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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.

The Chinese Postman Problem

There is a fantastic pdf resource from Suffolk Maths which goes into a lot of detail on this topic – and I will base my post on their resource.   Visit their site for a more in-depth treatment.

The Chinese Postman Problem was first posed by a Chinese mathematician in 1962.  It involved trying to calculate how a postman could best choose his route so as to mimise his time.  This is the problem that Kuan Mei-Ko tried to solve:

chinesepost1

How could a postman visit every letter on the graph in the shortest possible time?

Solving this requires using a branch of mathematics called graph theory, created by Leonard Euler.  This mathematics looks to reduce problems to network graphs like that shown above.  Before we can solve this we need to understand some terminology:

chinesepost3

Above we have 3 graphs.  A graph which can be drawn without taking the pen off the paper or retracing any steps is called traversable (and has a Euler trail).  Graph 1 is not traversable.  Graph 2 is traversable as long as you start at either A or D, and Graph 3 is traversable from any point that you start.  It turns out that what dictates whether a graph is traversable or not is the order of their vertices.

Looking at each letter we count the number of lines at the vertex.  This is the order.  For graph 1 we have 3 lines from A so A has an order of 3.  All the vertices on graph 1 have an order of 3.  For graph 2 we have the orders (from A, B, C, D, E in turn) 3, 4, 4, 3, 2.  For graph 3 we have the orders 4,4,4,4,2,2.

This allows us to arrive at a rule for working out if a graph is traversable.

If all orders are even then a graph is traversable.  If there are 2 odd vertices then we can find a traversable graph by starting at one of the odd vertices and finishing at the other.  We need therefore to pair up any odd vertices on the graph.

Next we need to understand how to pair the odd vertices.  For example if I have 2 odd vertices, they can only be paired in one way.  If I have 4 vertices (say A,B,C,D) they can be paired in 3 different ways (either AB and CD or AC and BD or AD and BC) .  The general term rule to calculate how many ways n odd vertices can be paired up is (n-1) x (n-3) x (n-5) … x 1.

So now we are ready to actually solve the Chinese Postman Problem.  Here is the algorithm:

chinesepost4

So, we can now apply this to the Chinese Postman Problem below:

chinesepost1

Step 1:  We can see that the only odd vertices are A and H.

Step 2:  We can only pair these one way (AH)

Step 3 and 4: The shortest way to get from A to H is ABFH which is length 160.  This is shown below:

 chinesepost5

 Step 5 and 6: The total distance along all the lines in the initial diagram is 840m.  We add our figure of 160m to this.  Therefore the optimum (minimum) distance it is possible to complete the route is 1000m.

Step 7:  We now need to find a route of distance 1000m which includes the loop ABFH (or in reverse) which starts and finishes at one of the odd vertices.  One solution provided by Suffolk Maths is ADCGHCABDFBEFHFBA.  There are others!

The Bridges of Konigsburg

Graph theory was invented as a method to solve the Bridges of Konigsburg problem by Leonard Euler.  This was a puzzle from the 17oos – Konigsburg was a Russian city with 7 bridges, and the question was, could anyone walk across all 7 without walking over any bridge twice.  By simplifying the problem into one of connected lines, Euler was able to prove that this was in fact impossible.

If you like this post you might also like:

Knight’s Tour – This puzzles dates over 1000 years and concerns the ways in which a knight can cover all squares on a chess board.

Game Theory and Tic Tac Toe – Tic Tac Toe has already been solved using Game Theory – this topic also brings in an introduction to Group Theory.

leonard

e’s are good – He’s Leonard Euler.

Along with pi, e is one of the most important constants in mathematics. It is an irrational number which carries on forever. The first few digits are 2.718281828459045235…

Leonard Euler

e is sometime named after Leonard Euler (Euler’s number).   He wasn’t the first mathematician to discover e – but he was the first mathematician to publish a paper using it.    Euler is not especially well known outside of mathematics, yet he is undoubtedly one of the true great mathematicians.  He published over 800 mathematical papers on everything from calculus to number theory to algebra and geometry.

Why is e so important? 

Lots of functions in real life display exponential growth. Exponential growth is used to describe any function of the form ax where a is a constant. One example of exponential growth is the chessboard and rice problem, (if I have one grain of rice on the first square, two on the second, how many will I have on the 64th square?) This famous puzzle demonstrates how rapidly numbers grow with exponential growth.

Sketch
y = 2x
y = ex
y = 3x

for between x = 0 and 3. You can see that y = ex is between y=2x and y = 3x on the graph, so why is e so much more useful than these numbers? By graphical methods you can find the gradient when the graphs cross the y axis. For the function y = ex this gradient is 1. This is because the derivative of ex is still ex – which makes it really useful in calculus.

The beauty of e.

e appears in a host of different and unexpected mathematical contexts, from probability models like the normal distribution, to complex numbers and trigonometry.

