The Barenaked Ladies’ “Snacktime” is on very heavy rotation in my house these days. It’s officially an album for children (which explains the heavy rotation, because if kids like something once, they like it for approximately the next billion times). However, a lot of it is laugh-out-loud funny for adults. For example, from the alternate alphabet song:
D is for djinn, E for Euphrates,
F is for fohn, but not like when I call the ladies.
But I digress.
The first song on the album is “789″, about the nefarious dealings of the number 7.
1, 2, 3, 4 and more makes 7
Why is six afraid of 7?
Cause 7 ate 9
Recently the eldest kid piped up: “Seven eats all the numbers. There are no more numbers after 8.” I asked why. “Well, seven ate nine, so it’s 7-8-10, so then seven ate ten, so it’s 7-8-11, so then seven ate 11, and then it just keeps going.”
So, the Barenaked Ladies just inspired my seven-year old to discover the principle of mathematical induction, which is one of the first techniques you learn when you venture into the land of advanced mathematics. The idea is that if you can prove that something is true for some integer n, and that it is also true for n+1, then it has to be true for all integers greater than n. So, for a simple (and somewhat silly) example, if you can first prove that if n>0 then n+1>0, and then you also prove that 1>0, then all positive integers are greater than zero. I remember having a hard time wrapping my head around this idea when I first bumped into it in high school (though I got over it in college after enough algebra classes with Michael Artin). I just find it pretty nifty that you can get the idea from a kid’s song.
August 5th, 2008 at 6:22 am
Did you use Artin’s textbook “Algebra”? If so, how did you like it? I love it, but people say students don’t like it, so I haven’t taken the leap and used it in my own classes.
August 5th, 2008 at 6:51 am
I used the notes that he was (apparently) developing into a textbook. His classes were phenomenal. I looooooved algebra as a result.
August 5th, 2008 at 9:14 am
Thank you!
It is possible you have saved my sanity…
August 5th, 2008 at 9:25 am
Finally! A good explanation for why Pluto is no longer a planet.
August 5th, 2008 at 10:48 am
Just a pet peeve, but maybe you should have said “The idea is that if you can prove that {IF something is true for some ARBITRARY integer n, THEN it is also true for n+1} and {the thing is true for some specific integer k}, then it has to be true for all integers greater than k.
Maybe the math majors can back me up on this. (sorry for nit-picking)
August 5th, 2008 at 11:31 am
I like Artin’s textbook, and he is a great teacher. However, by using his own book, the course did suffer a bit from the fact that you couldn’t get alternative viewpoints from the lectures and the text. This was more of an issue with the second half of the book–on rings and fields–than with the first half, at least for me. (On the other hand, I have little sympathy for any MIT student who can’t find another algebra textbook if they need a different take on the material.) The book also includes a fair amount more than Artin covered in his lectures.
August 5th, 2008 at 12:05 pm
That is totally awesome.
I think Artin’s class, called “Algebra I”, was the hardest class I ever took, but also fantastic!
August 5th, 2008 at 12:29 pm
I don’t know what’s more funny, the song or the fact that it was written by a band I used to play really loud to annoy my parents back in middle school.
August 5th, 2008 at 4:21 pm
That is one smart kid you got there!
August 5th, 2008 at 4:39 pm
“I remember having a hard time wrapping my head around this idea when I first bumped into it in high school…”
Really? I found it a pretty simple (if immensely cool) concept and I’m by no means a maths whiz.
August 5th, 2008 at 5:33 pm
wouldn’t there only be one number left, the number 8?
if 7 ate 9 and 6 was afraid
then wouldn’t that mean that 6 saw 8 heading towards it instead of after 10?
otherwise why would 6 be worried if 8 was always walking further and further away from it?
so I think that would mean that there would only be 1 number left at the end.
The number 8
After it ate 9,6,10,5,11,4,12,3,13…
Michael.
August 5th, 2008 at 5:44 pm
um let’s try that again:
wouldn
August 5th, 2008 at 6:41 pm
My favorite math joke is the one about Noah’s instructions to the animals when they departed the ark: “Go forth and multiply”. The snakes, who were back then called adders, insisted “We can’t multiply; we’re adders!” Noah’s reply: “Use logs”.
August 5th, 2008 at 8:30 pm
Teacher: Now we know A_1 is true. Now suppose that A_n is true..
Student: Why?
Teacher: Because we want to use induction.
Student: But what if it isn’t true?
Teacher: Let’s consider the possibility that it is.
Student: Why not consider the possibility that it isn’t? I mean, I thought we were supposed to *prove* this thing, and here you are *assuming* that it’s true! Anyway I bet you already *know* that it’s true…….
August 6th, 2008 at 4:18 am
Mark: there are several species of snake called adder.
August 6th, 2008 at 8:00 am
Another kid’s song that implies mathematical induction is:
One elephant began to play
Upon a spider’s web one day,
He found it such tremendous fun
That he called on another elephant to come.
Two elephants began to play…
(etc.)
It works even better in Spanish: the Spanish version of this song changes the third line, the reason for calling the n+1 elephant, to “Since they saw it resisted…”. So the song allows us to conclude explicitly by induction that a spider web can resist the weight of an infinite number of elephants!
August 6th, 2008 at 11:41 am
I think this is one of the most beautiful applicatons of induction:
Ramsey’s theorem. It is accessible to highs school students. A special case of Ramsey’s theorem is the Party Problem : If you invite too many people for a party, you’ll either have more than n people who all know each other or more than m people who do not know each other. For given n and m there exists a number
R(n,m), the so-called Ramsey’s number, which is the minimum number of people you must invite for the party for this to be true.
Surprisingly, even for small values of n and m, not much is known about the Ramsey numbers. E.g. the value of R(5,5) is not known.
August 7th, 2008 at 10:04 am
Joshua wrote:
I agree with your nit pick. Saying “if you can prove that something is true for some integer n, and that it is also true for n+1, then it has to be true for all integers greater than n”, as Julianne did, makes it sound like anything that is true for, say, 1 and 2, must also be true for all greater integers. In fact the key point is that being true for any n implies it’s true for n+1. This and the fact that it’s true for some specific n are what you need for induction.
Of course, Julianne surely knows this and was just being sloppy with language.
August 7th, 2008 at 10:30 am
Thanks for backing me up on that, TimG.
August 8th, 2008 at 6:43 am
[...] Mathematical Induction for Seven Year Olds | Cosmic Variance. [...]
August 8th, 2008 at 8:49 am
The problem with Mathematical Induction “is a Philosophical one” as far as this relates to proof and the deductive process? It would be somewhat of a contention raised, in terms of Proof and Intuition in the mathematical community?
August 10th, 2008 at 1:53 pm
The Almighty Bob wrote:
Maybe this is why Mark called his little story a “joke”.
August 10th, 2008 at 7:35 pm
Curry’s paradox for 7 year olds
September 13th, 2008 at 2:41 am
so if n > 0 and it is true that n+1 > 0 then 1 > 0, well that does not make sense to me because the only way that holds true is if n = 0 for n+1, there by confirming that n is not greater than zero. There has to be something wrong with this logic so please help me out.