Imaginary part - Definition and Overview

In mathematics, the imaginary part of a complex number z is the second element of the ordered pair of real numbers representing z, i.e. if <math>z=x+\mathrm{i}y<math> then the imaginary part of <math>z<math> is <math>y<math>.

In terms of the complex conjugate <math>\bar{z}<math>, the imaginary part of z is equal to <math>\frac{z-\bar{z}}{2\mathrm{i}}.<math>

The imaginary part of <math>z<math> is denoted <math>\mbox{Im}z<math> or <math>\Im z<math>. It is not holomorphic.

If a complex number is written as <math>re^{\mathrm{i}\theta}<math>, then the imaginary part is <math>r\sin\theta<math>. (See Euler's formula.)

See also

• Real part
• Imaginary number
• Complex number