Complex Numbers
Imaginary Unit
The imaginary unit, denoted by the symbol $i$ is a mathematical concept that is used to extend the number system to include the square root of -1. It is a fundamental concept in complex numbers.
$$\mathrm{Imaginary}\:\mathrm{unit}:\:\:\:{i}=\sqrt{−\mathrm{1}} \\ $$Important Relations in Complex Numbers
$$\mathrm{Complex}\:\mathrm{numbers} \\ $$ $$\mathrm{Imaginary}\:\mathrm{unit}:\:\:\:{i}=\sqrt{−\mathrm{1}} \\ $$ $$\mathrm{1}.\:\mathrm{Complex}\:\mathrm{number}\:{z}={a}+{ib},\:{a},\:{b}\:\mathrm{Real} \\ $$ $$\mathrm{2}.\:\mathrm{If}\:{z}_{\mathrm{1}} ={a}+{ib},\:\:{z}_{\mathrm{2}} ={c}+{id},\:\:\mathrm{then} \\ $$ $$\:\:\:\:\:\:{z}_{\mathrm{1}} +{z}_{\mathrm{2}} =\:\left({a}+{ib}\right)+\left({c}+{id}\right)=\left({a}+{c}\right)+{i}\left({c}+{d}\right) \\ $$ $$\:\:\:\:\:\:{z}_{\mathrm{1}} −{z}_{\mathrm{2}} =\:\left({a}+{ib}\right)−\left({c}+{id}\right)=\left({a}−{c}\right)+{i}\left({c}−{d}\right) \\ $$ $$\:\:\:\:\:\:{z}_{\mathrm{1}} \centerdot{z}_{\mathrm{2}} =\left({ac}−{bd}\right)+{i}\left({ad}+{bc}\right) \\ $$ $$\:\:\:\:\:\:\frac{{z}_{\mathrm{1}} }{{z}_{\mathrm{2}} }\:=\frac{{ac}+{bd}}{{c}^{\mathrm{2}} +{d}^{\mathrm{2}} }\:+{i}\:\frac{{bc}−{ad}}{{c}^{\mathrm{2}} +{d}^{\mathrm{2}} } \\ $$ $$\mathrm{3}.\:\mathrm{Real}\:\mathrm{and}\:\mathrm{Imaginary}\:\mathrm{Part} \\ $$ $$\:\:\:\:\:\:{z}={a}+{ib} \\ $$ $$\:\:\:\:\:\:\mathrm{Re}\left({z}\right)={a} \\ $$ $$\:\:\:\:\:\:\mathrm{Im}\left({z}\right)={b} \\ $$ $$\mathrm{4}.\:\mathrm{Conjugate}\:\mathrm{complex}\:\mathrm{numbers} \\ $$ $$\:\:\:\:\:\:\:{z}={a}+{ib} \\ $$ $$\:\:\:\:\:\:\:\mathrm{conjugate}\:\mathrm{of}\:{z}=\overset{−} {{z}}=\overline {{a}+{ib}}={a}−{ib} \\ $$ $$\mathrm{5}.\:\mathrm{Polar}\:\mathrm{representation} \\ $$ $$\:\:\:\:\:\:{z}={a}+{ib}={r}\left(\mathrm{cos}\:\varphi+{i}\mathrm{sin}\:\varphi\right) \\ $$ $$\:\:\:\:\:\:{r}=\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\:\:\:\left(\mathrm{modulus}\right) \\ $$ $$\:\:\:\:\:\:\varphi=\mathrm{arctan}\:\frac{{b}}{{a}}\:\:\left(\mathrm{argument}\right) \\ $$ $$\mathrm{6}.\:\mathrm{Modulus}\:{z}=\mid{z}\mid=\mid{a}+{ib}\mid=\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} } \\ $$ $$\mathrm{7}.\:\mathrm{Product}\:\mathrm{in}\:\mathrm{polar}\:\mathrm{notation} \\ $$ $$\:\:\:\:\:\:{z}_{\mathrm{1}} \centerdot{z}_{\mathrm{2}} ={r}{\mathrm{1}} \left(\mathrm{cos}\:\varphi{\mathrm{1}} +{i}\mathrm{sin}\:\varphi_{\mathrm{1}} \right)\centerdot{r}{\mathrm{2}} \left(\mathrm{cos}\:\varphi{\mathrm{2}} +{i}\mathrm{sin}\:\varphi_{\mathrm{2}} \right) \\ $$ $$\:\:\:\:\:={r}_{\mathrm{1}} \centerdot{r}_{\mathrm{2}} \left[\mathrm{cos}\:\left(\varphi_{\mathrm{1}} +\varphi_{\mathrm{2}} \right)+\mathrm{sin}\:\left(\varphi_{\mathrm{1}} +\varphi_{\mathrm{2}} \right)\right] \\ $$ $$\mathrm{8}.\:\mathrm{Conjugate}\:\mathrm{in}\:\mathrm{polar}\:\mathrm{notation} \\ $$ $$\:\:\:\:\:\:{z}_{\mathrm{1}} ={r}_{\mathrm{1}} \left(\mathrm{cos}\:\varphi_{\mathrm{1}} +{i}\mathrm{sin}\:\varphi_{\mathrm{1}} \right) \\ $$ $$\:\:\:\:\:\:\overset{−} {{z}}_{\mathrm{1}} ={r}_{\mathrm{1}} \left[\mathrm{cos}\:\left(−\varphi_{\mathrm{1}} \right)+{i}\mathrm{sin}\:\left(−\varphi_{\mathrm{1}} \right)\right] \\ $$ $$\mathrm{9}.