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Wednesday's calculus class was interesting. I was introducing the students to Taylor series. If you don't know what those are, don't worry; I'm not going to get technical here. The notation we use for Taylor series involves the concept of "factorial". This is easy to describe: if n is a positive whole number, n! (read "n factorial") is the product of the first n positive whole numbers. So 2! = 1x2, 3! = 1x2x3, and so on. But there's something odd that comes up; Taylor series involve the factorials of every positive whole number - including 1 - and the factorial of 0 as well. 1! and 0! are both equal to 1. Now that last bit usually sounds surprising to students, and I decided to talk about it for a while.
First off, I gave them a couple of justifications; I'll put one of them under a cut.
( Products )
That justification met with appreciative murmuring from some of the students. But then I revealed to them the great secret of mathematics.
We make stuff up.
Mathematical symbols are, at bottom, labor-saving devices. We define them to make it easier to talk about, write about, and think about mathematics. The question is not, "What is 0!?". The question is, "Is there a value we can assign to 0! which will make our lives as mathematicians easier?" The answer to that question is "Yes", and that value is 1. The justification under the cut is a rationalization - or, more charitably, an attempt to explain why it is that that choice does make life easier.
I then pointed out that mathematical notation is still in flux - that even in the last thirty years or so, new symbols have been introduced which make even very basic mathematics easier to deal with.
This fired the imaginations of some of the students.
Student 1: So, that's why anything to the power zero is 1?
Me: Right. It's the same kind of reasoning. (I went on to talk about the history of exponents.)
Student 2: So, just anybody can invent mathematical symbols?
Me: Yeah. Then you just have to convince enough people that your symbol is better. (S2 didn't look too pleased by that caveat.)
Student 3: So why can't we just say that 0/0 = 1?
Me: You can. But there's a problem. There's a nice rule that says a x (b/c) = (axb)/c, and if you say 0/0 = 1 and try to keep that rule, you can show that 1 = 2. So you'd have to sacrifice that nice rule. Would it be worth it?
I love teaching students that particular lesson. I think it's really the first time that most of them get a glimmering of what mathematics really is, and it usually seems to blow their minds. (When I get them into the history of mathematics class, I really get to drop the hammer on them, by mentioning a short book by Richard Dedekind, mid-19th century: Was sind, und was sollen, die Zahlen? - "What Are Numbers, and What Should They Be?" The audacity of that title delights me.)
First off, I gave them a couple of justifications; I'll put one of them under a cut.
( Products )
That justification met with appreciative murmuring from some of the students. But then I revealed to them the great secret of mathematics.
We make stuff up.
Mathematical symbols are, at bottom, labor-saving devices. We define them to make it easier to talk about, write about, and think about mathematics. The question is not, "What is 0!?". The question is, "Is there a value we can assign to 0! which will make our lives as mathematicians easier?" The answer to that question is "Yes", and that value is 1. The justification under the cut is a rationalization - or, more charitably, an attempt to explain why it is that that choice does make life easier.
I then pointed out that mathematical notation is still in flux - that even in the last thirty years or so, new symbols have been introduced which make even very basic mathematics easier to deal with.
This fired the imaginations of some of the students.
Student 1: So, that's why anything to the power zero is 1?
Me: Right. It's the same kind of reasoning. (I went on to talk about the history of exponents.)
Student 2: So, just anybody can invent mathematical symbols?
Me: Yeah. Then you just have to convince enough people that your symbol is better. (S2 didn't look too pleased by that caveat.)
Student 3: So why can't we just say that 0/0 = 1?
Me: You can. But there's a problem. There's a nice rule that says a x (b/c) = (axb)/c, and if you say 0/0 = 1 and try to keep that rule, you can show that 1 = 2. So you'd have to sacrifice that nice rule. Would it be worth it?
I love teaching students that particular lesson. I think it's really the first time that most of them get a glimmering of what mathematics really is, and it usually seems to blow their minds. (When I get them into the history of mathematics class, I really get to drop the hammer on them, by mentioning a short book by Richard Dedekind, mid-19th century: Was sind, und was sollen, die Zahlen? - "What Are Numbers, and What Should They Be?" The audacity of that title delights me.)
