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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.

For students taking their exams in 2021 there is a big change to the IB syllabus – there will now be 4 possible strands: IB HL Analysis and Approaches, IB SL Analysis and Approaches, IB HL Applications and Interpretations, IB SL Applications and Interpretations.

IB Analysis and Approaches

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There is a significant cross-over between the current SL and HL courses and the new Analysis courses.  The main differences are:

  1. The SL course will now be a complete sub-set of the HL course, and the HL exam will now include some of the same questions as the SL exam.  Previously whilst SL was almost a complete sub-set of the HL course, the questions on the HL paper were never the same as SL (and usually all significantly harder).
  2. There are a few small additions to the HL Analysis syllabus compared to the old HL syllabus – such as binomials with fractional indices, partial fractions and regression.   SL will be largely the same except that the unit on vectors has been taken out.
  3. The HL option unit has gone – and some of the old HL Calculus option has been added to the core syllabus (though only a relatively small proportion of it).
  4. HL students will instead do an investigation style Paper 3 – potentially with the use of technology.  This will lead students through an investigation on any topic on the syllabus.
  5. The Exploration coursework will remain – however the guidance is now that it should be 12-20 pages (rather than 6-12 previously).

What does this all mean?

Until we start to see some past papers it’s difficult to be too confident on this – but based on the syllabus and specimen paper I would say that the two new courses remain pitched at the same level as for the old SL and HL courses.  Therefore the Analysis and Approaches HL course is only suitable for the very best mathematicians who are looking to study either mathematics or a field with substantial mathematics in it (such as engineering, physics, computer science etc).  These students would usually have an A* at IGCSE and have also studied Additional Mathematics prior to starting the course.  The Analysis and Approaches SL course looks like it will still be a good quality mathematics course – and so will be aimed at students who need some mathematical skills for their university courses (such as biology, medicine or business).  These students would usually have an A* – B at IGCSE.

Resources for teachers and students

This will be a work in progress – but to get started we have:

General resources:

1) A very useful condensed pdf of the Analysis and Approaches formula book for both SL and HL.

2) An excellent overview of the changes to the new syllabus – including more detailed information as to the syllabus changes, differences between the two courses and also what 10 of the leading universities have said with regards to course preferences.

3) University acceptance.  Information collated by a group of IB teachers on university requirements as to which course they will require for different subjects (this may be not be up to date, so please check).

4) Christos Nikolaidis has put together a fantastic site which has full class notes on the entire Analysis syllabus, along with tests and exercises.  This is all free – and a superb resource.

5) Daniel Hwang has made a really top quality resource for preparing HL students for their Paper 3 investigation.  This has 25 different investigations and full worked solutions.  This is the best resource out there for Paper 3 – and Daniel has kindly allowed me to share it as a pdf download here.

6) There’s a brand new question bank called IB Taskmaster.  It has all original questions – which you can use to create practice tests on all topics.  All free to use – currently over 600 questions and markschemes available.

Specific resources for the new HL and SL syllabus content:

1. Linear correlation (previously only SL, now SL and HL)

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a) A worksheet (docx file) on using a GDC to calculate regression lines and r values.

2. Equation of regression line of x on y. (SL and HL)

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3. Sampling (SL and HL)

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4. Simple deductive proof (SL and HL)

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a) A deductive proof worksheet (docx file) with some simple examples of deductive proof.

5. Partial fractions. (HL)

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a) A Partial Fractions worksheet (docx file) with notes and some partial fraction questions.

6. Binomial expansion with fractional and negative indices (HL)

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a) A binomial expansion worksheet (docx file) requiring the use of fractional and negative indices, as well as use of the Maclaurin expansion.

7. More rational functions (HL)

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8. Graphing [f(x)]2 (HL)

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9. L’Hopital’s rule (Previously on the Calculus option now on HL)

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a) A Limits of functions worksheet (docx file) with some examples of simple limits and uses of L’Hopital’s rule. Markscheme here.

10. Euler method for differential equations (Previously on the Calculus option now on HL)

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a) A worksheet (docx file) with some questions using Euler’s method to solve differential equations.

11. Separating variables to solve differential equations (Previously on the Calculus option now on HL)

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a) A worksheet (docx file) with some questions separating variables to solve differential equations. Markscheme here.

12.  Solving differential equations by substitution (Previously on the Calculus option now on HL)

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a) A worksheet (docx file) with some questions using substitution to solve homogenous differential equations. Markscheme here.

13. Solving differential equations by the integrating factor method (Previously on the Calculus option now on HL)

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a) A worksheet (docx file) with some questions using the integrating factor to solve  differential equations. Markscheme  here.

14. Maclaurin series (Previously on the Calculus option now on HL)

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a) A worksheet (docx file) with some questions using the Maclaurin series. Markscheme here.

Investigation resources for Paper 3 [Higher Level]

  1. Old IA investigations

Standard Level

[Links removed – hopefully the IB will provide these resources elsewhere]

(a) All SL IA investigations from 1998 to 2009 : This is an excellent collection to start preparations for the new Paper 3.

(b) Specimen investigations: These are 8 specimen examples of IA investigations from 2006 with student answers and annotations.

(c) SL IA investigations 2011-2012: Some more investigations with teacher guidance.

(d) SL IA investigations 2012-2013: Some more investigations with teacher guidance.

(e) Koch snowflakes: This is a nice investigation into fractals.

Higher Level 

(a) All HL IA investigations from 1998 to 2009:  Lots more excellent investigations – with some more difficult mathematics.

(b) HL IA investigations 2011-2012: Some more investigations with teacher guidance.

(c) HL IA investigations 2012-2013: Some more investigations with teacher guidance.

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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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