There is no real number which squares to give \(-1\). However, we can define an object which squares to \(-1\). This is traditionally called \(i\) in mathematics and \(j\) in physics and engineering (to avoid confusion with \(i\) meaning an electrical current), so \(i^2=-1\). Thus we find that

  • the square roots of \(-1\) are \(i\) and \(-i\)
  • the square roots of \(-4\) are \(2i\) and \(-2i\)

and so on.

We call real multiples of \(i\) (that is, \(ai\) for some real number \(a\)) imaginary numbers.

We can visualise imaginary numbers as lying along a number line like the ordinary real number line, but at right angles to it. These two number lines together form a plane called the complex plane.