Saturday, April 27, 2013

Eratosthenes: The First Estimate of the Earth's Circumference

You might have thought that the Earth was proven round by Christopher Columbus in 1492. However, it was actually proven almost 2000 years earlier by a mathematician named Eratosthenes.

Eratosthenes was born in 276 BC in Cyrene, which was then a Greek colony. It would now be considered part of northern Libya. He was educated at Plato's school in Athens and went on to become chief librarian at the University of Alexandria.

Aside from his other contributions, he came up with a very close estimate of the circumference of the earth. First, he found out that at noon on the summer solstice, people in the Egyptian city of Syene noticed that there were no shadows from the sun. This is because Syene is very close to the Tropic of Cancer. This was confirmed by some tests he did by looking for brightness contrast inside of deep wells.

In his hometown of Alexandria, thought to be on the same meridian as Syene, he measured the angle of the sun's shadow on a pillar at noon of the next summer solstice. He found a 7° angle. This meant that when a circle is drawn to represent the circumference of the earth, Alexandria would be 7° around the circle.

Syene was known to be about 500 miles from Alexandria, and 7° is about one fiftieth of a circle. So, multiplying 500 x 50 gave Eratosthenes his estimate of 25,000 miles. This is a very impressive estimate, considering the modern distance is 24,901 miles.

It is extremely cool that you can determine the circumference of such a big thing without much technology at all. Just with a few mathematical principles, Eratosthenes could solve this daunting problem.

Saturday, April 20, 2013

Euler Paths and Circuits

This week, I wanted to talk about a topic that I have not yet mentioned on this blog. I wanted to introduce something called Graph Theory. This is a branch of Discrete Mathematics, along with Number Theory and Combinatorics.

Though I don't know much about graph theory, I was recently introduced to a graph theory property that I wanted to share. Since it is a visual area, I found a YouTube video that explains the concept pretty well.

I found it really interesting that you can always predict the possibility of an Euler Path and Euler Circuit. If anyone knows of a proof of what was mentioned in the video, please let me know. I would love to post it.

Saturday, April 13, 2013

Math in the News: Facebook Math Problem

A few weeks ago, I suddenly opened up my Facebook page to see that I was tagged by several people in a post about this article. Once I saw it, I immediately wanted to discuss it on Cool Math Stuff.

First, try to solve this math problem:

6 ÷ 2(1 + 2) = 

Even though it seems simple, there is lots of debate about the answer on social networking sites. Click here to view the article.

When I first saw the problem, I thought the answer was undoubtably 1. When I did the problem, I had multiplied by two while I was simplifying the parentheses. So, I assumed that people who got 9 as the answer were forgetting the order of operations.

Before I talk about the article, I was excited to see that this was a debate between people. Any time where the general public is talking about mathematics, especially in a argumentative way, is a huge accomplishment for society.

Reading the article, I was intrigued to find myself in the minority. I did not think of the parentheses as a substitute for the multiplication symbol, but a quick way to write two times the quantity one plus two.

What if the problem was written like this?

6 ÷ 2 • (1 + 2) =

If the problem were written like that, I would have gotten 9 as my answer. Since the 2 was separated from the parentheses, "Please Excuse My Dear Aunt Sally" would put the division first and then the multiplication.

However, without the dot there (more formally known as an interpunct), I do feel that the answer should be one. It seems like the most practical perspective on the problem.

Take an expression like 2a. It is implied that the 2 and the a are a single quantity. If you were using it in an actual problem, it is far more likely that you would be dividing, say 8, by the whole quantity 2a than 8 by 2, and then multiplying this by a.

Similarly, let's say you had 2(a + 3). Again, there would be no doubt that you should distribute the 2. This problem would then become 2a + 6, which seems like the correct thing.

If you ended up with 1 ÷ 2(a + 3), there would be no practical case where the one were to be divided by the two first.

Also, I think that the order of operations was put into place to make algebra and geometry more structured. For instance, say you had rhombus where the diagonals cross, or bisect, at a (3x3 + 9)° angle, and you had to find the measure of the sides which were of length 25 - 2x2. As you probably know, a rhombus is a quadrilateral with all four sides equal (a square has all four sides and all four angles equal).

There is a theorem that states that the intersection of the diagonals of a rhombus is always equal to 90°. I think this is pretty cool on its own, to know this angle all the time. So, we can assume that 3x3 + 9 must equal 90°.

