23.9: Appendix 23B - Complex Numbers

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A complex number $$z$$ can be written as a sum of a real number $$x$$ and a purely imaginary number $$iy$$ where $$i=\sqrt{-1}$$

$z=x+i y \nonumber$

The complex number can be represented as a point in the x-y plane as show in Figure 23B.1.

The complex conjugate $$\bar{z}$$ of a complex number z is defined to be

$\bar{z}=x-i y \nonumber$

The modulus of a complex number is

$|z|=(z \bar{z})^{1 / 2}=((x+i y)(x-i y))^{1 / 2}=\left(x^{2}+y^{2}\right)^{1 / 2} \nonumber$

where we used the fact that $$i^{2}=-1$$. The modulus $$|z|$$ represents the length of the ray from the origin to the complex number z in Figure 23B.1. Let $$\phi$$ denote the angle that the ray with the positive x -axis in Figure 23B.1. Then

$x=|z| \cos \phi \nonumber$

$y=|z| \sin \phi \nonumber$

Hence the angle $$\phi$$ is given by

$\phi=\tan ^{-1}(y / x) \nonumber$

The inverse of a complex number is then

$\frac{1}{z}=\frac{\bar{z}}{z \bar{z}}=\frac{x-i y}{\left(x^{2}+y^{2}\right)} \nonumber$

The modulus of the inverse is the inverse of the modulus;

$\left|\frac{1}{z}\right|=\frac{1}{\left(x^{2}+y^{2}\right)^{1 / 2}}=\frac{1}{|z|} \nonumber$

The sum of two complex numbers, $$z_{1}=x_{1}+i y_{1}$$ and $$z_{2}=x_{2}+i y_{2}$$ is the complex number

$z_{3}=z_{1}+z_{2}=\left(x_{1}+x_{2}\right)+i\left(y_{1}+y_{2}\right)=x_{3}+i y_{3} \nonumber$

where $$x_{3}=x_{1}+x_{2}$$, $$y_{3}=y_{1}+y_{2}$$. We can represent this by the vector sum in Figure 23B.2,

The product of two complex numbers is given by

$z_{3}=z_{1} z_{2}=\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)=\left(x_{1} x_{2}-y_{1} y_{2}\right)+i\left(x_{1} y_{2}+x_{2} y_{1}\right)=x_{3}+i y_{3} \nonumber$

where $$x_{3}=x_{1} x_{2}-y_{1} y_{2}$$ and $$y_{3}=x_{1} y_{2}+x_{2} y_{1}$$

One of the most important identities in mathematics is the Euler formula,

$e^{i \phi}=\cos \phi+i \sin \phi \nonumber$

This identity follows from the power series representations for the exponential, sine, and cosine functions,

$e^{i \phi}=\sum_{n=0}^{n=\infty} \frac{1}{n !}(i \phi)^{n}=1+i \phi-\frac{\phi^{2}}{2}-i \frac{\phi^{3}}{3 !}+\frac{\phi^{4}}{4 !}+i \frac{\phi^{5}}{5 !} \ldots \nonumber$

$\cos \phi=1-\frac{\phi^{2}}{2}+\frac{\phi^{4}}{4 !}-\ldots \nonumber$

$\sin \phi=\phi-\frac{\phi^{3}}{3 !}+\frac{\phi^{5}}{5 !}-\ldots \nonumber$

We define two projection operators. The first one takes the complex number $$e^{i \phi}$$ and gives its real part,

$\operatorname{Re} e^{i \phi}=\cos \phi \nonumber$

The second operator takes the complex number $$e^{i \phi}$$ and gives its imaginary part, which is the real number

$\operatorname{Im} e^{i \phi}=\sin \phi \nonumber$

A complex number $$z=x+i y$$ can also be represented as the product of a modulus $$|z|$$ and a phase factor $$e^{i \phi}$$

$z=|z| e^{i \phi} \nonumber$

The inverse of a complex number is then

$\frac{1}{z}=\frac{1}{|z| e^{i \phi}}=\frac{1}{|z|} e^{-i \phi} \nonumber$

where we used the fact that

$\frac{1}{e^{i \phi}}=e^{-i \phi} \nonumber$

In terms of modulus and phase, the sum of two complex numbers, $$z_{1}=\left|z_{1}\right| e^{i \phi_{1}}$$ and $$z_{2}=\left|z_{2}\right| e^{i \phi_{2}}$$, is

$z_{1}+z_{2}=\left|z_{1}\right| e^{i \phi_{1}}+\left|z_{2}\right| e^{i \phi_{2}} \nonumber$

A special case of this result is when the phase angles are equal, $$\phi_{1}=\phi_{2}$$ then the sum $$z_{1}+z_{2}$$ has the same phase factor $$e^{i \phi_{1}}$$ as $$z_{1}$$ and $$z_{2}$$

$z_{1}+z_{2}=\left|z_{1}\right| e^{i \phi_{1}}+\left|z_{2}\right| e^{i \phi_{1}}=\left(\left|z_{1}\right|+\left|z_{2}\right|\right) e^{i \phi_{1}} \nonumber$

The product of two complex numbers, $$z_{1}=\left|z_{1}\right| e^{i \phi_{1}}$$, and $$z_{2}=\left|z_{2}\right| e^{i \phi_{2}}$$ is

$z_{1} z_{2}=\left|z_{1}\right| e^{i \phi_{1}}\left|z_{2}\right| e^{i \phi_{2}}=\left|z_{1} \| z_{2}\right| e^{i \phi_{1}+\phi_{2}} \nonumber$

When the phases are equal, the product does not have the same factor as $$z_{1}$$ and $$z_{2}$$

$z_{1} z_{2}=\left|z_{1}\right| e^{i \phi_{1}}\left|z_{2}\right| e^{i \phi_{1}}=\left|z_{1}\right|\left|z_{2}\right| e^{i 2 \phi_{1}} \nonumber$

This page titled 23.9: Appendix 23B - Complex Numbers is shared under a not declared license and was authored, remixed, and/or curated by Peter Dourmashkin (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.