We have started a book study group at my institute, the idea being to work our way through Finnur Lárusson’s “Lectures on Real Analysis”. This is a really nice (and thin) Real Analysis book rigorously covering the same stuff as a standard one-variable calculus class. On Wednesday I tried to talk about the first two chapters. I took the opportunity to think more deeply about something that has intrigued me for a long time, namely the role of the Distributive Law.
What makes me think is my observation that many of the problems students have with algebra seems to revolve around the distributive law. I may write on the whiteboard (as part of some calculation):
5(3+x) = 15+5x
“Hey, what did you do!?”
Perhaps I write:
x+3x^2 = x(1+3x)
“What happened here!?”
The law can be used in two ways, of course, and there are names for those operations. In Swedish we say “multiplicera in” and “bryta ut” respectively. I’m not quite sure about the colloquial English expressions? Anyway, the question I’ve been asking myself is: Are there any deeper philosophical or cognitive reasons for these problems?
Some years ago I read in the n-Category Café a post about the fact that the Real Numbers support two binary operations: addition and multiplication and that these operations are closely connected to the exponential functions and the logarithmic functions. Of course, everyone knows that logarithms translate multiplication into addition, something which was indeed one initial motivation behind their invention. Not so well-known, perhaps, is the fact that astronomers in the 16:th century before the advent of logarithms, used trigonometric identities to simplify multiplication. The method went under the name of “Prostaphaeresis”. Now that can be understood through the relationships between trigonometric functions and exponential functions of complex arguments. Another story … but a related pedagogical observation is that logarithms cause even more severe problems for many students. There are certainly many reasons behind this, not the least the abstractness of logarithms and the un-symmetrical notation where we often write exponentials in the suggestive way e^x (based on notation for powers) but logarithms in the more obscure way log x.
My observation is that the connection between addition and multiplication is done via the distributive law at the axiomatic level. Then this connection comes back when we do logarithms and exponentials. Can there be a deep connection explaining the pedagogical problems here? Don’t know – have to think more. So instead back to what I actually tried to say at the seminar.
I started by stating the axioms for addition and multiplication respectively:
Let a, b and c be symbols for real numbers
A1 a + b = b + a
A2 a + (b + c) = (a + b) + c
A3 There is a real number 0 such that a + 0 = a for all a
A4 There is a real number b such that a + b = 0 for all a
M1 a · b = b · a
M2 a · (b · c) = (a · b) · c
M3 There is a real number 1 such that a · 1 = a for all a
M4 There is a real number b such that a · b = 1 for all a
Here my colleagues protested that I must exclude a = 0 in the last axiom.
No you dont’, if these are the only axioms you have. What we have are two Abelian groups with no connection whatsoever between them. Because what excludes a = 0 in the last axiom for multiplication is precisely the distributive law.
The Distributive Axiom
D a · (b + c) = a · b + a · c
In a way it is remarkable (as I read somewhere) that there is just one axiom connecting the operations of addition and multiplication. Note also that it is multiplication that distributes over addition and not the other way around. Are we pointing this out to the students?
Anyway, if you instead start your mathematics with the natural numbers Ν and the Peano axioms, then addition and multiplication must first be defined recursively from the successor operation and then step by step be extended to Z, then to Q and then to R.
As long as one of the numbers in the product a · b is a natural number, we can think of multiplication as repeated addition, and that is the basic intuition behind defining multiplication in Ν from addition. Now if we start with R directly, then we need something else that provides a connection between the two binary operations. That is precisely what the Distributive Law does.
If we don´t have it, then we cannot write a + a = 2 · a. Let’s prove it using D.
Th1 a + a = 2 · a
a + a = [M3] = a · 1 + a · 1 = [D] = a · (1 + 1) = [Def: 2 ≡ 1 + 1] = a · 2 = [M1] = 2 · a
The distributive axiom is crucial here. In this sense, it “captures the idea of multiplication as repeated addition” and generalizes it to both numbers in a · b being arbitrary.
Then for division by zero. Suppose there is a real number b such that
b · 0 = 1
Then 1 = b · 0 = b · (0 + 0) = b · 0 + b · 0 = 2 · b · 0 = 2 · 1 = 2
where I have only used axioms and Th1. A contradiction arises. Perhaps one should add that we must suppose that 1 ≠ 0.
Update: As a colleague was kind to point out I can arrive at the contradiction easier by just adding the additive inverse to b · 0 (axiom A4) to both sides of
b · 0 = b · (0 + 0) = b · 0 + b · 0
Then we get at once 0 = 1 and we see that we indeed need to assume 1 ≠ 0. Thanks Bengt!
Let me end with the pedagogical reflection. No one has any problems with addition. Multiplication is a little bit more difficult, but no students have any serious problems with it. The problems start precisely where the operations starts to interact through the distributive law. We as teachers takes it for granted being an absolute triviality. I don’t think it is. I think it is deep.
One operation gives us group theory. Two connected operations give us algebra. Is it a wonder that it is difficult, especially since we don’t really do any group theory as a preliminary?