Remember when you were in grammar school, and you were first learning about the square root, and how they told you that you couldn’t take the square root of a negative number? They lied.
OK, they didn’t exactly lie to you, though what they should have told you was “for now, you can’t take the square root of a negative number.” Eventually, though, you would have to come to grips with the fact that, sometimes, it was necessary to pretend that you could, like when you were faced with the equation
a2 + 4 = 0
In this case, you know that a2 = -4, so a = √-4. Now, understand, you still can’t take the square root of a negative number, but we can always imagine that we can. Now, √-4 = √-1 · √4. The square root of 4 is 2, and let’s call the square root of -1 i, the i standing for “imaginary.” (Saying that the i stands for “imaginary” might make a few mathematicians scowl, but screw ’em if they can’t take a joke.)
Numbers of the form a + bi, where a and b are real numbers, are called complex numbers, because they’re made up of a real part and an imaginary part. You can even plot them on graph paper, with the real part on the x axis and the imaginary part on the y axis. That makes your graph an Argand diagram.
That’s all I want to say about complex numbers, other than to say that they make it possible to solve some equations in science and engineering. Just take my word for it…