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Complex Numbers Polar Form


On the Argand Diagram page. We showed how a complex number can be drawn as a vector, and expressed in both Polar Form, and Cartesian Form.


CARTESIAN FORM:      z = a + bi

POLAR FORM:            z = r(cosθ + isinθ)

Where r is the modulus, the length of the vector.     r  =  \bf{\sqrt{a^2 \space + \space b^2}}

And  θ  is the argument Arg(z).  The angle the vector makes with the positive direction of the Real axis.
Introduction to Complex Numbers Polar Form.


Also shown, was how there is a need at times, to be careful when establishing the value of the argument θ.

If  a > 0b > 0,       θ = tan-1(\bf{\frac{b}{a}})

If  a > 0b < 0,       θ = tan-1(\bf{\frac{b}{a}})

If  a < 0b > 0,       θπ + tan-1(\bf{\frac{b}{a}})

If  a < 0b < 0,       θ = –π + tan-1(\bf{\frac{b}{a}})



It’s important to be comfortable converting between the 2 complex number forms of polar and cartesian.






Complex Numbers Polar Form,
Convert to Cartesian Form


First we’ll again look at one of the complex number examples from the Argand Diagram page,   z = 1 + i√3.

The vector for this complex number is nicely placed in the  1st  quadrant.

Standard complex number plotted.

Converting to Polar Form we have:

r  =  \bf{\sqrt{1^2 \space + \space ({\sqrt{3}})^2}}   =  2     ,      θ = tan-1(\bf{\frac{\sqrt{3}}{1}})  =  \bf{\frac{\pi}{3}}

z  =  2(cos(\bf{\frac{\pi}{3}}) + isin(\bf{\frac{\pi}{3}}))




Expanding the Polar Form:

Now this Polar Form can be expanded on slightly.

The angle  θ,  that the vector makes with the Real axis, is called the argument, denoted  Arg(z).


With the complex number above we established that it was  \bf{\frac{\pi}{3}}.

But  \bf{\frac{\pi}{3}}  could in fact be replaced by   \bf{\frac{\pi}{3}} + 2πk.

Where k is an integer.        kZ

k = 0,  +1,  + 2, …..


This is because  2π  is a full rotation on an Argand Diagram.
Taking you back to the same physical position you started from, the vector in this case.

Plotted complex number with the angle.


You can travel round the axis by  2π  any number of times, in either direction.

So with   \bf{\frac{\pi}{3}} + 2π = \bf{\frac{7\pi}{3}}       and       \bf{\frac{\pi}{3}}2π = –\bf{\frac{5\pi}{3}}.


The Polar Form of  1 + i√3  can also be:

z  =  2(cos(\bf{\frac{7\pi}{3}}) + isin(\bf{\frac{7\pi}{3}}))

z  =  2(cos(-\bf{\frac{5\pi}{3}}) + isin(-\bf{\frac{5\pi}{3}}))


But the original value of  θ,  in this case \bf{\frac{\pi}{3}}.

This is called the “principal value” of the argument Arg(z).


It is the unique value of the argument, that lies in the range,   –π  <  Arg(z<  π.
As \bf{\frac{\pi}{3}} does above.


Complex Number Arguments.



The range includes  π,  but not  –π,  so the argument of a negative real number has a principal value of  π.

Negative real number principal value.




Examples    


(1.1) 

Convert the complex number  z = 2 + i2
into Polar Form.

Solution   

r  =  \bf{\sqrt{2^2 \space + \space 2^2}}  =  √8  =  2√2

a > 0 , b > 0   =>   θ = tan-1(\bf{\frac{2}{2}})  =  \bf{\frac{\pi}{4}}

z  =  2√2(cos(\bf{\frac{\pi}{4}}) + isin(\bf{\frac{\pi}{4}}))




(1.2) 

Convert the complex number  z = –1 + i4

into Polar Form.

Solution   

r  =  \bf{\sqrt{({\text{-}}1)^2 \space + \space 4^2}}  =  √17

a < 0 , b > 0   =>   θ = π + tan-1(\bf{\frac{4}{{\text{-}}1}})  =  1.816 Radians

z  =  √17(cos(1.816) + isin(1.816))




(1.3) 

Convert the complex number  z = i5

into Polar Form.

Solution   

r  =  \bf{\sqrt{0^2 \space + \space 5^2}}  =  5

a = 0 , b > 0   =>   θ = \bf{\frac{\pi}{2}}

z  =  5(cos(\bf{\frac{\pi}{2}}) + isin(\bf{\frac{\pi}{2}}))




(1.4) 

Write down the Polar Form of  z = –4.

Solution   

r  =  \bf{\sqrt{({\text{-}}4)^2 \space + \space 0^2}}  =  4

As  z  is a negative real number. The argument  θ = π.

z  =  4(cos(π) + isin(π))

If  z = 4, the Argument would be 0.

As no angle is being made with the positive direction of the Real axis.

Positive Number Principal Value.




Following on from the examples shown on this page, the  complex number cartesian form  page shows how to convert from polar to cartesian form.
Along with introducing the exponential form of a complex number.





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