and also ELI16 what a an imaginary number is and how it works in real life
In: Mathematics
It helps to think of numbers as vectors, having two values… That is, they have a magnitude and a direction. You can think of them as arrows. Magnitude has no sign. Positive numbers point 👉 Negative numbers point 👈 180 degrees off. When you add vectors, you put them tip to tail. so 5👉 (5) plus 3👈 (-3) ends up at 2👉 (2). When you multiply vectors, you multiply the magnitudes and add the directions. this is not intuitive! But it explains why multiplying a negative by a negative gives a positive — you add 180 degrees and 180 degrees and get 360 degrees which is 0 degrees. Or you can think of it as rotating the grid to he other direction. Imaginary numbers point ☝️. That’s it, just 90 degrees. All the other rules apply… In particular, multiplying an imaginary number by another imaginary number has you add their directions, 90 degrees and 90 degrees to get 180 degrees — that is, a negative number. Once you understand vectors visually, all those weird rules like a negative squared is positive, or i squared is -1 — they just make sense because multiplication has that rotation step they never talk about. The best part is these concepts extend right into complex numbers, linear algebra, etc.
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