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. 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 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 integer n, is called De-Moivre’s Theorem.