# Complex Numbers Properties

The properties of complex numbers is important in mathematics.

List of Properties in Complex Numbers
1. √(-1) = i
2. -1 = i
3. z = a + ib is a complex number.
Real part of z = a & imaginary part of z = b
4. Negative of z = -z = - (a + ib) = -a - bi.
5. Conjugate of z = zbar = a - ib.
6. z.zbar = |z2| = a2 + b2
7. |z| = |zbar|
8. Zbarbar = z.
9. z = zbar -> purely real.
10. |z| = √(a2 + b2)
11. Rez = (z + zbar) / 2, Im z = (z - zbar) / 2
12. Rez <= |z|, Imz <= |z|
13. |z1.z2| = |z1||z2|
14. |( z1/z2)| = |z1| / |z2|
15. Triangle inequality: for any two complex numbers
z1,z2, |z1+z2|<=|z1|+|z2|.
16. |z1 - z2| >= ||z1| - |z2||
17. z = x + iy then |z| = √(x2 + y2) , arg z = tan-1 (y / x)
polar form: z = r (cos θ + isin θ)
18. arg (z1 z2) = arg z1 + argz2
19. arg (z1 / z2) = arg z1 - arg z 2
20. Euler's formula: eiθ = cos θ + isin θ
21. General rule for determining the argument θ, 22. Cube root of unity are
1, ( - 1 + i√3) /2, ( - 1 - i√3) /2
23. Fourth roots of unity are 1, i, -1, -i
24. Cube roots of unity form the vertices of an equilateral triangle.
25. Fourth roots of unity form the vertices of a square all lying on the unit circle.
26. De-Moivre's theorem: for any rational number n, cos n θ + isin n θ is a value or one of the value of (cos θ + isin θ)n.
27. Sixth roots of unity form the vertices of a hexaon all lying on the unit circle.
28. nth roots of unity form the vertices of a n-sided regular polygon all lying on the unit circle.