Complex Numbers – 1

Selasa, 17 Maret 2020

Natural numbers: n

Imaginary unit: i

Complex number: z

Real part: a, c

Imaginary part: bi, di

Modulus of a complex number: r, r₁, r₂

Argument of a complex number: φ, φ₁, φ₂

1.         

i1 = i	i5 = i	i4n + 1 = i
i2 = – 1	i6 = – 1	i4n + 2 = – 1
i3 = – i	i7 = – i	i4n + 3 = – i
i4 = 1	i8 = i	i4n = i

2.  z = a + bi

3.  Complex Plane

                   

                                     Figure

4.  (a + bi) + (c + di) = (a + c) + (b + d)i

5.  (a + bi) – (c + di) = (a – c) + (b – d)i

6.  (a + bi) + (c + di) = (ac – bd) + (ad + bc)i

7.

8.  Conjugate Compex Numbers

9.  a = r cos φ, b = r sin φ

Figure

10. Polar Presentation of Complex Numbers

  a + bi = r (cos φ + i sin φ)

11. Modulus and Argument of a Complex Number

     If a + bi is a complex numbers, then

12. Product in Polar Representation

    z₁ ∙ z₂ = r₁ (cos φ₁ + i sin φ₁) ∙ r₂ (cos φ₂ + i sin φ₂)

               = r₁r₂ [cos (φ₁ + φ₂) + (i sin (φ₁ + φ₂)]

13. Conjugate Numbers in Polar Representation

        

    

14. Inverse Complex Numbers in Polar Representation

15. Quotint in Polar Representation

16. Power of a Complex Numbers

         z_n = [r (cos φ + i sin φ)]ⁿ = rⁿ [cos (nφ) + i sin (nφ)]

17. Formula “De Moivre”

  (cos φ + i sin φ)ⁿ = cos (nφ) + i sin (nφ)

18. Nth Root of a Complex Numbers

  

where

k = 0, 1, 2, ... , n – 1.

19. Euler’s Formula

  e^ix = cos x + i sin x

Sumber

Labels: Mathematician

Thanks for reading Complex Numbers – 1. Please share...!