Calculating with Complex numbers
Como representar números complexos
Os números complexos podem ser representados de três formas: a forma algébrica, forma trigonométrica e com a fórmula de Euler, com todas as suas vantagens e desvantagens. Qual é a relação entre essas formas de representação?
Representations for Complex Numbers
Convert the component form into trigonometric form:
Ι Z Ι = r = √(x2 + y2)
with x = r cosϕ and y = r sinϕ
=> Z = r (cosϕ + i · sinϕ) and φ = arctan (y/x) the x- and y-coordinates are clearly defined.
Derivation of Euler’s form for complex numbers:
MacLaurin's series for eϕ: eϕ = 1+ φ + φ2 + φ3 + φ4 +…
. 1! 2! 3! 4!
Replace φ for j·φ and you get:
ejϕ = 1+ jφ + (jφ)2 + (jφ)3 + (jφ)4 +… = 1+ jφ - φ2 - jφ3 + φ4 +… =
. 1! 2! 3! 4! 1! 2! 3! 4!
ejϕ = 1 - φ2 + φ4 + j ( φ - φ3 + φ5 -… )
. 2! 4! 3! 5!
. |_________| |___________|
cos φ sin φ (according to the definition of the sine and cosine series)
=> ejϕ = cos φ + j sinφ
or with consideration of the length of the pointer follows: Z = r × e iϕ
Addition and subtraction of complex numbers
The addition and subtraction of complex numbers is most easily done with the Algebraic form. Simply add or subtract their real and imaginary parts:
Z1 = a + i·b => Z1 + Z2 = (a + c) + i (b + d)
Z2 = c + i·d Z1 - Z2 = (a - c) + i (b - d)
Multiplication and division of complex numbers
The easiest way to multiply or divide complex numbers is to use the exponential formula or Eulers form. Here, when multiplying, the amounts must be multiplied and the angles added. In division, the amounts are divided and the angles are subtracted:
Multiplication and division of complex numbers
The complex conjugate
You get the complex conjugate simply by changing the sign of the imaginary part of the complex number. It corresponds to a mirroring of the pointer on the real axis.
The complex conjugate
Note: Multiplying a complex number by its complex conjugate gives a real number. This allows complex parts to be removed from a system of equations.
Mnemonics
- For complex numbers, the terms 'greater than' or 'less than' are not defined.
- Two complex numbers are equal if their real and imaginary parts are equal.
- A complex number with the imaginary part equal to zero is a member of the real numbers.
- A complex number with zero real part is a member of the imaginary numbers.
- Two complex numbers are complex conjugate if they differ only in the sign of the imaginary part.