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The Cali Garmo does Math

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Archive for December, 2009

Time for a pop quiz! Say you are on a game show. You are presented with 3 different doors and are told that 2 of the doors contain goats and the 3rd door contains 6 quadrillion dollars! You are told to choose which door has the money, so you choose door number 1. At this point the game show host turns to you and says, “What a choice! Let’s look at what was behind door number 2. It was a goat! I’ll give you another chance since you know that the second door was a goat. Do you want to stay with door number 1 or switch?” What is the better choice: to stay with door number 1, to switch to door number 3, or do both doors have the same probability of having the goat/money?

Key points:

  • you choose a random door out of 3 doors (2 of which are ‘bad’ and 1 of which are ‘good’)
  • the game show host reveals that one of the other 2 doors is one of the ‘bad’ doors
  • you are asked whether you would like to keep the original door or switch
  • Which is better to stay, to switch, or are both the same chance?

This is called the Monty Hall problem because there used to be a game show back in the day that was put on by a guy named Monty Hall. I’m sure it was a super fun show!!!

[Solution after the holidays!]

Fun fact! In the real world, if you take a triangle and add up the angles, they can equal more than !

In school we always learn that the angles of a triangle always add up to , but that is not always the case, especially when you look at our planet! Don’t believe me? Let’s prove it!

Proof:
First let’s define what a triangle is. A triangle is 3 ‘straight’ lines that intersect at 3 points. By straight we mean a line that doesn’t curve to the right or to the left. We also notice that since we are using Earth as our reference point, we are doing all calculations on a sphere.
Ok, so let’s start! Pretend you are standing somewhere on the equator on Earth. Also, pretend you have outrageously long legs that allow you to travel really really far really quickly. Now walk north until you hit the North Pole. That’s 1 straight line, 2 more to go. So make a turn right and walk south all the way back down to the equator. That’s 2 lines, 1 more to go. Now in order to walk to our starting point we need to make a right turn. In fact we need to make a turn. (Cause we are heading directly south and now need to go directly west) Once we get to our starting point we notice that this final angle also equals . (Since we are looking west, but must turn north in order to ‘retrace’ the triangle) And that’s 3 straight lines. Adding up the angles we get:

Oh yeah! Take that geometry. I just kicked your butt!!! This type of geometry is called Spherical Geometry and a lot of mathematicians study it in order to better understand the universe.

And thats all from me until after the holidays. Armenian Christmas isn’t until January 6 so Ill be gone for the next 2-3 weeks and will return with some more math funness afterward. *lates*

Modulo! Mod you who? Modulo: A complicated way to say the remainder when you divide two numbers!

So what is modulo exactly and how does it work? The easiest way to break it down is to look at an equation:

What this means is that basically if you take a and divide m then you get a remainder of b. Let’s try a few examples to understand it better. Say and . Then So . Therefore we have the modulo equation: We can do the same with negative numbers! Let and let again. So now we have: . So we have and

Well, that was simple you may be thinking, but when would I use stuff like this in real life?! We actually use it DAILY without even knowing it! Want a quick example? Just look at a clock. What happens after you hit 12:59 (or 23:59 for some of you)? It goes straight to 1:00 (or 0:00). We use modulo to keep track of time! If its 3:00 right now and we go 30 hours in the future, then what time is it? Well we have and Then [or] and we have that the time is b which is 9!

What else can we do with modulo in the world of mathematics? Well it turns out that with such a simple concept we can actually compose a whole lot of theorems! Here are some theorems to keep you entertained. If you dare, try and prove them! (Note for those who try and attempt these, proofs require additional knowledge in division and how certain numbers divide other numbers.)

Theorem 1:
Let a, b, l and m be integers and let l and m be greater than 0. Now let . With this information we can show that

Theorem 2 (By John Wilson):
Let p be prime. Then

Theorem 3 (By Fermat):
Let p be prime and a be a whole number that is not divisible by p. Then

These theorems go on and on and there are hundreds that span from the concept of modulo. In the real world we also use modulo when it comes to creating/breaking codes, ISBN numbers, computer programming languages, and a ton of other places you probably never thought of!

