The renewed interest in this old post, reminded me of a more recent event:
Question: In how many ways you can chose a committee of three students from a class of ten students?
My expected answer: which is 120.
Alternative answer 1:(Lior) There are various ways: you can use majority vote, you can use dictatorship (e.g. the teacher chooses); approval voting, Borda rule…
Alternative answer 2: There are precisely four ways: with repetitions where order does not matter; with repetitions where order matters; without repetitions where order matters; without repetitions where order does not matter,
Alternative answer 3: The number is truly huge. First we need to understand in how many ways we can choose the class of ten students to start with. Should we consider the entire world population? or just the set of all students in the world, or something more delicate? Once we choose the class of ten students we are left with the problem of chosing three among them.
Here is a question from last year’s exam in the course “Basic Ideas of Mathematics”:
You buy a toaster for 200 NIS ($50) and you are offered one year of insurance for 24 NIS ($6).
a) Is it worth it if the probability that damage covered by the insurance will occur during the first year is 10%? (We assume that without insurance, such damage makes the toaster a “total loss”.)
b) Is it worth it if the probability that the toaster will be damaged is unknown?
As an additional test of your imagination, can you think of reasons why buying the toaster insurance would be rational?
This semester I am teaching an introductory course in mathematics for students in other departments. I taught a similar course last year entitled “Basic Ideas in Mathematics,” and this year, following a suggestion of my wife, I changed the name to “The Beauty of Mathematics”. Another change is that starting this year we have a general program in the university, called “Cornerstones“, (initiated by the rector,) whose purpose is to widen the education we offer to our students, and this course is part of the new program.
Talking about beauty rather than about basic ideas, combined with the new Cornerstone program have led many more students to enlist to the course this year, and subsequently the lectures will use computer presentations.
Of course, the challenge has become harder. I truly think mathematics is beautiful, but trying to convey its beautiful facets has never been easy. Also, I do not want to sweep under the rug the difficulty of mathematics, and the students will have to learn some basic mathematical skills and some abstract mathematics. Ideas and suggestions are most welcome.
What do you regard as a great example of the beauty of mathematics?
There will be is a separate page devoted to the course and the slides for the first lecture (in Hebrew) are linked there (and here). The first lecture was devoted to “numbers.”
Muhammad ibn Mūsā al-Khwārizmī
(From the Wikipedia article on zero.) In 976 Muhammad ibn Musa al-Khwarizmi, in his Keys of the Sciences, remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used “to keep the rows.”
The time has come to prove that the number of edges in every finite tree is one less than the number of vertices (a tree is a connected graph with no cycle). The proof is by induction, but first you need a lemma.
Lemma: Every tree with at least two vertices has a leaf.
A leaf is a vertex with exactly one neighbor.
Well, you start from a vertex and move to a neighbor, and unless the neighbor is a leaf you can move from there to a different neighbor and go on. Since there are no cycles and the tree is finite you must reach a leaf. Of course you describe the argument in greater detail and it seems that everybody is happy.
Theorem: A tree with vertices has edges.
After checking the basis for the induction we argue as follows: By the lemma has a leaf and once we delete and the edge containing it we are left with a graph on vertices. Now, has no cycles since did not have any. It takes some effort to show that is connected. You describe it very carefully. Once this is done you know that is a tree as well and you can apply the induction hypothesis.
Student: Now what about the case where is not a leaf? Continue reading