Chapter 5
Complex Numbers
Once, taking five from three had no answer, until negative numbers were admitted. Once, no fraction squared to exactly two, until irrational numbers joined. Squaring hits a wall of that kind now: multiply any number you know by itself and the result never comes out negative, so an equation as short as x squared plus one equals zero stalls there. Every earlier wall came down by enlarging the supply of numbers. Doing that once more brings in the complex numbers, and what one of them can do is not obvious at all.
What You'll Explore
4 lessons.
- The Imaginary Unit and Complex Numbers
Suppose one new number is admitted, with no place on the number line and a single job to do. Everything else has to be built from it: what such numbers look like, how they behave under repeated multiplication, and when two of them count as the same.
- Arithmetic of Complex Numbers
Addition and subtraction cause little trouble once numbers carry two parts, but a product raises a question: does multiplying two of them give back a number of the same kind, or does something new appear? Division sharpens that question, since its answer has to be written in two parts as well.
- The Complex Plane and Modulus
Real numbers each have a spot on a line, but a number with two parts has nowhere obvious to sit. Give it a whole plane instead and the operations from the last lesson start to look like movements you can see. Distance becomes a question worth asking too: how far is such a number from zero?
- Complex Roots of Quadratics
Every quadratic that once had no solutions is still sitting there, unfinished. Now that the square root of a negative has a meaning, the formula can run all the way through, and the answers it returns arrive in a pattern that is hard to miss once you have seen a few.