Advanced complex numbers

Beyond the rectangular form a + bi, a complex number can also be written in polar form, using its distance from the origin (its modulus) and the angle it makes with the positive real axis (its argument); this makes multiplying and finding powers of complex numbers easier, since De Moivre's theorem says raising a complex number in polar form to a power just multiplies the angle by that power and raises the modulus to that power. The Fundamental Theorem of Algebra guarantees that every polynomial of degree n has exactly n roots once you count complex roots and repeated roots, which is why familiar identities like the difference of squares, a^2 - b^2 = (a+b)(a-b), extend cleanly to complex numbers too, letting you factor expressions like x^2 + 4 as (x+2i)(x-2i).

This explanation is AI-generated and can occasionally be wrong. 🚩 Report