Here's my followup question and answer. We know i is the square root of -1, but what is the square root of -i ? I've always thought that you'd need another dimension to describe that, and another dimension for the square root of that unit, and so on.
But no. (Let's tackle sqrt(i) as a simpler case first.) We can answer sqrt(i) in terms of rotation as described in the article. i is a rotation from the unit vector by 90°, so applying that twice turns 1 into -1. What operation applied twice would result in i? This just clicked: a 45° rotation. Thus the unit vector at a 45° angle is the square root of i: 0.5√2 + 0.5√2 * i.
The mathematical approach bears that out. Follow the rules of complex number arithmetic to square 0.5√2 + 0.5√2 * i (multiply it by itself) and you do indeed get i.
And we can solve sqrt(-i) the same way. -i is a 270° rotation from the unit vector 1. So the square root of -i is a 135° rotation, or -0.5√2 + 0.5√2 * i.
Finally, a 270° rotation is equivalent to a -90° rotation. So -45° should also be a square root of -i, and indeed it is. Multiplying 0.5√2 + -0.5√2 * i by itself also gives you -i. We've arrived back at the axiom that all numbers have two square roots of opposite signs. 135° and -45° are the same vector pointing in exactly opposite directions.
Last question: What's the cube root of i? Easy: a 30° rotation. The 30° unit vector is 0.5√3 + 0.5i, and cubing that does indeed get you i.
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Wow, I got this too.
Here's my followup question and answer. We know i is the square root of -1, but what is the square root of -i ? I've always thought that you'd need another dimension to describe that, and another dimension for the square root of that unit, and so on.
But no. (Let's tackle sqrt(i) as a simpler case first.) We can answer sqrt(i) in terms of rotation as described in the article. i is a rotation from the unit vector by 90°, so applying that twice turns 1 into -1. What operation applied twice would result in i? This just clicked: a 45° rotation. Thus the unit vector at a 45° angle is the square root of i: 0.5√2 + 0.5√2 * i.
The mathematical approach bears that out. Follow the rules of complex number arithmetic to square 0.5√2 + 0.5√2 * i (multiply it by itself) and you do indeed get i.
And we can solve sqrt(-i) the same way. -i is a 270° rotation from the unit vector 1. So the square root of -i is a 135° rotation, or -0.5√2 + 0.5√2 * i.
Finally, a 270° rotation is equivalent to a -90° rotation. So -45° should also be a square root of -i, and indeed it is. Multiplying 0.5√2 + -0.5√2 * i by itself also gives you -i. We've arrived back at the axiom that all numbers have two square roots of opposite signs. 135° and -45° are the same vector pointing in exactly opposite directions.
Last question: What's the cube root of i? Easy: a 30° rotation. The 30° unit vector is 0.5√3 + 0.5i, and cubing that does indeed get you i.