Math Formulas ► Complex numbers ► Sets of Numbers ► Set Identities
Math Formulas ► Complex numbers
Definitions:
A complex number is written as a+bi where a and b are real numbers an i, called the imaginary unit, has the property that i2=−1.
The complex numbers z=a+bi and z−=a−bi are called complex conjugate of each other.
Formulas:
Equality of complex numbers
a+bi=c+di⟺a=c and b=d
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Addition of complex numbers
(a+bi)+(c+di)=(a+c)+(b+d)i
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Subtraction of complex numbers
(a+bi)−(c+di)=(a−c)+(b−d)i
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Multiplication of complex numbers
(a+bi)⋅(c+di)=(ac−bd)+(ad+bc)i
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Division of complex numbers
a+bic+di=a+bic+di⋅c−dic−di=ac+bdc2+d2+bc−adc2+d2i
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Polar form of complex numbers
a+bi=r⋅(cosθ+isinθ)
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Multiplication and division of complex numbers in
polar form
[r1(cosθ1+i⋅sinθ1)]⋅[r2(cosθ2+i⋅sinθ2)]=r1⋅r2[cos(θ1+θ2)+i⋅sin(θ1+θ2)]
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r1(cosθ1+isinθ1)r2(cosθ2+isinθ2)=r1r2[cos(θ1−θ2)+i⋅sin(θ1−θ2)]
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De Moivre's theorem
[r(cosθ+isinθ)]n=rn(cos(nθ)+isin(nθ))
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Roots of complex numbers
[r(cosθ+isinθ)]1/n=r1/n(cosθ+2kπn+isinθ+2kπn) k=0,1,…,n−1
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