[ExI] dicovery of irrational numbers

[Mike Dougherty]

On Wed, Dec 18, 2013 at 12:58 PM, spike <spike66 at att.net> wrote:

> 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.
>
>
>
fight fire with fire?

Draw a logarithmic spiral on a golden rectangle using a largish piece of
paper.

Now flip the paper over and draw the squares that correspond to the spiral
on the opposite side.

While keeping the spiral side hidden, cut the rectangle(s) into the
squares.  I love the sound of a heavy paper cutter making all those cuts.
 The beauty of that is you keep picking up the cutaway part and chopping
off smaller rectangles and leaving the various squares on the board part of
the cutter.  Repeat until you have too small of a rectangle to cut safely.

Now that you've made the square puzzle pieces, flip them over and
reassemble the rectangle so the spiral is visible.

I have no idea if he'll get logarithms just yet, but the inherent beauty of
the spiral is intuitive.  This golden ratio (phi) strikes me as a nearly
magical thing.

Infinite series and asymptotic approaching limits is cool, but a bit harder
to grasp.

Have you talked about the population of breeding rabbits that leads to the
Fibonacci series?  That's a series that's pretty intuitive (even in 2nd
grade) - then use the ratio of any two sequential numbers to show that the
later you go in the sequence the closer that ratio gets to phi.  So maybe
your approach to the irrational should also be asymptotic?  :)
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