cancel each other"?
Only in the sense that when you MULTIPLY them, you get 1.

Thus, in the vagueness of your language, you a) blur the distinction between addition and multiplication, and b) blur the distinction between the numbers 0 and 1.
----
The proper way is to keep these distinctions explicit, in that you call 0 "the additive neutral element" and 1 the "multiplicative neutral element", and also calls -a "the additive inverse of a" and 1/a "the multiplicative inverse of a"

-i is EQUAL to 1/i, i.e for complex numbers, it is true that the additive inverse of "i" equals the multiplicative inverse of i.
But, this is not a general feature of complex numbers, just a special case.

 

[economicsnerd]

That never occurred to me as a cute way of expressing $\{i,-i\}\subseteq\mathbb C$. They're the unique pair of complex numbers which are both multiplicative inverses and additive inverses. Of course, that's just a mild rephrasing of $$\{z\in \mathbb C:\enspace z^2+1=0\}=\{i,-i\}$$ but still a cute way of saying it.

 

[1MileCrash]

That never occurred to me as a cute way of expressing $\{i,-i\}\subseteq\mathbb C$. They're the unique pair of complex numbers which are both multiplicative inverses and additive inverses. Of course, that's just a mild rephrasing of $$\{z\in \mathbb C:\enspace z^2+1=0\}=\{i,-i\}$$ but still a cute way of saying it.

I agree, it's rather nice.