I remember taking a Philosophy of Math class as an undergrad, and it was the first time I ran into the question, is math real? Or maybe a more nuanced way of putting it is, what kind of reality does it have? This video surveys the most popular theories. Warning, in case you have a lot to do today, you can wonder about this one a long time.
Units have been going through my mind a lot lately. I ran into this on Futility Closet. It made me speculate what a Hendree unit might be? Doesn’t everyone deserve a unit named after them? Any suggestions?
I always find it interesting that a US presidential candidate could win the popular vote, but lose the election because of the electoral college. The question I started wondering the other day is what is the highest percentage of the popular vote that a US presidential candidate could theoretically win and still lose the election? Let’s also assume that miraculously everyone votes.
I found this via free tech for teachers. What a lovely combination of math and science working together to demonstrate something rather non-obvious: learning to run without shoes helps reduce the impact of running on your body. The one thing they didn’t show was what someone that learned to run without shoes looked like running in shoes. Would it be even smoother?
Commander James Lovell’s arithmetic scratch work in one of the Apollo 13’s manuals sold a little over a year ago for $388,375. Check out this Gizmodo article. Particularly poignant is the transcript of the radio dialogue between Lovell and Houston as he is asking them to check his work. Any slight error could have meant death for the astronauts aboard. Talk about high stakes testing! …also wondering if this is the most valuable mathematical artifact out there?
Lee over at PrimePuzzle pointed me to this cool set of dice that a mathematician devised that decides in one roll who (for up to four players) goes first. The Go First dice are designed to produce no ties, and give everyone an equal chance at going first. What’s cool is that they work for 2, 3, and 4 players.
Questions: how would one construct such a set, and can we construct a set for larger numbers of players? Also, these dice look to be 12-sided. Could the same be done with only 4-sided dice?
Check out this post on Mathematica’s blog. It’s one of the best I’ve read in a while. Ever wondered what the best hangman words are? Did you even know that you cared? It’s amazing how strange, interesting, and beautiful a smart and creative person can make the ordinary. I call this the Jerry Seinfeld school of mathematics. It reminded me of this Dan Meyer post, tackling the problem of which line to get into at the grocery store.
Start with the operation addition. We define an operation multiplication by iterating addition. For example, 2*3 is defined as 2+2+2. And in general,
Likewise, we can form exponentiation by iterating multiplication. E.g. 2^3 is defined as 2*2*2. And in general,
Addition, Multiplication, and Exponentiation are operations we are very familiar, but it isn’t hard to see that this chain of operations can be extended infinitely. The next operation, often called tetration, can be defined as a^^b = a^a^a…a (where a is being raised to itself b-many times) The whole sequence of operation is sometimes called the hyperoperation sequence.
All of this is a long preamble to a question. Once you know the facts of addition, you can gain any fact of multiplication or a fact of any subsequent operation down the hyperoperation chain. However, is it possible to discover the facts of addition from the facts of multiplication by reversing the iteration? For example, knowing “3*2=6″ gives me the addition fact “3+3=6″, but can I get all the addition facts?
We write our expressions with the understanding that a reader will order the operations involved using PEMDAS, or as I like to write with my students P/E/MD/AS. This is a convention. On Mars though a very different convention is observed. They use SA/DM/E/P to parse their expression.
Question 1: How would a Martian evaluate the following expression?
We, of course, get 50 using PEMDAS, but what would the Martians get?
Question 2: How could we translate into Martian, so they end up doing the operations in same order as we do? How can we rewrite the expression so they evaluate it to 50? (Note: in my answer I took the Martian use of nested parentheses to be outside in, instead of inside out like ours–they’re Martians after all, and I couldn’t get it to work any other way )