Operations with Complex Numbers

Adding complex numbers is easy because you just combine like parts: real with real, imaginary with imaginary, nothing crosses over. Multiplying is different because every term in the first number has to touch every term in the second. That means four products instead of two, the same FOIL pattern you'd use on any two binomials. The twist is the last term, which comes out as i times i. Since i² = -1, that term flips from imaginary to real. So a process that starts with four separate terms ends up with only two: a real part and an imaginary part, because one of the four pieces secretly belonged with the real numbers all along.

Division has no direct move the way multiplication does, so it borrows a trick: multiply top and bottom by the conjugate of the denominator. The conjugate just flips the sign of the imaginary part, and when you multiply a complex number by its own conjugate, the cross terms cancel and the i² term turns positive, leaving (real part)² + (imaginary part)², a plain real number. That clears i out of the denominator completely, so you're left dividing a complex numerator by an ordinary real number, which splits cleanly into a + bi form.

Both operations lean on the same fact, i² = -1, just used in different directions: multiplication uses it to simplify, division uses it to rationalize. This sits right next to the rest of the number-system questions on the test, where you're expected to treat i like a variable that obeys one extra rule, then clean up at the end.