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How do you prove the 3D complex number, ((sqrt(2)/2)i+(sqrt(2)/2)j)^2 = -1?

• So I'm extending complex numbers into 3D. Just like i is 1 unit north of the origin, j is 1 unit east of the origin, where j^2 = -1. Visually, I can see that ((sqrt(2)/2)i+(sqrt(2)/2)j)^2 = -1. However, when I factored it out, I got: -1+(1/2)ij+(1/2)ji. I know 3D complex numbers, multiplication isn't commutative, and visually, it makes sense that i*j = i, while j*i = j. So the equation would reduce to,-1+(1/2)i+(1/2)j which doesn't reduce to -1. Where have I gone wrong? Thanks!Add Photo/VideoSourceCancelAsk a questionusually answered in minutes!PhotoDetails

• Answer:

  You are confusing Cartesian coordinates with imaginary numbers. Coordinates are x, y, and z, all real numbers. Multiplying by i rotates a number line into the imaginary direction, not into another real coordinate.