Formulas and Results of Complex Numbers

1. $z = (a,b) = a + ib,{\text{ }}i = (0,1)$

2. $i = \sqrt { – 1} ,{\text{ }}{i^2} =- 1,{\text{}}{i^3} =- i,{\text{ }}{i^4} = 1,{\text{ }}{i^5} = i,{\text{ }} \ldots$

3. If $n$ is a positive integer, then ${(i)^{4n}} = 1,{\text{ }}{(i)^{4n + 1}} = i,{\text{ }}{(i)^{4n + 2}} =- 1,{\text{}}{(i)^{4n + 3}} =- i$

4. If $a + ib = 0$ then $a = b = 0$, and conversely

5. If $a + ib = c + id$ then $a = c$ and $b = d$

6. $(a,b) + (c,d) = (a + c,{\text{ }}b + d)$

7. $(a,b)(c,d) = (ac – bd,{\text{ }}ad + bc)$

8. ${z_1} + {z_2} = {z_2} + {z_1}{\text{ ;}}\forall {z_1},{z_2} \in \mathbb{C}$

9. ${z_1} \cdot {z_2} = {z_2} \cdot {z_1}{\text{ ;}}\forall {z_1},{z_2} \in \mathbb{C}$

10. ${z_1} + ({z_2} + {z_3}) = ({z_1} + {z_2}{\text{) + }}{{\text{z}}_{\text{3}}}{\text{ ;}}\forall {z_1},{z_2},{z_3} \in \mathbb{C}$

11. ${z_1} \cdot ({z_2} \cdot {z_3}) = ({z_1} \cdot {z_2}{\text{)}} \cdot {{\text{z}}_{\text{3}}}{\text{ ;}}\forall {z_1},{z_2},{z_3} \in \mathbb{C}$

12. $(0,0)$ is the additive identity.

13. $(1,0)$ is the multiplicative identity.

14. If $z = a + ib$ the multiplicative inverse of $z$ is ${z^{ – 1}} = \frac{a}{{{a^2} + {b^2}}} – i\frac{b}{{{a^2} + {b^2}}}$

15. The additive inverse of $z$ is $– z =- a – ib$

16. If $z = a + ib$, then $\overline z = a – ib$

17. $\overline {{z_1} + {z_2}}= \overline {{z_1}} + \overline {{z_2}} {\text{ ;}}\forall {z_1},{z_2} \in \mathbb{C}$

18. $\overline {{z_1} – {z_2}}= \overline {{z_1}} – \overline {{z_2}} {\text{ ; }}\forall {z_1},{z_2} \in \mathbb{C}$

19. $\overline {{z_1} \cdot {z_2}}= \overline {{z_1}}\cdot \overline {{z_2}} {\text{ ; }}\forall {z_1},{z_2} \in \mathbb{C}$

20. $\overline {\left( {\frac{{{z_1}}}{{{z_2}}}} \right)}= \frac{{\overline {{z_1}} }}{{\overline {{z_2}} }}{\text{ ; }}\forall {z_1},{z_2} \in \mathbb{C}$

21. If $\overline z = z$, then $z$ is a real number.

22. $\overline {\left( {\overline z } \right)}= z$

23. If $z = a + ib$, $a = \operatorname{Re} (z),{\text{ }}b = \operatorname{Im} (z)$

24. $z{\text{ }}\overline z= {(\operatorname{Re} z)^2} + {(\operatorname{Im} z)^2}$

25. If $z = a + ib$, the $\left| z \right| = \sqrt {{a^2} + {b^2}}$

26. $\left| z \right| \geqslant 0$

27. $\left| z \right| = \left| { – z} \right| = \left| {\overline z } \right|$

28. ${\left| z \right|^2} = z{\text{ }}\overline z$

29. $\left| {{z_1}{z_2}} \right| = \left| {{z_1}} \right|\left| {{z_2}} \right|$

30. $\left| {\frac{{{z_1}}}{{{z_2}}}} \right| = \frac{{\left| {{z_1}} \right|}}{{\left| {{z_2}} \right|}}{\text{, }}{z_2} \ne 0$

31. $\left| {{z_1}} \right| – \left| {{z_2}} \right| \leqslant \left| {{z_1} + {z_2}} \right| \leqslant \left| {{z_1}} \right| + \left| {{z_2}} \right|$

32. $\left| {{z_1} – {z_2}} \right| \geqslant \left| {{z_1}} \right| – \left| {{z_2}} \right|$

33. $\left| {\operatorname{Re} z} \right| \leqslant \left| z \right|$, $\left| {\operatorname{Im} z} \right| \leqslant \left| z \right|$

34. $\left| {{z_1} – {z_2}} \right| = \left| {{z_2} – {z_1}} \right|$

35. $\left| {\left| {{z_1}} \right| – \left| {{z_2}} \right|} \right| \leqslant \left| {{z_1} – {z_2}} \right|$

36. $z = r(Cos\theta + iSin\theta )$ is polar form of $z$, where $r = \left| z \right|{\text{ ;}}\theta = Ta{n^{ – 1}}\left( {\frac{b}{a}} \right) = \arg (z)$

37. If ${z_1} = {r_1}(Cos{\theta _1} + iSin{\theta _1})$ and ${z_2} = {r_2}(Cos{\theta _2} + iSin{\theta _2})$, then
• ${z_1}{\text{ }}{z_2} = {r_1}{r_2}[Cos({\theta _1} + {\theta _2}) + iSin({\theta _1} + {\theta _2})]$
• $\frac{{{z_1}}}{{{z_2}}} = \frac{{{r_1}}}{{{r_2}}}[Cos({\theta _1} – {\theta _2}) + iSin({\theta _1} – {\theta _2})]$
• $\arg ({z_1}{z_2}) = \arg {z_1} + \arg {z_2}$
• $\arg \left( {\frac{{{z_1}}}{{{z_2}}}} \right) = \arg {z_1} – \arg {z_2}$
1. $CiS\theta = Cos\theta + iSin\theta = {e^{i\theta }}$
2. ${(z)^0} = 1$
3. ${(z)^{m + 1}} = {z^m}z$
4. ${(z)^{ – m}} = {({z^{ – 1}})^m}{\text{, }}m \in {\mathbb{Z}^ + }$
5. ${({z^m})^n} = {(z)^{mn}}$
6. ${({z_1}{z_2})^n} = {({z_1})^n}{({z_2})^n}$
7. ${(Cos\theta + iSin\theta )^n} = Cos{\text{ }}n\theta + iSin{\text{ }}n\theta$. For all integers $n$ is called De-Moivre’s Theorem.