Math Circles 2018Complex Numbers, Lesson 3Wednesday, March 28, 2018

Rich Dlin

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Let’s Start With a Song!

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Utilitarian i

A long, long time ago, I can still remember howQuadratics used to make me cry.With factoring I’d solve a ton,Use the formula, to get it done,But I couldn’t solve x2 + 1 .The real numbers made me shiver,For no solution they’d deliver,Not positive, nor negative ...A working root could they give.And zero wasn’t worth a tryIt does not change with multiplySo I begin to question why,And then, I thought of i .

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Utilitarian i

Oooh... My, my utilitarian iWhen we square ya that don’t scare ya you are cooler than piAnd negative one comes out of the other side.Singing now I’m just like sixteen and ninenow I’m just like sixteen and nine

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Utilitarian i

Were you caught unawaresThat x2 + 1 is a difference of squaresThe complex numbers make it soAnd negative one really has a hootBecause it finally has a square rootAnd x2 + 1 can really be zeroNow we know that the discriminant’Bout the roots give us a giant hintWe know if they’re complexAnd how many x-interceptsi is the end of my quadratic woesNow parabolas are not my foesit was right there in front of my noseThe imaginary iI started singin’...

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Utilitarian i

My, my utilitarian iWhen we square ya that don’t scare ya you are cooler than piAnd negative one comes out of the other side.Singing now I’m just like sixteen and ninenow I’m just like sixteen and nine

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Warmup and Review

Here are the key things we discussed last week:

z = a + bi and z = a− bi are called conjugates.

When you multiply complex conjugates z and z , the result is alwaysreal, and equal to |z |2

If z 6= 0 then z−1 =1

|z |2z . This guarantees that zz−1 = 1.

We can determine values for (x + yi)pq by letting (x + yi)

pq = a + bi

I This means (x + yi)p = (a + bi)q

I We expand (x + yi)p and (a + bi)q,I We equate the real parts to the real parts and the imaginary parts to

the imaginary partsI This creates a system of equations we can solve.

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Warmup and Review

Annoying Problem

Evaluate (8 + 8√

3i)34

Let’s look at the solution to this together on the board...

Yeah. That was intense. And not totally fun.Maybe we can find a better way ...

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Complex Numbers and Geometry

The number z = x + yi is easy to think of as 2-dimensional. Here aresome results of this:

We can use the ordered pair (x , y) to uniquely represent z .

We could then plot points in the Complex Plane (aka an ArgandDiagram)

I When doing this, the horizontal axis is the Real axis andI the vertical axis is the Imaginary axis.I z is a reflection of z in the Real axisI |z | is the distance of the point (x , y) from the originI Addition is just adding the x ’s and adding the y ’s (the same as vector

addition, for those that have seen that).

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Complex Numbers and Geometry

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Polar Coordinates

In the Argand diagram, we can see that z could also be uniquely definedby an angle and radius:

The ordered pair (r , θ) defines aunique z .

r = |z | =√x2 + y2.

We measure θ counter-clockwisefrom the positive Real axis.

θ can be determined using yourfavourite inverse trig function(and your brain).

θ is called an argument for z .

Notice that if z = x + yi , then x = r cos θ and y = r sin θ.

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Polar Form

Polar Form

The polar form of a complex number, z , is

z = r(cos θ + i sin θ)

where r is the modulus of z and θ is an argument of z .

Examples: Converting From Standard (Rectangular) to Polar Form

1 Write z1 = 1−√

3i in polar form.

2 Write z2 = −1− i in polar form.

Let’s do these together on the board...

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Polar Form

Polar Form:

θ is not unique, due to the periodic nature of trig functions.

This is actually a very good thing in many ways, as we shall see.

Example

Convert z = cos15π

6+ i sin

15π

6to standard form.

Note that 15π6 = 5π

2 is coterminal with π2 .

sin π2 = 1 and cos π2 = 0

Therefore in this example, z = 0 + 1i = i .

Although θ is not unique, we typically choose θ so that 0 ≤ θ < 2π.

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Taylor Series Expansions

Let’s look at some stuff that appears magical together on the board...

What we saw are graphical examples of some cool calculus stuff. We willnot prove these, as they require some heavy calculus concepts, but theyare awesomely true.

sin x = x − x3

3!+

x5

5!− x7

7!+ · · ·

cos x = 1− x2

2!+

x4

4!− x6

6!+ · · ·

ex = 1 + x +x2

2!+

x3

3!+

x4

4!+ · · ·

Let’s look at some MORE stuff that appears magical together on theboard...

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Complex Exponentials

The Complex Exponential Function

We define e ix = cos x + i sin x .

Notes on this

We can write the polar form of z ∈ C as z = re iθ, where r = |z | andθ is an argument of z .

e iθ = e−iθ

I Because cos θ − i sin θ = cos(−θ) + i sin(−θ).

We can do cool things like write(2e i13π/6

)6in standard form.

I Let’s do that together on the board...

Some really amazing things are true.I Exponent laws:

F e ip × e iq = e i(p+q),F

(e iθ

)p= e ipθ, (which leads to DeMoivre’s Theorem ... let’s look at

that)

I Setting r = 1 and θ = π gives e iπ + 1 = 0

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Annoying Problem, Revisited

Example

Evaluate (8 + 8√

3i)34

Let’s use our new info to solve this together on the board...

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