Teaching Ideas

Read Euler, read Euler, he is the master of us all” written by Robin Wilson or “Euler: the master of us all” written by William Dunham are to show us how great and multifaceted Euler was as a mathematician. Indeed, he was. In this post, I want to write how great he was as an educator.

The first step was to ask the students to list “those numbers which arise from the sums of two squares” (Paper I), say up to 50 (Euler himself listed them up to 200 without any use of calculators, mobiles, computers and so on!). The next step was to let them see and come up with any general statement that might be true for all such numbers, or for the numbers which cannot be written as the sum of two squares. I believed whatever they come up with could be found somewhere in Euler’s texts, perhaps not the exact things, but for sure, something quite related. Thus, after examining the statements found in the class, I could direct them to the Euler’s text where he addresses the same statement or something similar. It was the plan and it went better than I expected.

The first observation* was that powers of 2 can be written as a sum of two squares:

​( 2=1^2+1^2, 4=2^2+0^2, 8=2^2+2^2, 16=4^2+0^2, … )​

We called our first observation Theorem 1.

Theorem 1: Any power of two can be written as a sum of two squares.

The students had some experience in “university mathematics”, but most of them felt no need to prove our first theorem since it was true for all the examples on the blackboard! Here again, Euler comes in rescue, saying “in this class (dissertation) of many such statements (propositions), which until now have been accepted without proofs, we (I) will furnish proofs of their truth” (Paper I).  Thus, I asked students to prove Theorem 1. To my surprise, they chose to prove it by mathematical induction (perhaps, because they had a lot of such proofs in the first semester). This is the way they did it:

Base: ( 2=1^2+1^2 )​

Assume that ( 2^k ) can be written as a sum of two squares.

​( 2^(k+1)=2. 2^k ) . Thus,( 2^(k+1) )can be written as a sum of two squares.

It was not easy for students to see why the product (here, ( 2^(k+1) ) does not automatically inherit the property of its factors (here, the factors are 2 and ( 2^k ) and the property is being a sum of two squares). Again Euler has something to say about this inheriting phenomenon. For example, a number that is a sum of two squares but neither of its factors is a sum of two squares. I used a silly example, giving that 3 and 5 are prime numbers (the property here is being a prime) but their product is not a prime number! This discussion put forward three options:

(i) Ignore proving Theorem 1 (that indeed wasn’t an option)

(ii) Choose a different direction to prove it.

(iii) Amend our failed proof.

The students chose the last option and this brought us to our next theorem.

Theorem 2: If ( m ) and( n )are two numbers, each of which is the sum of two squares, then their product $latex mn$ will also be the sum of two squares.

Interesting, Euler suggests a number of simpler propositions that are special cases of Theorem 2 before giving the general form, because by it (special cases) this (the general case) will be more easily observed (Paper I).  In fact, one of his lemmas (special cases) was enough to complete our proof by induction: If a number ( m ) is a sum of two squares, then so will be ( 2m ). However, we decided to proceed by proving the general case. Then, in addition to correcting our mathematical induction, we wrote 65 and 1105 (both suggested by Euler) as sums of two squares, using our proof of Theorem 2.

Euler has it all. He plays with examples, draw conclusions, warn you not to rely on them, seek proofs, guess when you might think wrong or overgeneralize, discuss them, give counterexamples and so on.

I haven’t yet distributed his papers in the class. This post was just about the first session. I’ll complete this story.

*Euler himself considered the square numbers first.

Artin Algebra!

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