Imaginary part

Fraktur I symbol

An illustration of the complex plane. The imaginary part of a complex number $z=x+iy$ is $y$.

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 $z=(x,y)$, or equivalently, $z=x+iy$, then the imaginary part of $z$ is $y$. It is denoted by $Im(z)$ or $\Im${z}, where $\Im$ is a capital I in the Fraktur typeface. The complex function which maps $z$ to the imaginary part of $z$ is not holomorphic.

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

For a complex number in polar form, $z=(r,\theta )$, or equivalently, $z=r(\cos \theta +i\sin \theta )$, it follows from Euler's formula that $z=re^{i\theta }$, and hence that the imaginary part of $re^{i\theta }$ is $r\sin \theta$.

In electric power, when a sine wave voltage drives a "linear" load (in other words, a load that makes the current also be a sine wave), the current $I$ in the power wires can be represented as a complex number $I=x+jy$ (engineers use $j$ to indicate the imaginary unit rather than $i$, which also represents current). The "real current" $x$ is related to the current when the voltage is maximum. The real current times the voltage gives the actual power consumed by the load (often all that power is dissipated as heat). The "imaginary current" $y$ is related to the current when the voltage is zero. A load with purely imaginary current (such as a capacitor or inductor) dissipates no power; it merely accepts power temporarily then later pushes that power back on the power lines.

See also

• Real part
• Imaginary number