[ExI] dicovery of irrational numbers

[spike]

Funny thing happened on the way to the forum: yesterday I was teaching my
son about irrational numbers.  He learned a couple weeks ago how to convert
any fraction to a decimal, so every fraction either terminates or repeats.
He entertained himself looking for the  ratio which has the longest
repeating pattern.  

 

Yesterday I used his compass to draw a radius and a half circumference, then
asked him to guess the ratio of their lengths.  First he guessed 3.  So I
constructed half a hexagon to show him it was more than three.  He next
guessed 4, so I constructed a 4R path that goes way outside the half circle.
So he guessed 3 and a tenth, which I found interesting in itself.

 

I then went on to explain irrational numbers, but his reaction really
surprised me: it really upset that little second grader.  His world just got
a bit messier.  The existence of numbers that cannot be expressed as a ratio
bothers the hell out of him.  I was vaguely reminded of the ancient Greeks'
reaction when they were confronted with the concept.  According to legend,
Hippasus discovered irrationals while on a sea voyage.  It is said to have
upset the other Pythagoreans so badly they hurled him overboard.  I don't
know if that is true, or if he was jettisoned for some other reason such as
for being a generic asshole, but I find it interesting that my own son would
be so bothered by the whole concept.

 

I am open to suggestion for how to present irrationals in a more friendly
way, especially since I have more bad news for that lad: most numbers are
irrational.  Depending on how you count them of course.  There are
infinitely many rationals of course, but there are infinely many more
irrationals, since you can give me *any* two rational numbers and I can give
you infinitely many irrationals that fit between them, larger than the
smaller and smaller than the larger.  Of course, I can also give you
infinitely many rational that fit between the two as well, but between each
of those between the two are infinitely many irrationals.

 

Aint math cooool?  {8-]

 

spike

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