\:\mathrm{Quotient}\:\mathrm{in}\:\mathrm{polar}\:\mathrm{notation} \\ $$ $$\:\:\:\:\:\:\frac{{z}_{\mathrm{1}} }{{z}_{\mathrm{2}} }\:\:=\:\frac{{r}_{\mathrm{1}} }{{r}_{\mathrm{2}} }\left[\mathrm{cos}\:\left(\varphi_{\mathrm{1}} −\varphi_{\mathrm{2}} \right)+\mathrm{sin}\:\left(\varphi_{\mathrm{1}} −\varphi_{\mathrm{2}} \right)\right] \\ $$ $$\mathrm{10}.\:\mathrm{Power}\:\mathrm{of}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{number} \\ $$ $$\:\:\:\:\:\:\:\:{z}^{{n}} =\left[{r}\left(\mathrm{cos}\:\varphi+{i}\mathrm{sin}\:\varphi\right)\right]^{{n}} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:={r}^{{n}} \left(\mathrm{cos}\:{n}\varphi+{i}\mathrm{sin}\:{n}\varphi\right) \\ $$ $$\mathrm{11}.\:\mathrm{Euler}’\mathrm{s}\:\mathrm{formula} \\ $$ $$\:\:\:\:\:\:\:\:\:{e}^{{ix}} =\mathrm{cos}\:{x}+{i}\mathrm{sin}\:{x} \\ $$ $$\mathrm{12}.\:\:\mathrm{De}\:\mathrm{Moiver}\:\mathrm{Formula} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\left(\mathrm{cos}\:\varphi+{i}\mathrm{sin}\:\varphi\right)^{{n}} =\mathrm{cos}\:{n}\varphi+\mathrm{sin}\:{n}\varphi \\ $$Roots of Complex Numbers
$$\mathrm{If}\:{z}={r}\left(\mathrm{cos}\:\varphi+{i}\mathrm{sin}\:\varphi\right),\:\mathrm{then} \\ $$ $$\:\:\:\:\sqrt[{{n}}]{{z}}=\sqrt[{{n}}]{{r}}\left(\mathrm{cos}\:\frac{\varphi+\mathrm{2}\pi{k}}{{n}}+\mathrm{sin}\:\frac{\varphi+\mathrm{2}\pi{k}}{{n}}\right),\:\mathrm{where} \\ $$ $$\:\:\:\:{k}=\mathrm{0},\mathrm{1},\mathrm{2},….,\left({n}−\mathrm{1}\right) \\ $$ $$ \\ $$Roots of Unity
$${z}=\mathrm{1}=\mathrm{cos}\:\mathrm{0}+{i}\mathrm{sin}\:\mathrm{0} \\ $$ $$\sqrt[{\mathrm{3}}]{\mathrm{1}}=\left(\mathrm{cos}\:\frac{\mathrm{2}\pi{k}}{\mathrm{3}}+{i}\mathrm{sin}\:\frac{\mathrm{2}\pi{k}}{\mathrm{3}}\right),\:{k}=\mathrm{0},\mathrm{1},\mathrm{2} \\ $$ $$\mathrm{Three}\:\mathrm{cube}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{unityare} \\ $$ $$\mathrm{1},\:\mathrm{cos}\:\frac{\mathrm{2}\pi}{\mathrm{3}}+{i}\mathrm{sin}\:\frac{\mathrm{2}\pi}{\mathrm{3}},\mathrm{cos}\:\frac{\mathrm{4}\pi}{\mathrm{3}}+{i}\mathrm{sin}\:\frac{\mathrm{4}\pi}{\mathrm{3}}\:\:\:\:\:\mathrm{or} \\ $$ $$\mathrm{1},\:\frac{−\mathrm{1}+{i}\sqrt{\mathrm{3}}}{\mathrm{2}},\:\frac{−\mathrm{1}−{i}\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$ $$\mathrm{Cube}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}\:\mathrm{are}\:\mathrm{also}\:\mathrm{denoted}\:\mathrm{by} \\ $$ $$\mathrm{1},\:\omega,\:\omega^{\mathrm{2}} \:\:\:\mathrm{where}\:\omega\:=\:\frac{−\mathrm{1}+{i}\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$Complex Number in Argand Plane
The Argand plane is a two-dimensional Cartesian coordinate system where the horizontal axis represents the real numbers, and the vertical axis represents the imaginary numbers. Thus, a complex number $z = a + bi$ can be represented as a point in the Argand plane, where “$a$” is the real part and “$b$” is the imaginary part.
In the Argand plane, the origin (0, 0) represents the complex number $0 + 0i$. The positive direction along the real axis corresponds to positive real numbers, while the positive direction along the imaginary axis corresponds to positive imaginary numbers.
![](https://www.tinkutara.com/wp-content/uploads/2023/06/argand.png)
Problems in Complex Numbers
Complex Number Problems: https://www.tinkutara.com/?s=complex
Algebra Problems: https://www.tinkutara.com/category/maths/algebra/
Physics
Chemistry
Computing