How do u make a m look like a 3..............Dr. King..............give me a R........an L.........and a 16 with an eye in it.........F (i).......Italian seafood...................3 spins........in the counterclockwise direction.............a backwards clock..........the devils code..............i cubed......................which is - square root of - 1........or just - i........................

If we multiply by -i twice, the first multiplication would turn 1 into -i, and the second turns -i into -1. So there’s really two square roots of -1: i and -i.

This is pretty cool. We have some sort of answer, but what does it mean?

• i is a “new imaginary dimension” to measure a number
• i (or -i) is what numbers “become” when rotated
• Multiplying i is a rotation by 90 degrees counter-clockwise
• Multiplying by -i is a rotation of 90 degrees clockwise
• Two rotations in either direction is -1: it brings us back into the “regular” dimensions of positive and negative numbers.

Numbers are 2-dimensional. Yes, it’s mind bending, just like decimals or long division would be mind-bending to an ancient Roman. (What do you mean there’s a number between 1 and 2?). It’s a strange, new way to think about math.

We asked “How do we turn 1 into -1 in two steps?” and found an answer: rotate it 90 degrees. It’s a strange, new way to think about math. But it’s useful. (By the way, this geometric interpretation of complex numbers didn’t arrive until decades after i was discovered).

Also, keep in mind that having counter-clockwise be positive is a human convention — it easily could have been the other way.

Finding Patterns

Let’s dive into the details a bit. When multiplying negative numbers (like -1), you get a pattern:

• 1, -1, 1, -1, 1, -1, 1, -1

Since -1 doesn’t change the size of a number, just the sign, you flip back and forth. For some number “x”, you’d get:

• x, -x, x, -x, x, -x…

This idea is useful. The number “x” can represent a good or bad hair week. Suppose weeks alternate between good and bad; this is a good week; what will it be like in 47 weeks?

So -x means a bad hair week. Notice how negative numbers “keep track of the sign” — we can throw (-1)^47 into a calculator without having to count (”Week 1 is good, week 2 is bad… week 3 is good…“). Things that flip back and forth can be modeled well with negative numbers.