Hey everyone,

It’s been a while since I’ve updated so I’ll write a new post summarizing what’s been happening over the past few months. First of all, my Facebook account has been disabled since they’ve accused me of using a fake name. Of course, I have done no such thing and am upset and frustrated since I have a lot of pictures on there that I would have liked to keep for myself. Due to the enormous size of Facebook I’ll probably never see what I had on my page again and I don’t know if I would want to use a service that cannot accurately distinguish real accounts from fake ones. On the bright side, now that my account’s gone I’ll probably have more time to do something productive since Facebook is hardly the place to do that. I also wrote a lengthy blog entry detailing my experience at the Memorial Park Criterium, but after uploading my photos, WordPress crashed on me and all my drafts were lost. So remember folks, save your data often or else you will regret it!

Anyway, over the last few months I’ve been traveling all across Texas looking for work and racing bicycles. Fortunately, I had some success as I placed well in my remaining races (3rd at Chappell Hill, 7th at State Championships), but more on that later in other blog posts. I also landed a job so now that I’ve become employed I’ve had to rearrange my priorities and projects to suit my career. And some more good news – I achieved a perfect score on my second actuarial exam, FM/2! I had studied very hard the month before I took the test, but not so hard the month before that. I’m just glad that I probably got every single question right on that exam. I was pretty cautious during the test and took my time, using the entire three hours I was allotted for that exam. I didn’t feel entirely prepared but in retrospect I probably studied more than most people do for that test, since I read the entire textbook in the syllabus and did every single problem in the book – twice. Unfortunately my main goal was to read the text twice, but I felt so lazy for only reading it once, go figure! Perhaps I’m too cautious – given that there are an infinite number of possible questions – you can never be fully prepared!

I had to put Python aside and I probably won’t be able to go back to it until maybe the summer of 2011.  Fortunately, I’m learning two other programming languages in its place, C++, and VBA (Visual Basic for Applications), so the time I spent on python has definitely not gone to waste. C++ is harder, and I would have liked to learn Python first but I’ll have to make do since that’s what we use at work. I also finished a computer science textbook, a textbook on Microsoft Excel, and another introductory textbook on C++, as well as a couple of legal documents regarding my work. I’m currently reading my second textbook on C++,  and a textbook on VBA as well. Check out the projects page for more details.

Hey everyone,

Today’s problem actually consists of a potentially infinite number of problems! I’ve been learning some programming in my spare time, so I’ve written my first program using Python, a powerful high-level scripting language that can do several things. My first program is called “Gene’s Multiplication Tutor.” It simply picks random numbers for the user to multiply, lets the user pick the number of problems the user wants to do, and tells the user how many problems the user answered correctly. To download the program, click the following link:

Gene’s Multiplication Tutor

The link will take you to a page that hosts the file. Click “Click here to start download..” and save it to the desired directory. After you have the file, you will need to open up a command prompt or terminal window, cd to the directory at which the file is located, and type in:

python mult3.py

at the prompt to start the program. Of course, you will need a python interpreter in order to run the program. The program will first ask you if you wish to proceed, and if you wish to proceed, it will ask you how many problems you wish to do. After inputing the number of problems you wish to do, the program will ask you the questions, and will tell you whether or not you got them right after each answer. After all the questions have been answered, the program will tell you how many you answered correctly, and then will return to the menu.

Here are some screenshots:

Access the program through the terminal window.

It looks like I missed one. Bummer!

Enjoy!

Hey everyone,

I posted a new page called Projects which describes the subjects that I’m learning during my spare time. You can locate the page at the top of this site (it’s the third tab),

Enjoy!

Hey everyone,

Today’s problem is a proof derived from mathematical fallacy. See if you can spot the error! (Hint: you’re not Chuck Norris).

Problem 7:

Find the error in the following “proof.” Let x = y. Then:

$latex displaystyle begin{aligned} x^2&=xy,\ x^2-y^2&= xy-y^2,\ (x+y)(x-y)&= y(x-y),\ x+y&=y,\ 2y&=y,\ 2&=1. end{aligned}$

Solution 7:

We know that something is obviously wrong with this proof since we know that 2 ≠ 1. It took me a while to solve this, but after a few minutes of staring at the problem I think I’ve found the solution. Consider the following lines:

$latex displaystyle (x+y)(x-y) = y(x-y),$

$latex displaystyle x+y=y$

Here, the proof writer attempts to divide by zero. In other words:

$latex displaystyle (x+y)(x-y)(x-y)^{-1} = y(x-y)(x-y)^{-1}$

However, x = y, x – y = 0, so:

$latex displaystyle (x-y)^{-1}=text{undefined}.$

I don’t have a solutions manual for these problems, so feel free to disagree if you’re not convinced. I am, however, quite certain that this is a correct solution.

Note: I used a new command today, the begin{aligned} environment to align the equal signs in my first box of $latex LaTeX$ equations, and the target=”blank” attribute to open up links in a new window.

Hey everyone,

I didn’t write a problem yesterday, so today I’m going to write about two famous unsolved problems in mathematics: the Twin Prime Conjecture and Goldbach’s Conjecture. These two problems are so famous because they’re simply stated, yet so seemingly difficult to solve.

Problem 4:

A twin prime is a prime number that differs from another prime number by two. A twin prime can refer to a single prime number with this property, or it can refer to a pair of prime numbers that differ by two. Are there an infinite number of twin primes? The Twin Prime Conjecture states that there are infinitely many primes p such that p+2 is also prime. Is the conjecture true?

Solution 4:

As of today, there are no proofs of the Twin Prime Conjecture accepted by the mathematical society. This is one of the oldest unsolved problems in mathematics and no one (that we know of) has been able to solve it for thousands of years! It’s my wild guess that there are an infinite number of twin primes. Mathematicians have made some progress, and have shown that there are infinitely many n such that at least two of n, n+2, n+4, and n+6 are prime.

Problem 5:

Goldbach’s Conjecture states that every even number greater than two can be written as the sum of two primes. Is Goldbach’s Conjecture true?

Solution 5:

Again, no one knows the answer to this conundrum. There’s a large prize of $1,000,000 (I fathom much more has been spent trying to solve the problem) offered to the solver! The term “solver,” however, is erroneous, because I suspect the final proof will have been the cumulative result of several mathematicians’ works over several centuries.

I don’t want to close this post without solving a problem, so today I’ll write a bonus problem proving the fact that the product of two negative numbers is a positive number.

Problem 6:

Prove that the product of two negative numbers is a positive number.

Solution 6:

If a and b are any numbers, then:

$latex displaystyle (-a) cdot (-b) +[-(a cdot b)] = (-a) cdot (-b) + (-a)cdot b$

(Associative Property)

$latex displaystyle (-a) cdot (-b) +[-(a cdot b)] = (-a) cdot [(-b)+(b)]$

(Distributive Property)

$latex displaystyle (-a) cdot (-b) +[-(a cdot b)] = (-a) cdot 0$

(Since –b + b = 0)

$latex displaystyle (-a) cdot (-b) +[-(a cdot b)] = 0$

(Since –a x 0 = 0)

$latex displaystyle (-a) cdot (-b) = (a cdot b)$

(Add a x b to both sides)

That’s why the product of two negative numbers is a positive number! Ever since kindergarten I’ve taken this fact for granted, but I’m grateful for coming across this proof now, for the sake of better understanding.

Note: I used a new command today, cdot, which specifies the dot commonly used to denote multiplication.