Complex Numbers Summary

Complex Analysis · Axiom Academy

Unit 1 recap — the imaginary unit, the complex plane, arithmetic, modulus and argument, the conjugate, and polar form. The imaginary unit satisfies i^2 = -1 , and its powers cycle with period 4 : Every complex number z = a + bi is a point in the complex plane — real part on the horizontal axis, imaginary part on the vertical. Add and subtract componentwise; multiply by expanding (FOIL) and using i^2 = -1 ; divide by multiplying top and bottom by the conjugate. The modulus is the distance to the origin; the argument is the angle from the positive real axis. In polar form , multiplication multiplies the moduli and adds the arguments — a scaling plus a rotation. Core Concept The Imaginary Unit i Defining i as a square root of -1 extends the reals so that every equation like x^2 + 1 = 0 has a solution. A complex number is then z = a + bi with a,b real. Reduce a power: take the exponent (e.g. i^ 27 =i^ 3 =-i ). Core Concept The Complex Plane Each complex number is a point (or a vector from the origin) on the Argand diagram: the horizontal axis carries the real part a , the vertical axis the imaginary part b . Real axis: numbers with b = 0 . Imaginary axis: numbers with a = 0 . Add and subtract by combining real with real and imaginary with imaginary. Multiply by expanding and replacing i^2 with -1 . Divide: multiply top and bottom by the conjugate to clear i from the denominator. Watch out for: the -bd term — it comes from i^2 = -1 . Core Concept Modulus & Argument

This is the written version of the interactive lesson above. See the full Complex Analysis course.