Pumpkin Patch and 18 month check up

I never figured us for a family-outing-to-the-pumpkin-patch-kind-of-family, but we're game for stuff that other people will organise and it was really fun. Thanks, Rebecca! Thanks also for remembering and lending your camera.

Adam started out really grouchy because
a) He was strapped into the car for so long. and
b) We arrived around 11:30, which is his natural nap time.

But after a short while, he really got into crawling between the pumpkins and finding compfy pumpkin "chairs." He also enjoyed the wagon ride from "Fia" (Sophia Caranci). I'm quite happy about how the blue coat contrasts with the pumpkins. I didn't plan it that way (this year) but I probably will in the future.

Also. At his 18 month-check up, Adam got his last two vaccinations until he's four (not counting flu, which we'll do again every year, probably). He was highly uncooperative with the weighing and measuring but the stats are 28 pounds (75th percentile) and 35 inches (99th + percentile). The old site I used to use to get more accurate percentiles now requires registration, so fuggetaboudit.

caught red-handed

Adam thought that 10am was a better time for bath than after dinner. We had to do a whole proper bath because this soap will give him a nasty rash.

not as yummy as I was expecting.

O hey! Mom! Camera!

uh. We're all good, right? right? I'm smiling pretty, see?

Thanksgiving

hanging out with Clare and Becky

hanging out with Dad

a cool moment

When I went to get Adam up from his nap yesterday afternoon, there was a bright, narrow sunbeam shining through his otherwise dark room. I showed it to him by passing my hand and his hand through it a few times.

"What is it, Adam?" I asked.

"Bubbles."

Bubbles? I looked closer and indeed the small specks of dust in the air were reflecting all the colours of the rainbow.

I thought I understood exponents ...

until I asked my 12U students to make a conjecture about whether an odd or even function taken to a natural number power becomes odd, even, or neither. I didn't ask for a proof, it was just a little 2-mark thinking question.

In case you need reminding: x^2 is even because (-x)^2 = x^2. sin(x) is odd because sin(-x)=-sin*(x).

Anyway, it got me thinking: what does a real number exponent do to the symmetry? What does a real number exponent MEAN anyway? I thought I had a handle on rational number exponents. But even then, the domain (valid x values) gets messed up. And now my real problem (though not real enough for me to look up the "answer....")

What about irrational (or better yet, transcendental) exponents? I'd say, "Who cares? just approximate to a rational," but it comes up in polar coordinates all the time. e^ipi anyone? My graphing calculator seems to tell me that
x^2pi
(x^pi)^2
(x^2)^pi
are all different. Try a table of values. What does -0.7^2pi even MEAN? I don't know how to calculate that by hand. Excel should work for those lacking graphing calculators.
Actually: I was putting x^2pi into my calculator wrong. It's graphed the same as (x^2)^pi.

How's my calculator doing it anyway? Taking an approximation of pi, converting it to a rational, then doing the exponent? Then the different curves (domain aside) are simply artefacts of rounding errors that blow up due to exponentiation? Or Dion suggests that it might be interpolating between two nearby rationals. Rob thinks it probably depends on the calculator but likely it just chooses one nearby rational.

Maybe it all makes sense if exponents are seen as the inverse of logarithms (which were invented first, BTW). But then I would need to understand logarithms.

Discuss.

Uh yeah, whatever, Karen.