imaginary number

An imaginary number is any real multiple of the imaginary unit i — numbers like 3i, -7i, or 0.5i. They are the numbers you land on when you scale i up or down, just as the real numbers are what you get by scaling 1. The name “imaginary” is a historical leftover that sounds spookier than it is; these numbers are perfectly legitimate and show up constantly in physics and engineering.

Concretely, an imaginary number has the form bi where b is a real number. Its square is always a negative real number or zero: (bi)^2 = b^2 times i^2 = -b^2. So squaring an imaginary number can never give a positive result — that is exactly the gap in the real numbers that i was built to fill.

A subtle point about terminology: some authors call bi a “pure imaginary number” to stress that there is no real part attached. Whether 0 itself counts as imaginary is a matter of convention; it is both real and imaginary since 0 = 0i.