Complex Numbers

To begin, please make note of the fact that the name “imaginary number” might give you the impression that these numbers don’t mean anything — that is very far from the truth!  Instead it indicates that there was once a time when people didn’t know what they could measure, so one person in particular (Decartes — remember him?) disparaged them as imaginary.  He was wrong.  They are as meaningful as the real numbers, just in a slightly different context.  They help us to measure many things relating to electricity, and they give us a beautiful and elegant way to change our perspective — to rotate.

A term for the number system we will now begin studying is the “complex numbers.”   For historical reasons, we still call the part of the number that denotes the root of -1 imaginary.


  1. \rhd Introduction to the Imaginary Numbers (5:20) Introduces the number whose square is -1, called i, which is *not* wacky, in spite of Sal’s claim
  2.  \rhd Simplifying roots of negative numbers (4:04) Simplify numbers like \sqrt{-52}.
  3. \star Simplifying roots of negative numbers
  4. \rhd Powers of the imaginary unit (6:20) Shows how to compute any integer power of i, such as i^{2000}.
  5. \rhd i as the principal root of -1 (6:45) Illustrates the hugely important sequence in which we must simplify radicals including negative numbers and why
  6. \rhd Introduction to Complex Numbers (4:43) Adding real and imaginary numbers to get complex numbers