What are the two main features of a vector?

Magnitude (length) and Direction (angle)

How do we define the length of a complex number a + bi ?

Absolute value:

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a+bi = a 2 +b2

Another name for the absolute value (or magnitude) of a complex number is the MODULUS.

To relate the modulus to circular trigonometry, if the complex number represents a point on a circle centered at the origin, then the modulus is equivalent to the radius of the circle containing that point.So … given a complex number z, such that z = a + bi,

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z = a 2 +b2 =r

The other feature of a vector is its direction, which is measured as an angle. A complex number is referenced by an angle as well.The angle associated with a complex number is found in the same trigonometric way that the direction angle is found for a vector.These angles are measured between 0° and 360°,or between 0 and 2π. (Recall that you must consider the quadrant!)

The angle θ for a complex number z = a + biis referred to as the ARGUMENT and is found using

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θ=tan −1 b

a

⎛ ⎝ ⎜

⎞ ⎠ ⎟

For example:

Given z = 3 + 4i, determine the modulus and the argument of z.

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z =5, and θ =53.1°

We need to use the combination of the modulus (r) and the argument (θ) of a complex number in order to write complex numbers in trigonometric form, aka POLAR FORM.

TRIGONOMETRIC FORM of a complex number:

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z =a+bi

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=r cisθ

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= r cosθ( )+ r sinθ( )i

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=r cosθ + i sinθ( )

For example:Given z = 3 + 4i, re-write z in trigonometric form (aka polar form).

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z =r =5, and θ =53.1°From our previous example we know that

Therefore, z = 3 + 4i = 5 cis 53.1°.

Finally, think about how to reverse the process. How do you go from trigonometric (polar) form to standard complex number form (rectangular)?

For example, write the complex number in rectangular form: 6 cis 120° = ___________

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6(cos120+ i sin120)

=6−1

2+i

3

2

⎛

⎝ ⎜

⎞

⎠ ⎟

=−3+3 3 i

Given two complex numbers in polar form: &

Then, €

z1 =r1 cisθ1

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z2 =r2 cisθ2

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z1⋅ z2 = r1⋅ r2 cis θ1 + θ2( )

What about division of two complex numbers in polar form?

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z1z2

=r1

r2cis θ1 − θ2( )

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z1z2

=

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z1⋅ z2 =

Given a complex number in polar form:

Then,

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z=r cisθ

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z( )n= r( )

ncis n⋅θ( )

Note: when completing these operations, if your new angle goes outside of the range of 0° to 360°, you will need to answer with a coterminal angle that is between 0° and 360°

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z( )n=

How is raising a number to a power related to multiplication?

Since raising a number to a power represents repeated multiplication, the rule for raising a complex number to a power is an extension of the rule for multiplying complex numbers.

Given two complex numbers in polar form: &

Then,

DeMoivre’s Theorem:

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z1 =r1 cisθ1

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z2 =r2 cisθ2

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z1 ⋅z2 =r1 ⋅r2 cis θ1 +θ2( )

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z1( )n

= r n( ) cis nθ( )

Note: If (nθ) is too large, subtract 360 so that it falls within the acceptable range.

One final reminder about polar form of complex

numbers:

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0° ≤ θ < 360° or 0 ≤ θ < 2πThis is important because if any of the operations you

perform cause the angle to go outside of this range, then you must adjust it (using coterminal angle rules).

Finding roots of a complex number requires just a little bit more trigonometry knowledge.

2nd Recall from early trig lessons that coterminal angles are angles that differ by 360°. Therefore, if the argument of a complex number is 88°, then 88 + 360 = 448° is a coterminal angle and will need to be used when finding roots.

1st Every non-zero complex number has exactly n nth roots. In other words, a complex number has 4 fourth roots, and nine ninth roots, etc.

DeMoivre’s Theorem works for finding roots of complex numbers, too.

It uses roots and division, instead of powers and multiplication.

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zn = rn cisθ

n+

360k

n

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Example: Find all complex cube roots of Record answers in polar form.

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z=8 − 8 3 i

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z3

= 163

cis 100° = 163

cis 220° = 163

cis 340°

Quick Check:Please answer these questions on a clean sheet of paper & turn in when complete.

Given and

1st convert both numbers to polar form2nd find the product of z and w (in polar form)

3rd find the quotient of z and w (in polar form)

4th find z6 (in polar form, then convert to rectangular)

5th find the complex cube roots of w (in polar form)

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z =1+4i

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w=−6i