Without relying on a background in calculus, analysis, abstract algebra etc., is there an elementary way to extend the definitions of basic operations from $\mathbb{Z}$ to $\mathbb{R}$. For example, the "repeated addition is multiplication" approach coincides with the typical definition in $\mathbb{N}$, but this clearly isn't sufficient (it's straight up incorrect.)

How can one teach multiplication without depending on an axiomatic treatment of the operation? Same with exponentiation, I clearly do not want to rely on $\ln$ or limits. This has more or less immediate import for students, who are say, learning about $\pi$. Take the commutative property of multiplication:

Clearly, if you have 5 baskets of 3 apples, it is abundantly clear that $3\cdot 5=5 \cdot 3$. However, while one can have $3 \pi$'s or $3/4(\pi)$'s but the reverse order doesn't have an easy justification (intuitively or otherwise.)

alternative answer: Is there a way to convince students that the basic laws of commutativity, associativity, etc. still work? I think we would have to certainly undo the "repeated operation $-$" mentality first. But then how can I "fill the void," and explain what each is doing?

One geometric possibility is to describe $\times: \mathbb{R}^2 \to \mathbb{R}$, as taking two points on the $y$-axis and $x$-axis and defining $x\times y$ to be the $x$-intercept of line parallel to the segment joining $x$ and $(0,1)$, but also going through the point $(0,y)$. This geometric approach might help, although we would want to show that it coincides with our current definition, it has the properties we want, etc.

Here is a resource that continues in this way: http://www.math.csusb.edu/faculty/pmclough/MP.pdf


Let me offer some thoughts, in the hope that others will do better.

It all depends, doesn’t it, on what your students’ understanding is of the behavior and nature of real numbers? Let’s suppose you make a good correspondence between real numbers and lengths of segments. Then you can define multiplication by means of ratios: $ab$ is defined to be the thing that fits into the proportionality $a:1 = ab:b$, and this can be gotten computationally and defined by using similar triangles. Then if they believe the basic facts about similarity, they should be able to see (or prove) that you get the same thing with the proportionality $a:1=ab:b$.

For exponentiation, I think you’ll have to be satisfied with integer (or, conceivably, rational) exponents. The definition of $3^{\sqrt2}$ must certainly wait for analytic sophistication on the part of the student.

  • $\begingroup$ thank you for your answer. I think that using triangle similarity is indeed useful, I had considered doing this, and I think it is a nice approach. Have you tried to explain multiplication in this way, successfully? Here is a resource that I just found, using the construction I give in my OP (I wonder if this is secretly where I remember seeing that definition in the first place...) math.csusb.edu/faculty/pmclough/MP.pdf $\endgroup$ – Andres Mejia Jul 8 '16 at 15:04
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    $\begingroup$ I fear that almost all the students in our schools here in the States are perfectly happy to accept all sorts of things as known or as given. Indeed, the problem is more often to persuade them that many false relations, like $(a+b)^2=a^2+b^2$, really are false. $\endgroup$ – Lubin Jul 9 '16 at 2:47
  • $\begingroup$ I am also tutoring in the U.S, although sometimes rather advanced students (in New York for the most part.) I believe your correct, but once you say something like "multiplication is not repeated addition," you really do get questions concerning what it "actually is," in the words of a student. Also, +1 for freshman's dream, I thought it was a hoot when it worked out in commutative rings with prime characteristic (to show the frobenius endomorphism.) $\endgroup$ – Andres Mejia Jul 9 '16 at 3:47
  • $\begingroup$ I've accepted this answer, since it doesn't appear that anymore will be coming. This is still on my mind, however. $\endgroup$ – Andres Mejia Oct 7 '16 at 13:32

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