Complex numbers

You can see a simple implementation of a complex number in Java.

imaginary unit

By definition

i2 = -1

Because no real number satisfies this equation, i is called an imaginary number.

complex number

Complex number c is expressed as

c = α + i*β

Where

• α is called real part of complex number
• β is called imaginary part of complex number
• i is imaginary unit

You can visualize a complex number as point (α, β) on the complex plane.

basic operations

• a = b, if α_a = α_b and β_a = β_b
• a + b = (α_a + α_b, β_a + β_b)
• a*b = (α_a*α_b - β_a*β_b, α_a*β_b + α_b*β_a)
• a/b = (  (α_a*α_b + β_a*β_b)/(α_b² + β_b²), (α_b*β_a - α_a*β_b)/(α_b² + β_b²)  )

Modulus

• |a| = sqrt(α² + β²)
• |a*b| = |a|*|b|
• |a+b| <= |a| + |b|

de Moivre's identity

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

Euler's formula

ei*x = cos(x) + i*sin(x)

sin(x) = (eix - e-ix) / 2i

cos(x) = (eix + e-ix) / 2

tan(x) = (e-ix - eix) / (eix + e-ix)

trigonometric form

c = |c| * ( cos(φ) +i*sin(φ) )

exponential form

c = |c|*ei*φ