Complex Algebra Introduction

Posted By on December 21, 2014


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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¯=abi are called complex conjugate of each other.

Formulas:

Equality of complex numbers

a+bi=c+dia=c  and  b=d

Addition of complex numbers

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

Subtraction of complex numbers

(a+bi)(c+di)=(ac)+(bd)i

Multiplication of complex numbers

(a+bi)(c+di)=(acbd)+(ad+bc)i

Division of complex numbers

a+bic+di=a+bic+dicdicdi=ac+bdc2+d2+bcadc2+d2i

Polar form of complex numbers

a+bi=r(cosθ+isinθ)

Multiplication and division of complex numbers in polar form

[r1(cosθ1+isinθ1)][r2(cosθ2+isinθ2)]=r1r2[cos(θ1+θ2)+isin(θ1+θ2)]
r1(cosθ1+isinθ1)r2(cosθ2+isinθ2)=r1r2[cos(θ1θ2)+isin(θ1θ2)]

De Moivre’s theorem

[r(cosθ+isinθ)]n=rn(cos(nθ)+isin(nθ))

Roots of complex numbers

[r(cosθ+isinθ)]1/n=r1/n(cosθ+2kπn+isinθ+2kπn)  k=0,1,,n1

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Posted by Akash Kurup

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