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squigs t1_j6ai3sm wrote

People always fixate on the "square root of -1" thing, but that's not what imaginary numbers are for.

Essentially, numbers give us forward. Negative Numbers give us backwards, and imaginary numbers give us "sideways"

Imaginary numbers give us numbers in 2 dimensions. We don't just look at imaginary numbers on their own, but as a pair. We have a "real" part and an "imaginary" part. Something like 3+4i .

Now, we can convert real numbers to imaginary numbers by multiplying by i. We're basically rotating them. What happens if I rotate something twice? We end up reversing it.

Another way of saying we rotate twice is we multiply by i twice. i x i. Or i². We get -1. Rotate again (multiply by i again), and we get i³ = -i. Rotate again and we're back to where we started. i⁴ = 1.

Like I said earlier, there's an imaginary part and a real part. So 3 + 4i. Rotate by multiplying by i we get 3i + 4 i² = 3i - 4.

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sirtimes t1_j6bxu2f wrote

This is probably the best explanation - it definitely helps to be visualizing the number line for this. I really enjoy the 3blue1brown YouTube series on this concept

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The_camperdave t1_j6c4gmm wrote

> Essentially, numbers give us forward. Negative Numbers give us backwards, and imaginary numbers give us "sideways"

So what gives us a vertical? SQRT(-i)?

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squigs t1_j6c9r2c wrote

This is where my answer is a bit messy. Imaginary numbers can be thought of that way, but using them for 2d space is an application.

We do have quaternions. This gives us 3 imaginary numbers, we call i, j and k, where squaring any of them gives -1, and ij = k, jk = i and ki =j. Weirdly, we lose commutivity here. ij = k, but ji = -k. Reversing the order gives a negative result.

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