The glee of (geek) children

[drwex]

This AM my younger informed me that he has named a pet stuffed cephalopod "Octo-Pi".

He carefully explained that this means "pi to the eighth" but confessed he's having trouble calculating it. I offered to help him with that tonight; I hope he's in a mood to accept approximations.

Comments

[sariel-t.livejournal.com]

Excellent.

[shades-of-nyx.livejournal.com]

That's excellent!
Geek smalls make me smile.

[chienne-folle.livejournal.com]

Apple, tree, no distance. :-)

[drwex]

While I'm happy to take credit for my kids' geekiness, this interest in mathematics is very unlike his father. I am reasonably certain I'm the most innumerate person ever to get an MIT Ph.D.

[gsh]

Teach him about logarithms, and how ln (pi^8) = 8*ln(pi)

[sweetmmeblue.livejournal.com]

I talked to him about the infinite nature of pi and how people would write the way you have it in (). He wants to work it out with the first 4-5 digits but he also wants to use a calculator.

[vibrantabyss.livejournal.com]

So, who gets to explain to him that the proper plural is "octopedes", with "octopuses" as a secondary option?

Sadly "octopi" is like "five" (ie right out)

[wotw]

Use the fact that Pi^2 is extremely close to 10. This is enough to make an excellent approximation to Pi^8.

[wotw]

This is a little more ambitious, but I bet he can handle it, and he'll learn something:

1) Ask him to think about what fraction of numbers have no square factors (other than 1).

2) Make an estimate by seeing how many numbers up to 10 have no square factors (2,3,5,6,7,10) --- okay, that's about 60%.

3) Tell him that the exact answer is known to be 6/Pi^2.

3a) Hope that he doesn't ask you for a proof of 3).

4) From 2) and 3), we have 6/Pi^2 is approx .6. Have him figure out what this tells him about Pi^2.

5) Now that he knows Pi^2 is about 10, have him estimate Pi^8 = (Pi^2)^4=10^4.

___________


Extra credit:

6) Instead of using the numbers from 1 to 10, have him use the numbers from 1 to 100. If I just counted right (which I'm not sure of), he'll find that 63 are square-free. (First have him count the multiples of 4, then the multiples of 9, etc, then correct for double-counting --- e.g. you don't want to count 36 as both a multiple of 4 and of 9.)

7) Use this to get an even 'better" estimate of Pi^8.

8) Discover that your estimate just got worse, not better.

9) Talk about how taking a bigger sample can sometimes make your results less accurate --- but taking even *bigger* samples (e..g. counting all squarefree numbers up to a million, or a billion) is guaranteed to give you better accuracy.

[vibrantabyss.livejournal.com]

I am familiar with that as the chances that two numbers are relatively prime. It was cool to lay it out and realize those are one and the same.

[drwex]

https://www.youtube.com/watch?v=wFyY2mK8pxk&feature=youtube_gdata