References and an Originality Question
In 1976 (or 79?) I saw Richard Feynman in guest lectures at McGill
University describe the addition and multiplication of vectors in the
plane using parallelograms for addition and the rule add angles,
multiply lengths for multiplication. Since then I have wondered how
to transform his presentation in a development of complex numbers that
could be given before the start of trigonometry. This essay is the last
result - clearer and better than previous ones. It shows how complex
numbers may be geometrically introduced with a level of rigour sufficient
for most, before the introduction of periodic trigonometric functions.
Since Feynman's presentation, I have looked for easier and simpler ways
to introduce complex numbers. Chapters 22 and 23 followed in 1995-6. That
included a geometric proof of the distributive law. After after finding
and exploring several other ways to interpret and derive the distributive
law, I found a simple rotate a triangle-midpoint development given in
this site section.
Fehr's development of the complex numbers in his 1951 textbook, a
textbook I finally met in 2010, introduces the complex numbers with the
aid of rectangular and polar coordinates, and the leaves the proof of the
distributive law as an exercise. So the site development here of complex
numbers is not new. Except the line segment, mid-point rotation proof of
the distributive law given in this site section and announnced in fall
2009 is so simple, earlier knowledge of it would have provided simpler
routes for instruction in complex numbers and trignometry, routes not
seen. Is the mid-point rotation proof may be original? That is the originality
question.
References
-
Mathematical Thought from Ancient to Modern Times, by Morris
Kline, as three volumes (1990, published by Oxford University
Press).
In volume 2, Chapter 27, the third section called The Geometrical
Representation of Complex Numbers briefly describes the approach
of Caspar Wessel (1745-1818). Part of Wessel's work (translated into
English) is reproduced in David Eugene Smith's 1929 work A Source
Book in Mathematics, Dover 1959 Reprint.
-
What is Mathematics, R. Courant & H. Robbins, Oxford
University Press, Fourth Edition.
Classic Work. This may be taken a prequel to the discussion in the
1950s of what should be taught in pre-university mathematics. Very
readable for undergraduate students in mathematics. The geometric
interpretation (or representation) of complex numbers assumes the
addition theorems (angle sum formulas) for sine and cosines in order
to show how to multiply complex numbers using moduli and angles.
-
Secondary Mathematics, A Functional Approach for Teachers, H. F.
Fehr, D. C Heath and Company Boston 1951.
The chapter pp254-296 on complex number systems and trigonometry gives
as a exercise for students (!) the task of giving a geometric proof of
the distributive law for complex numbers when multiplication is defined
by multiplying moduli and adding angles. This site December 2009 proof
below give the most recent and simplest site proof of the distributive
law, the simple proof I have looking for since seeing Feynman in 1976
describe physics in terms of adding and multiplying vectors in the
plane. I should have known about Fehr work earlier. My copy was a fall
2009 gift from a McGill University colleague.
Addendum: That exercise, the task of providing a
geometric proof of the distributive law - the key element of this
essay, exercises raises the question of why complex numbers were not
geometrically developed before trigonometry in the course designs of
the 1950s or 60s. Some inquiry or research is now needed to
determine what proofs were available then. It appears that the
development in this essay or webpage represents a mix of inventions or
re-inventions, with which is which not clear to its author, me.
-
A History of Algebra from al-Khwarizmi to Emmy Noether, B. L.
van der Waerden. Springer Verlag, ISBN 3-540-13610-X, 260+ pages.
Page 178 says the following regarding complex numbers: Euler ... did
not give a satisfactory definition. Clear, geometrical definitions
... were given by Caspar Wessel in 1997, by Jean Robert Argand in
1806, by John Warren in 1828, and by Carl Fredrick Gauss in 1831.
...William Rowen Hamilton defined (1843) the complex numbers as pairs
of real numbers subject to ... rules of addition and multiplication.
Augustin Cauchy interpreted (1847) the complex numbers as residue
classes of polynomials,..., modolo x2 +1
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
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Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
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70
Calculus Starter Lessons
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They cover basic topics in ways likely to complement your
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if one or more explanations is not to liking, try another. It may
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
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if the come easy, may be deceptive - provide a too light and not
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calculus and more generally in the first year of college. Bon
Appetite.
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