For a good introduction to these types of numbers look to the following book: Elementary Number Theory and its applications by Kenneth H. Rosen (AT&T, 2005)

Say you have an infinite number of camels walking through the desert and a little baby camel is born, how many camels are there now? Surely there are still an infinite number, and surely since we added 1 more camel this infinity is greater than the original infinity. But is it even possible for infinity to be greater than infinity? So confusing!

The biggest error most people make when talking about infinity is to think of infinity as a number. In reality, infinity is not a number, it is a concept. By definition infinity is a quantity without end. So if you take a number and you keep adding 1 to it, then you are going toward infinity, but you never actually hit infinity since infinity is not a number.

So since infinity is not a number, what happens when you ‘add 1’ to infinity. As we saw earlier if we keep adding one we still have an infinite amount. So by adding 1 we are still at infinity. Now looking at the number of camels, even though we added 1 camel to the group, we still have an infinite number of camels and so the ‘number of camels’ has not changed. It’s a weird concept to grasp, but this concept (created by Georg Cantor) helps us understand infinity.

So then we are naturally inclined to ask, can we ever make a certain infinity greater than another infinity? Weird concept to think about, but thanks to Cantor and a bunch of other super smart mathematicians, we now know the answer is yes!

This concept is created in Set Theory where we look at the number of elements in a set. Cardinality is just the number of elements in a set, and a set is just a collection of objects. So if you have a group of 10 people, then the cardinality of the group is 10. The cardinality of the number of months is 12, the cardinality of the number of weeks is 52, the cardinality of the set of birthdays is 366. Well how about the cardinality of a set like the natural numbers? The natural numbers are the numbers 1, 2, 3, 4… So what is the cardinality since it seems like there is no end to the number of items in the set.

This is where the concept of infinity comes into play. Since the number of items in the set of natural numbers never ends, there are an infinite number of them. We call this type of infinity, countably infinite. The reason we call it countable is because we as humans can sit there and count them. We may never be able to hit the end (since there is none), but we can count them. (We can first say 1, then say 2, then say 3, etc. And if you say 1 number per second for 70 years you’d hit 2,209,032,000 !)

Well are there more countably infinite sets? Yes, integers and rational numbers are also countable! We can count integers by counting them in this way: 0, 1, -1, 2, -2, 3, -3, etc. How about rational numbers?! There seems to be a crazy amount of rational numbers, and in between 0 and 1 there are an infinite number, so they must not be countably infinte, surely. But they are!

How did mathematicians find this out? They took the rational numbers (which are all integers and fractions) and put them into a grid like the following:

Now in order to make it countable just go in diagonals. So our sequence would be: . Then in order to get all the negative numbers, just switch in between the two (positive, then the same number negative like we did for the integers.)

Now how about the set of all numbers (aka real numbers, which is the rational numbers with the irrational numbers such as ) It turns out that the real numbers are actually not countable. We can see this because there is no way to sit and count the real numbers. For example let us start with the number 1. What number do we choose next? 1.01 is wrong, because 1.001 is less and is also a real number. But we can’t choose that since 1.0001 is also a real number, so no matter what number we choose there is no logical next number to choose since we can keep finding a smaller number closer to 1.

Wait, now we have 2 different types of infinity! Countable and uncountable infinities. The cool thing is that if you look further into set theory you’ll find out that uncountable infinity has a higher cardinality than countable infinity! So there are the same number of whole numbers as rational numbers, but there are more real numbers than any other type of number. So cool!

If you want further information on Set Theory to be able to see different types of infinity I recommend A transition to advanced mathematics by Douglas Smith, Maurice Eggen, & Richard St. Andre (Books/Cole 2001) Have fun counting!

Do you have an unquenchable thirst for mathematics? From Euclid to Newton to Einstein, you want it all. You want to know the details about topologies and about the intriguing nature of prime numbers and Fibonacci numbers. Or maybe you are sitting in your algebra class confused as all else and can’t understand a word the teacher says. Well you are at the right place! The Cali Garmo was placed upon this world to help you understand math like you’ve never understood it before, learn concepts you’ve never learned before, and run free in a world you’ve never experienced before. So turn on your noggin, hunker down with pen and paper and get ready to experience the funnest subject on the planet in this weekly blog!