Day 2, MAT115

• You will have a quiz today over our readings of "From Fish to Infinity" (and the accompanying video), and the portion of the comic book covering sieving primes.

  The prime factorization will be the subject of next week's quiz.

• You have a reading assignment for next time, and some sample prime factorizations to do (use trees...).

• Definitions:
  • The counting numbers are the positive integers: 1, 2, 3, .... These are also known as the natural numbers.
  • A counting number is prime if and only if it has two distinct divisors among the counting numbers: numbers 1 and itself.
  • If a counting number has more than two divisors, then it is called composite.

• Theorem (prime decomposition): every natural number (other than 1) is either prime, or can be written as a product of primes in a unique way (the order of the primes aside).

Yikes! We can use trees to break composites down into their prime factorization: let's do an example or two:

1. 42
2. 1??

Hopefully you checked out the Mathemalchemy comic book (in particular the part about the chipmonks), and saw how the chipmonks explain the sieving of the primes (p. 28 of 40).

Let's review the sieve for the following:

The only two adjacent primes, The triplet primes, The twin primes, "Spirals", and The unprimiest looking primes!:)

• Is 113 prime? (Only need to check primes up to 7, since \(\sqrt{113}\approx 10.63\), and the next prime is 11 which is greater than that.)
• Is 329 prime? (Only need to check primes up to 17, since \(\sqrt{329}\approx 18.14\), and the next prime is 19 which is greater than that.)

This list would allow you to test any number for primeness up to 10000 (because its square root is 100...).

Seems a rather silly question, doesn't it? We learn something of that, anyway -- at least about six fish. But

• There is an interesting philosophical question posed by the author: "Where exactly do numbers come from?"
  1. "Did humanity invent them?"
  2. "Or discover them?"
  What do you think?

• Strogatz suggests that "...numbers are shortcuts for counting by ones...." (p. 5): so we might say that "six is a shortcut for counting by ones -- six times?"

  (You know that you're not supposed to use the thing you're defining in the definition!)

  Furthermore, "...Humphrey might realize he can keep counting forever." (p. 5) -- something that will be very important when we come to talk about infinity.

  Since we have a unique symbol for every counting number (we use a "base 10" system of 10 digits, which form the building blocks), we can count up to anything, merely by following the order of the numbers: as we "count", we associate an object with a number. The last number we land on tells us "how many". That's how we count!:)

• The key concept between the idea of number and counting is the idea of "one-to-one correspondence": we can think of it as perfect matching.

  Definition: a one-to-one correspondence is an association between two sets, so that each member of one set has a unique partner in the other set (and vice versa).

  So each penguin gets a unique fish dinner, and each fish dinner gets a unique penguin to eat it: that's how we know that everyone's happy! (Except for the fish....) And one of the keys here is that we don't have to know that there were six fish dinners -- so long as there's a one-to-one correspondence, there were exactly as many fish dinners as penguins (and vice versa).

  We know that the two sets of objects (penguins and fish dinners) were exactly the same in number -- we just don't know the number. But when we count, we know how many penguins there are -- and what to tell the kitchen about the amount of food to prepare.

  It is the one-to-one correspondence between the fish and the numbers 1 through 6 (as shown in the video) that is the essense of "counting". It is dependent, of course, on the order of our numbers; that we present them in the proper order: 1, 2, 3, ....

  

• Did you notice Humphrey using any other cues to help him keep track of the number of fish?

• How would Humphrey's method have worked had there been
  • 20 penguins?
  • 60 penguins?
  • 13 penguins?
  How about Ernie's method?

  These vocal cues leads us to your next readings: Rock (maybe acorns would be better!) Groups, and The Loneliest Numbers (a bit of a summary of primes).

• It is said that
  1. One is the loneliest number, but that
  2. Two can be as bad as one; it's the loneliest number since the number one.
  3. I can prove that all counting numbers are interesting, using order!
  It turns out that prime numbers tend to get lonelier and lonelier as they move off to infinity. And yet mathematicians suspect that there are an infinite number of the so-called "twin primes" (such as 3 and 5, 11 and 13, 137 and 139, and so on....). As long as you're floating out there in the vastness of space, it's nice to have a partner I suppose.

  Other numbers seem very gregarious; they play well with other numbers (e.g. 6, which seems particular friendly with 2 and 3; or 12, which has lots of friends: 2,3,4,6!).

  But how can we understand "6" without understanding "5" as well? (and thus 4, 3, 2, 1,...0?) We'll discover that 0 was pretty hard to understand from early on!

• Maybe this is a way to make that explicit: let's draw numbers as diagrams, which we will call graphs.

  Definition: a graph is a collection of points ("vertices") as well as a collection of arcs ("edges" -- each of which joins two points).

  An edge can even join a vertex to itself, which is called a loop in a graph.

  (This graph has three vertices, and seven edges -- four of them loops.)

  Definition: If every vertex in a graph is connected to every other (different) vertex by a single arc, we call it a complete graph.

  Each number, considered as a complete graph, encompasses all the numbers that go before it:

  

  This picture of "four" (which is known as "\(K_4\)") has an unfortunate feature: its edges appear to overlap in the center -- you might mistakenly believe that there's another vertex there.

  We can alleviate the problem by pulling \(K_4\) out of the plane (two dimensions), and turning it into a three-dimensional animal (one of which I've brought with me today:

  The tetrahedron (one of the Platonic solids, which we'll study down the road) is actually also the complete graph with four vertices:

  
  1. Do you see the complete graph with three vertices inside of it?
  2. Do you see the complete graph with two vertices inside of it?
  3. Do you see the complete graph with one vertex inside of it?
  Now you build the complete graphs with five vertices and six vertices, and show your neighbor.

• By the way, trees are examples of graphs. Our trees so far have points which are numbers, and the arcs represent that one number is a factor of another number:

  Trees have two properties that graphs don't possess in general:

  • They have a special point (vertex), called "root";
  • They have no "circuits" -- no paths that lead off from a point and then arrive back eventually, as in the tetrahedron.
  That means that there's a unique path between any two vertices (which is not true for tetrahedron, because there are circuits in that graph).

• As long as we're talking about counting, let's see if we can figure out a pattern to the number of connections (arcs, or edges) the complete graphs have as the number of points grow.

  Each time we add a new point (vertex), we have to connect it to the other points (vertices): so how do the number of connections grow with the number of points? We want a formula:

  \(arcs(n\ vertices) = ....\)

  To get the answer, we start with a table, and try to figure out the pattern (remember, mathematicians are pattern lovers!).

  \(K_n\) \(arcs(K_n)\) 1 0 2 1 3 3 4 6 5 ?

  This problem is related to another story, about a little boy who became the greatest mathematician of all time.... Carl Friedrich Gauss ("the Prince of Mathematicians").

  The story is told in a different way in one of your readings (The Loneliest Numbers).

You may not have the readings (or watch the video!:) at your desk.