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This topic, Complex Numbers, arrives when our students (who are extension 2) have much of the background to start to put the whole beautiful structure together. For some the elegant formality of the formal constructions they may study as undergraduates, if they take the pure mathematics path, will provide that. Some of the reading we have proposed could help them get a glipse of this earier.

But for others there needs to be something else.

According to [B. L. van der Waerden, A History of Algebra, Springer Verlag, NY 1985.],

“Cardano was the first to introduce complex numbers a + √−b into algebra, but had misgivings about it.”

In Chapter 37 of Ars Magna the following problem is posed:

“To divide 10 in two parts, the product of which is 40”.

It is clear that this case is impossible. Nevertheless, we shall work thus:

We divide 10 into two equal parts, making each 5.

These we square, making 25.

Subtract 40, if you will, from the 25 thus produced, as I showed you in the chapter on operations in the sixth book leaving a remainder of -15,

the square root of which added to or subtracted from 5 gives parts the product of which is 40.

These will be 5 + √−15 and 5 − √−15.

Putting aside the mental tortures involved, multiply 5 + √−15 and 5 −√−15

making 25 − (−15) which is +15. Hence this product is 40.

perhaps re the lesson connecting to Argand Plane, use:

There are indications that Gauss had been in possession of the geometric representation of complex numbers since 1796, but it went unpublished until 1831, when he submitted his ideas to the Royal Society of Gottingen. Gauss introduced the term complex number

If this subjet has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed. Had +1, -1 and √ −1, instead of being called positive, negative and imaginary (or worse still, impossible) unity, been given the names say,of direct, inverse and lateral unity, there would hardly have been any scope for such obscurity.

just like calculus was re-defined in terms of limits to eliminate “impossible” infinities, so complex numbers are redefined considering groups and sets … but this requires a great deal more theory:

Augustin-Louis Cauchy (1789-1857) initiated complex function theory in an 1814 memoir submitted to the French Acad´emie des Sciences. The term analytic function was not mentioned in his memoir, but the concept is there. The memoir was published in 1825. Contour integrals appear in the memoir, but this is not a first, apparently Poisson had a 1820 paper with a path not on the real line. Cauchy constructed the set of complex numbers in 1847 as R[x]/(x2 + 1)

We completely repudiate the symbol √ −1, abandoning it without regret because we do not know what this alleged symbolism signifies nor what meaning to give to it.

… from pp2 to top of numbers … that the ancient pyth, cubis and quadratics are linked here with complex

Then consider Everett, Lockhart for the creation of numbers ‘so far’.

Consider Shrödinger, Gleick, Gallileo and finally Penrose for narratives of science, mathematics and the excitment of the journey. Penrose is about complex numbers but only for a few. Yet keep him in mind … one of my fellow pure maths majors was there, in part, because he was given a copy in school.

Consider Weeks, if you want something leading up to Penrose, or really building the spatial imagination that will help in this topic. Consider how all these ideas were developed, the stories are very real, they can make the mathematics come alive.

So … for some we have them in the formality, providing we do not neglect that. For others, I hope the historical journey will help justify. Next we looked into the practical, the concrete applications.

<p>way by revealling the way we built these structures we might find ways to help teach them. It is very relevant to consider with Complex Numbers, as with Calculus

<p>And these stories, especially the more recent ones where the narrative of discovery is known, can make great reading … for the teacher (perhaps arming them with some nice resposes to curly questions), and for our students. Mathematics is creative, it is exploration, imagination, discovery and above all “just making it up” … in a very special way of course.

<p>Here we propose an introductory lesson.

<p>After we have introduced the new number, as an invention, with the suggestion that this new invention did prove usefull … so it is stillbaround centuries later.

<p>As a teacher consider some of the historical background. Use it to build on the “just make it up”

schrodinger | gleick-chaos | galileo | copernicus | penrose-physics

… a rough outline of the overview:

(I see everything we have found so far linking from here … see the site outline, so lesson exerpts and I have abparagraph or so onbeach book I have mentioned)

A review of the syllabus document reveals a topic which focusses on the abstract, there are not any specific applications mentioned.

This presents some challenges, and opportunities.

What the syllabus outlines is algebraic, geometric and trigonometric aspects of complex numbers. Their power is they unify what, till this stage in the students journey through mathematics, had been somewhat distinct areas.

Generally geometry is one topic, algebra another, split into their strands. There have been opportunities where they intersect, Pythagorus theorem keeps cropping up through the years with triangles, and into algebra. Quadratics and cubics relate to area and space.

The history of our development of these ideas can help understand the relationships, and by revealling the way we built these structures we might find ways to help teach them.