Euler’s Identity is frequently voted the most beautiful equation of all time by mathematicians, it links 5 of the most important constants in mathematics together into a single equation.

euler4

Infinite fraction: e can be represented as a continued infinite fraction can students you spot the pattern? – the LHS is given by 2 then 1,2,1 1,4,1 1,6,1 etc.

e=2+{\cfrac  {1}{1+{\cfrac  {1}{{\mathbf  2}+{\cfrac  {1}{1+{\cfrac  {1}{1+{\cfrac  {1}{{\mathbf  4}+{\cfrac  {1}{1+{\cfrac  {1}{1+\ddots }}}}}}}}}}}}}}=1+{\cfrac  {1}{{\mathbf  0}+{\cfrac  {1}{1+{\cfrac  {1}{1+{\cfrac  {1}{{\mathbf  2}+{\cfrac  {1}{1+{\cfrac  {1}{1+{\cfrac  {1}{{\mathbf  4}+{\cfrac  {1}{1+{\cfrac  {1}{1+\ddots }}}}}}}}}}}}}}}}}}.

Infinite sum of factorials: e can also be represented as the infinite sum of factorials:

e=\sum_{n=0}^\infty \frac{1}{n!}

A limit: e can also be derived as the limit to the following function.  It was this limit that Jacob Bernoulli investigated – and he is in fact credited with the first discovery of the constant.

\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n,

Complex numbers and trigonometry :  e can be used to link both trigonometric identities and complex numbers:

e^{{ix}}=\cos x+i\sin x,\,\!

You can explore more of the mathematics behind the number e here.

If you enjoyed this post you might also like:

Ramanujan’s Beauty in Mathematics Some of the amazingly beautiful equations of Ramanujan.

Screen Shot 2018-03-19 at 5.29.06 PM

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.

leonard

e’s are good – He’s Leonard Euler.

Having recently starting a topic on the exponential function, I was really struggling to find some good resources online – which is pretty surprising given that e is one of the most important and useful numbers in mathematics. So, here are some possible approaches.

1) e memorisation challenge.

This is always surprisingly popular – and a great starter which reinforces both that e is infinite and also that it’s just a number – so shouldn’t be treated like other letters when it comes to calculus.

5 minutes: How many digits of e can students remember?

Recital at the front. You can make this easier by showing them that  2. 7 1828 1828 45 90 45  they only need to remember 2.7 and then that 1828 repeated, followed by the angles in a triangle – 45, 90, 45.  Good students can get 20 places plus – and for real memory champions here are the first 1000 digits .

2) Introduction to Leonard Euler

Euler is not especially well known outside of mathematics, yet is undoubtedly one of the true great mathematicians. As well as e being named after him (Euler’s number), he published over 800 mathematical papers on everything from calculus to number theory to algebra and geometry.

30-40 minutes – The Seven Bridges of Königsberg
This is one of Euler’s famous problems – which he invented a whole new branch of mathematics (graph theory) to try and solve. Here is the problem:

The city of Königsberg used to have seven bridges across the river, linking the banks with two islands. The people living in Königsberg had a game where they would try to walk across each bridge once and only once. You can chose where to start – but you must cross each bridge only once:

euler

The above graphic is taken from the Maths is Fun resource on Euler’s bridge problem.  It’s a fantastically designed page – which takes students through their own exploration of how to solve similar problems (or as in the case of the 7 bridges problem, understanding why it has no solution).

3) Learning about e

30 minutes – why e ?

This is a good activity for students learning about differentiation for the first time.

First discuss exponential growth (example the chessboard and rice problem ) to demonstrate how rapidly numbers grow with exponential growth – ie. if I have one grain of rice on the first square, two on the second, how many will I have on the 64th square?

Next, students are given graph paper and need to sketch  y = 2^x y = e^x y = 3^x for between x = 0 and 3.  Students can see that y = e^x is between y=2^x and y = 3^x on the graph, so why is e so much more useful than these numbers?   By graphical methods they should find the gradient when the graphs cross the y axis.    Look at how the derivative of e^x is still e^x – which makes it really useful in calculus.   This is a nice short video which explains graphically why e was chosen to be 2.718…

4) The beauty of e.

10-30 minutes (depending on ability), discussion of some of the beautiful equations associated with e and Euler:

a) Euler Identity – frequently voted the most beautiful equation of all time by mathematicians, it links 5 of the most important constants in mathematics together into a single equation.

euler4

b) e as represented as a continued infinite fraction (can students spot the pattern? – the LHS is given by 2 then 1,2,1  1,4,1  1,6,1 etc.

e

c) e as the infinite sum of factorials:

e=\sum_{n=0}^\infty \frac{1}{n!}

d) e as the limit:

\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n,

So, hopefully that should give some ideas for looking at this amazing number. (The post title will be lost on anyone not a teenager in England in the 1990s -to find out what you’re missing out on, here’s the song).

If you enjoyed this topic you may also like:

Cesaro Summation: Does 1 – 1 + 1 – 1 … = 1/2? – a post which looks at the maths behind this particularly troublesome series.

A Maths Snooker Puzzle – a great little puzzle which tests logic skills.

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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