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Complex Numbers as Matrices – Euler’s Identity

Euler’s Identity below is regarded as one of the most beautiful equations in mathematics as it combines five of the most important constants in mathematics:

I’m going to explore whether we can still see this relationship hold when we represent complex numbers as matrices.

Complex Numbers as Matrices

First I’m I’m going to define the following equivalences between the imaginary unit and the real unit and matrices:

The equivalence for 1 as the identity matrix should make sense insofar as in real numbers, 1 is the multiplicative identity.  This means that 1 multiplied by any real number gives that number.  In matrices, a matrix multiplied by the identity matrix also remains unchanged.  The equivalence for the imaginary unit is not as intuitive, but let’s just check that operations with complex numbers still work with this new representation.

In complex numbers we have the following fundamental definition:

Does this still work with our new matrix equivalences?

Yes, we can see that the square of the imaginary unit gives us the negative of the multiplicative identity as required.

More generally we can note that as an extension of our definitions above we have:

Complex number multiplication

Let’s now test whether complex multiplication still works with matrices.  I’ll choose to multiply the following 2 complex numbers:

Now let’s see what happens when we do the equivalent matrix multiplication:

We can see we get the same result.  We can obviously prove this equivalence more generally (and check that other properties still hold) but for the purposes of this post I want to check whether the equivalence to Euler’s Identity still holds with matrices.

Euler’s Identity with matrices

If we define the imaginary unit and the real unit as the matrices above then the question is whether Euler’s Identity still holds, i.e:


First I can note that:

Next I can note that the Maclaurin expansion for e^(x) is:

Putting these ideas together I get:

This means that:


Next I can use the matrix multiplication to give the following:

Next, I look for a pattern in each of the matrix entries and see that:

Now, to begin with here I simply checked these on Wolfram Alpha – (these sums are closely related to the Macluarin series for cosine and sine).

Therefore we have:

So, this means I can write:

And so this finally gives:

Which is the result I wanted!  Therefore we can see that Euler’s Identity still holds when we define complex numbers in terms of matrices.  Complex numbers are an incredibly rich area to explore – and some of the most interesting aspects of complex numbers is there ability to “bridge” between different areas of mathematics.

Plotting the Mandelbrot Set

The video above gives a fantastic account of how we can use technology to generate the Mandelbrot Set – one of the most impressive mathematical structures you can imagine. The Mandelbrot Set can be thought of as an infinitely large picture – which contains fractal patterns no matter how far you enlarge it.  Below you can see a Mandelbrot zoom – which is equivalent to starting with a piece of A4 paper and enlarging it to the size of the universe!  Even at this magnification you would still see new patterns emerging.

The way the Mandelbrot set is formed in the first video is by using the following iterative process:

Zn+1 = Zn2 + c

Here Z is a complex number (of the form a + bi) and c is a constant that we choose.  We choose our initial Z value as 0.  Z1 = 0. We then choose a value of c (which is also a complex number) and see what happens when we follow the iterative process.

Let’s choose c = 2i +1. Z1 = 0

Zn+1 = Zn2 + 2i +1
Z2 = (0)2 + 2i +1
Z2 = 2i + 1

We then repeat this process:
Z3 = Z22 + 2i +1
Z3 = (2i+1)2 + 2i +1
Z3 = (2i)(2i) + 2i + 2i + 1 + 2i +1
Z3 = 6i-2 (as i.i = -1)

As we continue this process Zn spirals to infinity.

What we are looking for is whether this iterated Z value will diverge to infinity (i.e get larger and larger) or if it will remain bounded. If diverges to infinity we colour the initial point 2i+1 as blue on a complex axis. If it remains bounded we will colour it in black. In this case our initial point 2i+1 will diverge to infinity and so it will be coloured in blue.

So, let’s use Geogebra to see this is action.  The Geogrebra online program for this is here.

We choose a value for c.  Let’s say c = 0.23 + 0.42i.  Z1 = 0
Zn+1 = Zn2 + 0.23 + 0.42i.
Z2 = (0)2 + 0.23 + 0.42i.
Z2 = 0.23 + 0.42i.

Z3 = Z22 + 0.23 + 0.42i.
Z3 = (0.23 + 0.42i.)2 + 0.23 + 0.42i.
Z3 = 0.1065 + 0.6132i
Z4 = (0.1065 + 0.6132i)2 + 0.23 + 0.42i.
Z4 = -0.13467199 + 0.5506116i

We carry on with this iterative process and plot the points that we get each time.  We can see the (0.23, 0.42), (0.1065, 0.42) and (-0.13467199, 0.5506116) correspond to the first coordinates on the spiral after (0,0).  We can see that as this process continues we see a convergence to a point close to (0.05, 0.45).

Screen Shot 2019-04-25 at 3.09.17 PM

If we choose another starting value for c: c = 0.17 + 0.56i we get the following diagram:

Screen Shot 2019-04-25 at 3.09.33 PM

Again we have a stable spiral which spirals around a geometric shape and does not diverge to infinity.

