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.