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The Cali Garmo does Math


Archive for January, 2010

A prime number by definition is a number p>1 such that p has no positive integer divisors other than 1 and p.

Wait… What?

Confuzzling! Let’s try to figure out what that means by breaking it down into smaller parts. Let’s first look at the section ‘is a number p>1’. This part is actually saying 2 things in 1. It’s first stating: let the ‘prime number’ be represented by the symbol ‘p’. So ‘p’ is our variable. It also states that p must be greater than 1. So, so far, p can be 1.1, 2, 940,300, , etc.

The next part says ‘such that p has no positive integer divisors other than 1 and p’. First let’s look at what a ‘positive integer’ is. An integer is any whole number. Basically a number that doesn’t have any decimal part, and no fractions when simplified. So 2, 55, -7, 3.0, , etc. are all integers. A positive integer means basically that the number must be greater than 0. Next let’s see what ‘divisor’ means. A divisor is an integer that divides another number into an integer. So since 2 divides 4 into 2, then 2 is a divisor of 4. Since 3 divides 99 into 33, then 3 is a divisor of 99. So a ‘positive integer divisor of p’ is a number that divides p, is a whole number, and is greater than 0. So if we let p be 4, then 1, 2, and 4 are all positive integer divisors of p.

But, we are stating that p does NOT have any positive integer divisors except for p and 1. So 4 is not prime since 2 is also a positive integer divisor. 2 is a prime number since it is greater than 1, and it’s only positive integer divisors are 1 and 2. 9 is not prime even though it is greater than 1 since it’s positive integer divisors are 1, 3, and 9. 6.5 cannot be prime since it is not an integer.

So by going number by number we can get a huge list of prime numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 …
To name the first few. You can go and look at the long list of prime numbers provided by sloane which is an online encyclopedia of integer sequences (or integer patterns). Here is a link: [Sloane]

At about this point you may want to exclaim ‘Wait a second! What about the number 1?! It can only be divided by 1 and itself too, why is 1 not classified as a prime?’ Originally the number 1 was considered a prime number, but a lot of mathematical concepts that have to deal with primes usually turned out funky when dealing with the number 1. When 1 was considered a prime, the concepts worked perfectly sometimes, but failed at other times, but they all succeeded if the number 1 was not considered a prime or a composite. So instead of keeping it as a prime number, it was eventually relegated to be just a number, neither prime nor composite (explained in a few!) That is why if you pay close attention to our definition we say our prime p has to be greater than 1.

So we mentioned that the number 1 cannot be composite either, what’s composite?! A composite number is just an integer c > 1 such that c is not prime. With all the information I gave you above, I’m sure you can figure this one out =D

So that’s the main concept behind prime numbers. Want to learn more of have any questions? Leave a comment!

Questionably yours,
The Cali Garmo

Before the holidays I posted a question to you all to see what you thought. The question was the famous Monty Hall problem where there are 3 doors and a contestant is told 1 has a fabulous prize while the other 2 have goats. The contestant selects a random door and the host reveals one of the doors that have a goat and gives you a chance to switch doors if you want or not. The question was: is it better to keep the original door, switch doors, or are the odds the same? Here’s the link if you want a refresher.

So most people will look at this problem and say the chances are the same whether you switch or keep the same door. The logic usually follows like this: There are 2 doors left. One has a present, the other a goat, so there is a 50% chance that either of them will have a goat. Thus each door has the same probability of having the prize.

Although this is sound logic, there is only one big mistake, the assumption that the 3rd door doesn’t matter. Let’s start from the beginning and work our way up in order to see what the answer is. Since we first choose a random door let’s pretend it’s door A. We now have 3 options: the present is behind door A, the present is behind door B, the present is behind door C.

  1. If the present is behind door A: We chose door A, so the host then chooses either door B or C (both have goats so it doesn’t matter) and opens it. If we switch to either B or C (whichever the host does not choose) then we lose, if we stay on A then we win.
  2. If the present is behind door B: We chose door A, so the host must select door C to reveal the goat. So if we switch to B we win, and if we stay on A then we lose.
  3. If the present is behind door C: We chose door A, so the host must select door B to reveal the goat. So if we switch to C we win, and if we stay on A we lose.

Since those are the only doors available those are only ways of playing. Now let’s look at the winning ratio. If we stayed the same we would have won 1 game. If we switched then we would have won 2 games. So by switching doors you have increased your chances of winning to 66%! So remember to always switch when given a choice!