> So let’s picture multiplication by –1 as half a rotation, anticlockwise, around a circle (in our case, the circle passes through 1 and –1). It’s actually a rotation by 180 degrees.
Sorry but this just didn't work for me. My mental model is to use +/- as directional indicators. For me multiplication by -1 is same as multiplication by +1 but in the opposite direction.
I would rather think of "i" as rotation by 90°. In this realm +/- stand for anti/clockwise. So +/- continue to stand for direction and "i" tells me if I need to rotate or not.
It neatly lines up with different operations. As an example (+i) × (-i) == -i² == 1. In terms of movements you are rotating 90° anti-clockwise followed by 90° clockwise bringing you back where you started.
On the other hand, Nautilus's article is bit too much of mental gymnastics for me to follow their reasoning.
> So let’s picture multiplication by –1 as half a rotation..It’s actually a rotation by 180 degrees..What happens if we only do half of this rotation? It’s halfway to multiplying by –1, which you can think of as the same as multiplying by √–1...
I'm curious. I seem to be missing something, because I feel like what you're saying tracks very much with the "alternative interpretation". That is, I don't see the difference (in my mind's eye) at all. Is there maybe a different way to explain what you perceive to be critically different?
For me it is about expressing a concept as compactly as possible.
In case of Nautilus they first ask us to imagine -1 as 180° and then proceed say half of it as 90° rotation and finally give it a name i. For some this may work fine as a way to understand. However it gets awkward to further explain. If 𝑖 stands for half of -1 (as they say) then does that mean 𝑖 == -(1/2)? If not then why not? and so on.
Compare that with:
𝑖 is a unit of rotation which is 90°. That's it. Using this as a base it's easier to explain why 𝑖² = -1 (two units of anti-clockwise rotation) or why 𝑖³ = -i (three units of anti-clockwise rotation is same as one unit of clockwise rotation) and so on.
More fundamentally, they ask us to imagine an operation on 1-d (-1) as happening through 2-d (180° rotation). It seems convoluted way to introduce 𝑖. I'd rather explain 𝑖 from the first principles and proceed to show its consequences such as 𝑖² = -1.
Thank you for your response. I see where you are coming from. I think because I don't interpret their text as saying that i is "half of -1" but "half of the rotation that would lead to -1" I consider your description equivalent. In fact I am struggling to interpret their text to say "half of -1". Regardless, I see your point.
Sorry but this just didn't work for me. My mental model is to use +/- as directional indicators. For me multiplication by -1 is same as multiplication by +1 but in the opposite direction.
I would rather think of "i" as rotation by 90°. In this realm +/- stand for anti/clockwise. So +/- continue to stand for direction and "i" tells me if I need to rotate or not.
It neatly lines up with different operations. As an example (+i) × (-i) == -i² == 1. In terms of movements you are rotating 90° anti-clockwise followed by 90° clockwise bringing you back where you started.
On the other hand, Nautilus's article is bit too much of mental gymnastics for me to follow their reasoning.
> So let’s picture multiplication by –1 as half a rotation..It’s actually a rotation by 180 degrees..What happens if we only do half of this rotation? It’s halfway to multiplying by –1, which you can think of as the same as multiplying by √–1...