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Use the definition of addition to show that the real and imaginary parts can be expressed as a sum/differenceof a complex number and its conjugate. z z z 2 and z z z 2 .

z z a b a b 2 a 2 z . Similarly, z z a b a b 2 b 2 z

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Complex numbers can also be expressed in an alternate form, polar form , which we will find quite useful. Polar form arises arises from the geometric interpretation of complex numbers.The Cartesian form of a complex number can be re-written as a b a 2 b 2 a a 2 b 2 b a 2 b 2 By forming a right triangle having sides a and b , we see that the real and imaginary parts correspond to the cosine and sine of the triangle's base angle. We thus obtain the polar form for complex numbers. z a b r θ r z a 2 b 2 a r θ b r θ θ b a The quantity r is known as the magnitude of the complex number z , and is frequently written as z . The quantity θ is the complex number's angle . In using the arc-tangent formula to find the angle, we must take into account the quadrant in which the complex number lies.

Convert 3 2 to polar form.

To convert 3 2 to polar form, we first locate the number in the complex plane in the fourth quadrant. The distance from the originto the complex number is the magnitude r , which equals 13 3 2 2 2 . The angle equals 2 3 or -0.588 radians ( 33.7 degrees). The final answer is 13 33.7 degrees.

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Euler's formula

Surprisingly, the polar form of a complex number z can be expressed mathematically as

z r θ
To show this result, we use Euler's relations that express exponentials with imaginary arguments in terms of trigonometric functions.
θ θ θ
θ θ θ 2
θ θ θ 2 The first of these is easily derived from the Taylor's series for the exponential. x 1 x 1 x 2 2 x 3 3 Substituting θ for x , we find that θ 1 θ 1 θ 2 2 θ 3 3 because 2 -1 , 3 , and 4 1 . Grouping separately the real-valued terms and the imaginary-valued ones, θ 1 θ 2 2 θ 1 θ 3 3 The real-valued terms correspond to the Taylor's series for θ , the imaginary ones to θ , and Euler's first relation results. The remaining relationsare easily derived from the first. We see that multiplying the exponential in [link] by a real constant corresponds to setting the radius of the complex number to the constant.

Calculating with complex numbers

Adding and subtracting complex numbers expressed in Cartesian form is quite easy: You add (subtract) the real parts andimaginary parts separately.

± z 1 z 2 ± a 1 a 2 ± b 1 b 2
To multiply two complex numbers in Cartesian form is not quite as easy, but follows directly from following the usual rules of arithmetic.
z 1 z 2 a 1 b 1 a 2 b 2 a 1 a 2 b 1 b 2 a 1 b 2 a 2 b 1
Note that we are, in a sense, multiplying two vectors to obtain another vector. Complex arithmetic provides a unique wayof defining vector multiplication.

What is the product of a complex number and its conjugate?

z z a b a b a 2 b 2 . Thus, z z r 2 z 2 .

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Division requires mathematical manipulation. We convert the division problem into a multiplication problem by multiplyingboth the numerator and denominator by the conjugate of the denominator.

z 1 z 2 a 1 b 1 a 2 b 2 a 1 b 1 a 2 b 2 a 2 b 2 a 2 b 2 a 1 b 1 a 2 b 2 a 2 2 b 2 2 a 1 a 2 b 1 b 2 a 2 b 1 a 1 b 2 a 2 2 b 2 2
Because the final result is so complicated, it's best to remember how to perform division—multiplying numerator and denominator by thecomplex conjugate of the denominator—than trying to remember the final result.

The properties of the exponential make calculating the product and ratio of two complex numbers much simpler when the numbers are expressed in polar form.

z 1 z 2 r 1 θ 1 r 2 θ 2 r 1 r 2 θ 1 θ 2
z 1 z 2 r 1 θ 1 r 2 θ 2 r 1 r 2 θ 1 θ 2 To multiply, the radius equals the product of the radii and the angle the sum of the angles. To divide, the radius equalsthe ratio of the radii and the angle the difference of the angles. When the original complex numbers are in Cartesianform, it's usually worth translating into polar form, then performing the multiplication or division (especially in thecase of the latter). Addition and subtraction of polar forms amounts to converting to Cartesian form, performing thearithmetic operation, and converting back to polar form.

When we solve circuit problems, the crucial quantity, known as a transfer function, will always beexpressed as the ratio of polynomials in the variable s 2 f . What we'll need to understand the circuit's effect is the transfer function in polar form. For instance, supposethe transfer function equals

s 2 s 2 s 1
s 2 f
Performing the required division is most easily accomplished by first expressing the numerator and denominator each inpolar form, then calculating the ratio. Thus,
s 2 s 2 s 1 2 f 2 -4 2 f 2 2 f 1
s 2 s 2 s 1 4 4 2 f 2 f 1 4 2 f 2 2 4 2 f 2 2 f 1 4 2 f 2
s 2 s 2 s 1 4 4 2 f 2 1 4 2 f 2 16 4 f 4 f 2 f 1 4 2 f 2

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Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
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