Today I finished the 160 exercises in the review of algebra that I’ve been working through. This is an important achievement for me. Over the years, I’ve made repeated stabs at reviewing math, but I’ve never been able to tolerate it long enough to finish anything. To finish such a marathon problem set is positive proof that this time is different.
There isn’t much to talk about mathematically. One of the final exercises was to prove that $|ab|=|a||b|$, which I was prompted to do using the fact that $|a^b|=|a|^b$.1 Yet surely the latter statement is proven using the former. Isn’t it? In the end, I decided that doing the proof that way would be silly, and opted for a somewhat clunky casewise argument instead.
Anyway, happy Algebra Completion Day, everyone!
- To do (see comments). ↩︎
And Happy ACD to you, too!
What a huge thing! I absolutely celebrate Algebra Completion Day with you!
With regard to the Problem you mention, good spotting. The hint in the book is wrong. The solution in the book doesn’t use Equation 3, $|a^b|=|a|^b$. What it uses is Equation 13, $|a|=\sqrt{a^2}$, which is right above Equation 3 in the book. I won’t write down the solution they give, in case you want to do it with the right hint. It’s pretty slick.
Still smiling and celebrating Algebra Completion Day.
Ah-ha. That makes more sense. I will see what I can come up with.
I’m actually wondering now whether it’s possible to prove that $|ab|=|a||b|$ from the fact that $|a^b|=|a|^b$. I will think about that, too.