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Number Sets

# Complex Numbers

Set of complex numbers: $$\mathbb{C}$$
Imaginary unit: $$i$$
Complex numbers: $$z$$, $${z_1}$$, $${z_2}$$
Real numbers: $$a$$, $$b$$, $$c$$, $$d$$

Absolute value of a complex number: $$r$$, $${r_1}$$, $${r_2}$$
Argument of a complex number: $$\varphi$$, $${\varphi_1},$$ $${\varphi_2}$$
Whole numbers: $$k$$
Natural numbers: $$n$$

1. Algebraic form of complex number $$z = a + bi$$
2. Powers of the imaginary unit
1. Complex plane
1. Equality of complex numbers
$$a + bi = c + di$$, if $$a = c$$ and $$b = d$$
$$\left( {a + bi} \right) + \left( {c + di} \right) =$$ $$\left( {a + c} \right) + \left( {b + d} \right)i$$
3. Subtraction of complex numbers
$$\left( {a + bi} \right) – \left( {c + di} \right) =$$ $$\left( {a – c} \right) + \left( {b – d} \right)i$$
4. Multiplication of complex numbers
$$\left( {a + bi} \right)\left( {c + di} \right) =$$ $$\left( {ac – bd} \right) + \left( {ad + bc} \right)i$$
5. Division of complex numbers
$$\large\frac{{a + bi}}{{c + di}}\normalsize =$$ $${\large\frac{{ac + bd}}{{{c^2} + {d^2}}} + \frac{{bc – ad}}{{{c^2} + {d^2}}}\normalsize} i$$
6. Conjugate of a complex number $$\overline {a + bi} = a – bi$$
7. Modulus $$r$$ and argument $$\varphi$$ of a complex number
$$z = a + bi,$$ $$r = \sqrt {{a^2} + {b^2}} ,$$ $$\varphi = \arctan \large\frac{b}{a}\normalsize$$
1. Polar form of complex number
$$z = a + bi =$$ $$r\left( {\cos \varphi + i\sin \varphi } \right)$$
2. Product of complex numbers in polar form
$${z_1} \cdot {z_2} =$$ $${r_1}\left( {\cos {\varphi _1} + i\sin {\varphi _1}} \right) \cdot$$ $${r_2}\left( {\cos {\varphi _2} + i\sin {\varphi _2}} \right) =$$ $${r_1}{r_2}\big[ {\cos \left( {{\varphi _1} + {\varphi _2}} \right) }+$$ $${ i\sin \left( {{\varphi _1} + {\varphi _2}} \right)} \big]$$
3. Conjugate of a complex number in polar form
$$\overline {r\left( {\cos \varphi + i\sin \varphi } \right)} =$$ $$r\left[ {\cos \left( { – \varphi } \right) + i\sin \left( { – \varphi } \right)} \right]$$
4. Inverse of a complex number in polar form
$${\large\frac{1}{{r\left( {\cos \varphi + i\sin \varphi } \right)}}\normalsize} =$$ $${\large{\frac{1}{r}}\normalsize}\left[ {\cos \left( { – \varphi } \right) + i\sin \left( { – \varphi } \right)} \right]$$
5. Division of complex numbers in polar form
$${\large\frac{{{z_1}}}{{{z_2}}}\normalsize} =$$ $${\large{\frac{{{r_1}\left( {\cos {\varphi _1} + i\sin {\varphi _1}} \right)}}{{{r_2}\left( {\cos {\varphi _2} + i\sin {\varphi _2}} \right)}}}\normalsize} =$$ $${\large\frac{{{r_1}}}{{{r_2}}}\normalsize}\big[ {\cos \left( {{\varphi _1} – {\varphi _2}} \right) }+$$ $${ i\sin \left( {{\varphi _1} – {\varphi _2}} \right)} \big]$$
6. Exponentiation of a complex number
$${z^n} =$$ $${\left[ {r\left( {\cos \varphi + i\sin \varphi } \right)} \right]^n} =$$ $${r^n}\left[ {\cos \left( {n\varphi } \right) + i\sin \left( {n\varphi } \right)} \right]$$
7. De Moivre’s formula $${\left( {\cos \varphi + i\sin \varphi } \right)^n} =$$ $$\cos \left( {n\varphi } \right) + i\sin \left( {n\varphi } \right)$$
8. Nth root of a complex number
$$\sqrt[\large n\normalsize]{z} =$$ $$\sqrt[\large n\normalsize]{{r\left( {\cos \varphi + i\sin \varphi } \right)}} =$$ $$\sqrt[\large n\normalsize]{r}\Big( {\cos {\large\frac{{\varphi + 2\pi k}}{n}\normalsize} + i\sin \large\frac{{\varphi + 2\pi k}}{n}} \Big),\;$$ $$k = 0,1,2, \ldots ,n – 1$$
9. Euler’s formula $$\exp \left( {ia} \right) = \cos a + i\sin a$$
10. Exponential form of complex number $$z = r\exp \left( {i\varphi } \right)$$
11. $$\sin a = \large\frac{{\exp \left( {ai} \right) – \exp \left( { – ai} \right)}}{2}$$
12. $$\cos a = \large\frac{{\exp \left( {ai} \right) + \exp \left( { – ai} \right)}}{2}$$
13. $$\tan a = \large\frac{{\exp \left( {ai} \right) – \exp \left( { – ai} \right)}}{{\exp \left( {ai} \right) + \exp \left( { – ai} \right)}}$$
14. $$\cot a = \large\frac{{\exp \left( {ai} \right) + \exp \left( { – ai} \right)}}{{\exp \left( {ai} \right) – \exp \left( { – ai} \right)}}$$