I am going into Honors Algebra II and while I am fine at using imaginary numbers in a formulaic sense I never understood them conceptually. I can’t tell if they exist just to make math work better or because there is an actual logical way to understand them
In: 16
Another way to think about them: you know that when you multiply by a positive number, you stretch or shrink the number line. When you multiply by -1 you flip the number line over. If you flip it over twice you get back where you started, because (-1)*(-1) = 1.
So, what could you do that if you do it *twice* you flip the number line over? Give it a quarter turn! So, turn the number line by π/2 radians (90º) and now you have the imaginary axis. Of course, now there’s no reason not to allow sums of one real and one imaginary number, which gives you the complex plane.
Everything we said before still works: multiplying by a positive real number scales the whole plane. Multiplying by -1 flips everything over. And multiplying by i rotates the plane by a quarter turn.
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