And these stories, especially the more recent ones where the narrative of discovery is known, can make great reading … for the teacher (perhaps arming them with some nice resposes to curly questions), and for our students. Mathematics is creative, it is exploration, imagination, discovery and above all “just making it up” … in a very special way of course.

And this topic, complex numbers, arrives when our students (who are extension 2) have much of the background to start to put the whole beautiful structure together. For some the elegant formality of the formal constructions they may study as undergraduates, if they take the pure mathematics path, can provide that. But for others there needs to be something else.

Complex numbers, at this stage are the keystone that links the strands, but that is a very formal thing, and it was not the original motivation. It is something discovered later.

The introduction of ‘i’ is hard.

We have hammered the impossibility of taking the square root of a negative number overnand over agian. This may have been a mistake, but we have. But in this respectbwe are just like the mathemeticians of 1530s.

Bagni considered this, and suggests that the historical order could inform or teaching. He tested this on students. The quadratic setting, with the final answer as ‘i’ just hits rejection, this is not allowed, ‘i’ is not the solution … there is no solution. But going via a complex number to arrive at a real solution, a solution that can be seen to ge correct independently, is much more believable. The historical narrative shows the resistance, Bagni confirms the effect on students. He proposes using the historical artefacts of Borrelli’s publication that first clearly laid out the system.

Here we propose an introductory lesson.

After we have introduced the new number, as an invention, with the suggestion that this new invention did prove usefull … so it is stillbaround centuries later.

As a teacher consider some of the historical background. Use it to build on the “just make it up”

Then consider Everett, Lockhart for the creation of numbers ‘so far’.

Consider Shrödinger, Gleick, Gallileo and finally Penrose for narratives of science, mathematics and the excitment of the journey. Penrose is about complex numbers but only for a few. Yet keep him in mind … one of my fellow pure maths majors was there, in part, because he was given a copy in school.

Consider Weeks, if you want something leading up to Penrose, or really building the spatial imagination that will help in this topic. Consider how all these ideas were developed, the stories are very real, they can maie the mathematics come alive.

So … for some we have them in the formality, providing we do not neglect that. For others, I hope the historical journey will help justify. Next we looked into the practical, the concrete applications.

Contemporary usage on the cutting edge of physics aside … that stuff is amazing, but tough.

We looked at were else it is used.

This video lays out the linkage to another step in our exquisite structure … but matrices are for another year, now is not the moment to make anothe great leap of invention! (in a better world, perhaps a syllabus less obsessed with 17th and 18th century mechanics, we might have less calculus and more modern mathematics in mathematics advanced, maybe even a little something in stage 5)

A lot of the places that complex numbers solve certain algebraic situations requires to much setting up, too much from outside the syllabus. We have started with the cubic, and these were some of the original motivations, but mostly they are rather esoteric.

But there are practical science applications.

A lesson can be created by pulling suitable tasks out of a method working with mechanical vibrations. We would show this video first, then consider one or two parts as an exercise in compkex numbers.

A lesson could also be derived from some of the calculations involved in electrical circuits. The power of this area is that complex numbers convert a difficult feedback problem involving differential equations into a much more straightfoward problem in geomerty, within the Argand plane.

initial proposal … what have I left out?

• overview … note “outline research used” is the report required, with links to resources
  • 1000 words for the “report” as a website is VERY short … but must be a 1000wd report on research done, with links to resources.
• issues
  • getting over years of destructive drill: “you cannot take square root of negative number”
  • finding an accessible use-case outside the classroom
• opportunities (a part of the use-cases, but more ‘pure’)
  • linking algebra, geometry and trigonometry in a way nothing else can
    • complex plane is the cartesian plane and more
    • then the unit circle, viewed in the complex plane, opens up an immense vista
  • the equation … e^iπ+1=0 … links every core number (it in a sense defines complex indices … but I need to think about this statement further, it must be introduced exactly in the right way, at the right time, probably best as we start to look at the unit circle just after multiplication as add angles, multiply lengths and an exploration of many roots of 1 and -1 … in a careful extension of indices … this is a tricky jump)
• resources … these are the “links” from the “report”
  • lesson excerpt 1
  • lesson excerpt 2
  • lesson excerpt ….
  • Borrelli’s explanation, images, translation, modern form
  • The story of the invention, disbelief, struggle, personal conflict
  • Something re modern physics, something reasonably accessible
  • ……
  • A set of links for the more adventurous, more advanced
    • Penrose … “only for the very game!”
    • ???
• syllabus links
  • previous knowledge
    • the starting point
    • consideration of the order in the year… esp wrt vectors
  • where it leads
    • these are extension 2, many will do undergrad maths, a summary of what is next
  • academic reference list