imaginary part

The imaginary part of a complex number is the real number multiplying i. In a + bi, the imaginary part is b. Carrying the map picture forward, if the real part says how far east or west a number sits, the imaginary part says how far north or south — it is the vertical coordinate.

The standard notation is Im(z), and for z = a + bi we have Im(z) = b. Crucially, the imaginary part is the coefficient b, a real number — it is not bi. So the imaginary part of 3 + 4i is 4, not 4i. A complex number is purely imaginary exactly when its real part is zero and its imaginary part is not.

This convention trips people up constantly, so it is worth saying twice: Im(z) strips off the i and returns the plain real coefficient underneath.