A.M. Big group today. Ran with Jeff, Mary Ann, Daniel, Chauncy, and Matt Anderson. The trail was slippery, so we did not go very fast. We went to Daniel and Chauncy's turnaround (3 miles), then back, then Daniel and Chauncy got off the trail with 0.46 to go, and the rest of us continued. Matt turned off to BYU about a half mile later. Jeff and Mary Ann finished 10 miles with me in 1:21:58. Then ran with Benjamin and Jenny. Benjamin and I ran 2 miles in 19:49, Jenny ran the first 1.5 with us in 15:42.
Benjamin and I started trigonometry. We are currently working on having him pass the pre-test for BYU Math 112. During breakfast I reminded him about the factorial function, had him play around with the calculator to appreciate how fast it grows, then explained the series for the sine, cosine, and the exponential function. He was excited about the pattern. Then I decided the most natural way to teach him the derivation of the sine and cosine of the sum would be by using the Euler's formula exp(i*x) = cos(x)+i*sin(x), so we went through that decorating the white board in the play room with geek writing.
Afterwards I contemplated how nicely everything happens to work out in the world of numbers. It is not just pretty. Even though math comes from an imagination of man, there appears to be some underlying internal beauty that goes beyond man's ability to create it. How odd. Man invents a set of simple and fairly obvious rules that roughly model the world around him, and then it turns out that those simple and obvious rules define a beautiful world with numerous laws of its own that manage to exist in perfect harmony with each other. A mathematician begins to discover those laws rather than create them even though he starts in a world of his own imagination. The fact that this is even possible to me is plain evidence that God created our world. If He had not, math would not work. The patterns would not be there to begin with. And if we were not His children, we would not have been able to spot and abstract those patterns.
P.M. 1 mile with Julia in 10:23.