Category: Math trivia

To the 0th power

On twitter I got involved in a discussion on exponents. At first it started with square roots being the same as ‘to the power of 0.5’. The first tweet commented on a blog that stated suprise about the fact that students did not know why the square root of a number was equivalent to ‘to the 0.5th power’. Of course this can be readily proved by using the definition of sum of exponents. I certainly agree with the fact that a lot of students (and teachers?) don’t know why some rules apply. But to me, this is closely related to all the rules that we use. So later on the discussion became more general, and was about when you assume certain properties and when you don’t assume them. For example. proving $g^0=1$ can be done by using the rule for exponents $g^a/g^b=g^{a-b}$ and so if $a=b$ then $g^0=1$ . But you could also say that, just like $g^a/g^b=g^{a-b}$ , $g^0=1$ could be assumed when proven. If you would state: ‘but I want $g^0=1$ to be proven from the most basic rules and assumptions’ this seems a fair request.

However, then I would say it is not necessary to assume the rule $g^a/g^b=g^{a-b}$ but that ${g^a} \cdot {g^b} = g^{a+b}$ is enough. Because let $a=0$ then ${g^0} \cdot {g^b}$ which is (under this law) $g^{0+b}=g^b$ . So ${g^0} \cdot {g^b}= g^b$ and so $g^0$ must be 1.

Proofwiki has an excellent list of the (inter-)dependency of exponent laws. I know it perhaps is nitpicking if I say ‘assuming less is better’ but some of the rules and laws we commonly use are not always that trivial. For students this is a nice way to approach it. It even has a link to a great question by a student whether we have to define or prove $g^0=1$ . Actually I don’t really care whether you prove it or not, but I do think some ‘feel’ for assumptions, axioms, dependencies etc. is useful for everyone. This was the only point I was trying to make.

A special case involves $0^0$ which is indeterminate.

@sndrclsn (who also made this nice picture) tweeted about his great movie bij James Tanton on “zero to the zero”-th power.

It sparked many more musing, like plotting $x^y$ in Wolfram and in Google. But there are other views, like this one stating that $0^0=1$ and also the Math Forum has some interesting quotes.

Math trivia

Elfenland and Graph Theory

Post author By cbokhove
Post date March 25, 2012
2 Comments on Elfenland and Graph Theory

Quite a while back I supervised a class project on graph theory. One inspiration was the boardgame Elfenland. Back then I made quite an elaborate series of lessons involving graph theory and that game. I remembered this because of a tweet involving this post on math strategies for games. Unfortunately I couldn’t find all the documents. On a more modest basis I have recreated some of the activities.

Elfenland is a game that involves using cards to travel from one point to the other. The board is below.

The first step is to transform this board to a graph. There are some difficulties and/or assumption:

If one can travel from A to B I have added a vertices and an edge.
I did not model any weights or “one way” edges (like 17 to 16, 16 to 15, for labels see below).

I then inputted the graph in Graph Magic (http://www.graph-magics.com/).

Now, assuming that one has to start and end in the capital at vertex 17, It is clear that because of vertex 7, a Hamiltonian circuit cannot be found, as vertex 9 is passed at least twice. So, I excluded vertex 7. Calculating the circuit for the remainder of the graph would suffice, just as long as the player would visit vertex 7 when arriving in vertex 9. The result was:

In this optimal route edges between 17,16 and 15 are not used so the problem of “one way” edges does not have a direct consequence for my strategy. In the next steps weights were added according to the Elfenland rules. In the real game, chance plays a role as travel is done by using playing cards. I have no time to improve it in this occasion, I’ll leave that to you. I think it is a nice introduction to some Graph Theory concepts.

Math trivia

Lovely equation

Post author By cbokhove
Post date January 13, 2012

On this weblog I encountered a great equation that can be solved in a creative way (the dots denote that the expression in repeated infinitely). I have reproduced it here:

(I) $\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}=4$