If we choose another starting value for c: c = -0.25 + 0.64i we get the following diagram:

Screen Shot 2019-04-25 at 3.10.04 PM

If we choose another starting value for c: c = 0.11 + 0.59i we get the following diagram:

Screen Shot 2019-04-25 at 3.09.43 PM

However, If we choose another starting value for c: c = 0.3 + 0.68i we get the following diagram:

Screen Shot 2019-04-25 at 3.10.12 PM

This time we can see that the orbit of points does not converge, but instead it diverges to infinity.

We can then colour in each point – simply categorising whether the value of c leads to an orbit which diverges or remains bounded.  Black means it remains bounded, blue that it has escaped to infinity.  So, below we can see that when we do the iterative process with c = 0.39+ 0.63i our orbit will escape to infinity (as it is coloured blue)

Screen Shot 2019-04-25 at 3.13.45 PM

If we do this exercise in much finer detail we arrive at the following picture:

Screen Shot 2019-04-25 at 3.14.02 PM

This is the Mandelbrot Set – and will keep producing fractal patterns as you zoom in to infinity.

IB Revision

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2019-07-27 at 10.02.40 AM

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.
 

 

 

 

Mandelbrot and Julia Sets – Pictures of Infinity

The above video is of a Mandelbrot zoom.  This is a infinitely large picture – which contains fractal patterns no matter how far you enlarge it.  To put this video in perspective, it would be like starting with a piece of A4 paper and enlarging it to the size of the universe – and even at this magnification you would still see new patterns emerging.

Screen Shot 2015-02-04 at 3.34.57 PM

To understand how to make the Mandelbrot set, we first need to understand Julia sets.  Julia sets are formed by the iterative process:

Zn+1 = Zn2 + c

Here Z is a complex number (of the form a + bi) and c is a constant that we choose.  So, for example if we choose Z1 = 1+i and c = 1 then:

Z2 = Z12 + 1
Z2 =(1+i)2 + 1
Z2 = 2i + 1

We then repeat this process:
Z3 = Z22 + 1
Z3 = (2i+1)2 + 1
Z3 = 4i-2

and so on – what we are looking for is whether this iterated Z value will diverge to infinity (i.e get larger and larger) or if it will remain bounded. If diverges to infinity we colour the initial point 1+i as red on a complex axis. If it remains bounded we will colour it in black. In this case our initial point 1 + i will diverge to infinity and so it will be coloured in red.

Next we do this for every single point in the complex plane – each time seeing what happens when we iterate it many times. Each time we colour it in as red if it diverges and black if it remains bounded. Once we have done that we will have a picture which represents what happens to every point in the complex plane. This then is our Julia set.

For example the Julia set for c = 1 looks like this:

Screen Shot 2015-02-04 at 4.10.09 PM

 

This is because every single complex number when iterated by  Zn+1 = Zn2 + 1 will diverge to infinity (get infinitely big).

Not very interesting so far, but different values of c provide some amazing patterns.

Screen Shot 2015-02-04 at 4.13.42 PM

This above pattern is generated by c = 0.376 – 0.1566i.

Screen Shot 2015-02-04 at 4.15.27 PM

and this pattern is for c = 0.376 – 0.1566i.

Screen Shot 2015-02-04 at 4.17.35 PM

and this one is c = -0.78 + 0.1i.

Screen Shot 2015-02-04 at 4.24.45 PM

This last one for c = 0.4 + 0.1i looks different to the others – this one has patterns but they are not connected together as in the other examples.

Mandelbrot Set

This brings us on to how to calculate the Mandelbrot set.  We calculate every possible Julia set for all complex numbers c, and then for every Julia set which is connected then we colour the c value in black, and every value of c which the Julia set is disconnected we colour the c value in red.  We then have a new plot in the complex plane of c values.  This gives us the Mandelbrot set shown below:

Screen Shot 2015-02-04 at 4.31.23 PM

 

Don’t worry if this seem a bit complicated – it is!  You can play around making your own Julia sets by choosing a c value at this online generator.  You might also like towatch the Numberphile video on the same topic:

 If you enjoyed this post you might also like Dan Pearcy’s post on this topic which explains how Geogebra can be used to generate these sets.  Also PlusMaths have a number of posts on this amazing subject

IB Revision

Screen Shot 2018-03-19 at 4.35.19 PM

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2019-07-27 at 10.02.40 AM

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

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IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

  1. Investigation essentials,
  2. Marking criteria guidance,
  3. 70 hand picked interesting topics
  4. Useful websites for use in the exploration,
  5. A student checklist for top marks
  6. Avoiding common student mistakes
  7. A selection of detailed exploration ideas
  8. Advice on using Geogebra, Desmos and Tracker.

Available to download here.

IB Exploration Modelling and Statistics Guide


IB Exploration Modelling and Statistics Guide

A 60 page pdf guide full of advice to help with modelling and statistics explorations – focusing in on non-calculator methods in order to show good understanding. Includes:

  1. Pearson’s Product: Height and arm span
  2. How to calculate standard deviation by hand
  3. Binomial investigation: ESP powers
  4. Paired t tests and 2 sample t tests: Reaction times
  5. Chi Squared: Efficiency of vaccines
  6. Spearman’s rank: Taste preference of cola
  7. Linear regression and log linearization.
  8. Quadratic regression and cubic regression.
  9. Exponential and trigonometric regression.

Available to download here.

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