3x3 + 9 = 90
3x3 = 81
x3 = 27
x = 3

We must then plug this into the expression equal to the side measure to find our answer. First, let's look at the expression.

25 - 2x2

What are we being asked to do? Since two is a coefficient for x2, we would square the x first, then multiply by 2, and then subtract that from 25. This would be agreed upon by all algebra teachers. What happens if we plug the 3 back in?

25 - 2(3)2

If there was no order of operations, we would probably do this problem from left to right. This would give us:

25 - 2(3)2

You can probably already tell that 4761 is way bigger than this side length would be intended to be. Even if you didn't make this estimate, this attempt at the simplification was nowhere close to the agreed order. By implementing the order of operations, there is no debate that the order should be PEMDAS. This clarifies lots of algebra when you are substituting terms into equations.

What if the expression were 108 ÷ 2x2? I still think most of us would agree that the 2xis meant to be its own term, and would therefore be divided as a quantity into the 108.

But after substitution, we get the following:

108 ÷ 2(3)2
108 ÷ 2(9)

If we did the 108 ÷ 2 first, we would end up with the wrong answer. Even though the surface definition of PEMDAS would ask for this to be done first, it is not a practical approach to the problem.

However, I did see that the SAT or ACT would expect students to receive 9 as an answer. Since there is clearly not a correct perspective to have, I would encourage you to comment what you thought the answer to be and why. Just like with Pi vs Tau, this is a post that is actually a lot of fun to debate about.

Saturday, April 6, 2013

Triangular Day: The Oblong Numbers

I don't know if you noticed, but today is a triangular day. It is April 6th, and six is the third triangular number.

You might remember from December when I brought up figurative families. The triangular numbers are a figurative family because each number can be represented by a triangular array of dots. Each array forms a geometric figure.

I also talked about square numbers as being a figurative family. Each number in that sequence can be represented by a square array, which is another geometric figure.

What about a rectangle? Considering that any number can be turned into a rectangular array, it would not be as interesting of a family. However, what about a rectangle whose array has just one more row than it does columns. For instance:

•  •  •  •
•  •  •  •
•  •  •  •

This has four rows and three columns, making it fit into this family. These numbers are called the "oblong numbers."

The oblong numbers go as follows:

2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132...

Here is a little property of oblong numbers. Divide each number in that sequence by two. You get:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66...

You end up with the triangular numbers. Why is this? Remember the formula used to find any triangular number?

Tn = n(n + 1)/2

We said that a number is oblong if its array had one more row than it had columns. So, an oblong number must be factorable into two numbers that are just one apart. We can denote these two factors as n and n+1. This makes the oblong number equal to:

On = n(n + 1)

After dividing by two, we ended up with the triangular numbers.

I enjoy trying to find ways to combine these different families together. For instance, take the first triangular number, first square number, and first oblong number. This would be 1, 1, and 2 respectively. We will add the first two together, and then subtract the last one.

1 + 1 - 2 = 0

Let's try that with some more:

T1 + S1 - O1 =     1 + 1 - 2         = 0
T2 + S2 - O2 =     3 + 4 - 6         = 1
T3 + S3 - O3 =     6 + 9 - 12       = 3
T4 + S4 - O4 =     10 + 16 - 20   = 6
T5 + S5 - O5 =     15 + 25 - 30   = 10
T6 + S6 - O6 =     21 + 36 - 42   = 15

Do you see the pattern? Each of the answers is a triangular number. In fact, it is one less than the triangular number used in the example.

Why is this? When I saw it, I first thought that we could check by using algebra. Take the explicit formula for each sequence:

Tn = n(n + 1)/2
Sn = n2
On = n(n + 1)

Our goal is to end with Tn-1, so we will use n(n - 1)/2 to denote that.

[n(n + 1)/2] + [n2] - [n(n + 1)] = [n(n - 1)/2]

Let's simplify all of the brackets so we don't have parentheses to work with.

[n(n + 1)/2] + [n2] - [n(n + 1)] = [n(n - 1)/2]
[1/2n+ 1/2n] + [n2] - [n+ n] = [1/2n- 1/2n]

Now, we will combine like terms.

[1/2n+ n- n2] + [1/2n - n] = 1/2n- 1/2n
1/2n- 1/2n = 1/2n- 1/2n

And there is our proof. Though oblong numbers are not the family of a regular polygon, it is still an interesting sequence to look at.