The Complex Plane and Multiplication by i

Start with why i times i equals -1 at all. It is not an accident, it is a definition built to make rotation work. If multiplying by i once turns a point 90 degrees, multiplying by i twice should turn it 180 degrees - straight to the opposite side of the origin. Landing on -1 (and in general on the negative of whatever you started with) is exactly what a 180 degree turn does. So i times i = -1 is the rotation behaving consistently, not a random rule.

To see the rotation, you need a plane first. A complex number a + bi gets plotted with a as the horizontal coordinate and b as the vertical one, just like (a, b) in ordinary coordinate geometry, except the vertical axis tracks the imaginary part instead of a second real variable. Once a + bi is a point, multiplying by i has an algebraic effect you can compute directly: i(a + bi) = ai + bi² = ai - b = -b + ai. The new point is -b + ai. Compare coordinates: the old horizontal value a became the new vertical value, and the old vertical value b became the new horizontal value, flipped in sign. That swap-and-flip is precisely a 90 degree counterclockwise turn about the origin.

Because rotation only changes direction, never distance, multiplying by i never changes how far a point sits from the origin. That distance, √(a² + b²), is called the modulus, and it connects this topic to the next idea the test cares about: treating complex numbers as vectors with length and direction rather than just algebraic expressions.