Today we will discuss some of the deeper implications of common and advanced mathematics. We will start with some basic assumptions, shattering them to pieces, before moving on to advanced themes.

- Two and two always makes four. This is basic and we learn it early and repeat it often. The truth, however, is not so simple. One factor which is often omitted or ignored is the units involved. I present a simple example to illustrate: Two
*apples* and two *oranges* makes neither four *apples* nor four *oranges*. The only way to make our beloved cliche true here is to derive a common unit from the given units. Two *apples* and two *oranges* makes four *pieces of fruit*. But what about two *apples* and two *boots*. Not four *apples*, *boots*, or *fruit*. We have to default to the empirical unit *objects* here to make this work. Two *apples* and two *boots* is four *objects*. Unless *boots* is a verb… Then things get hairy.
- Well, if two and two isn’t always strictly four, what happens now with fractions? Well, I’ll give you a bitter example from real life (because these things are always more useful when applied to everyday situations rather than the abstract). A
*half* plus a *half* is a *whole* right? Not always so! I have two *halves* of an associate degree. So how many degrees do I have? Well, of course, I have none. Go figure. (At least it was only community college tuition we’re talking about here.)
- So we’ve established that numbers do not always combine the way we would like them to, but, at least on their own, we can count on them to remain true to themselves, yes? No, I’m afraid not. Here is what I call the Bulson Proof, as it was handed down to me by my high school math teacher Mrs. Bulson.

x=y

x*x=x*y

x^{2}=xy

x^{2}-y^{2}=xy-y^{2}

(x+y)(x-y)=y(x-y)

x+y=y

x+x=x

2x=x

2=1

Whoa! We’ve just been stabbed in the back by our former friends one and two, who we thought we could count on, in good times and bad, to be just who we always thought they were, but here we see they have been fooling us all this time. They’re completely interchangeable! This throws the whole entire number theory right into the garbage disposal!

Or it presents a pretty compelling reason, other than the whims of your math teacher and her text books, why you’re not allowed to divide by zero (or any variable not explicitly stated to be not equal to zero).

Now we’ll move on to word problems, because rarely in life are you thrown a bunch of numbers requiring your crunching (unless you are often thrown a bunch of numbers requiring your crunching, in which case you have a boring job which you should be grateful for since computers are probably quicker and more accurate than you are). Read the question carefully before answering. This is an important test taking skill.

*Considering the average human walks at a rate of about 3 miles per hour, and marathon winners complete the 26.2 mile run in about 2 1/2 hours, which is equivalent to running roughly 10 miles per hour, would it be faster for me to walk 10 miles or to run 10 miles?*

Think about the question before rushing to answer. You don’t want to miss easy questions like this on any upcoming exams you may have because you failed to understand the question being asked. Do you have your answer?

Okay. Here is the correct answer: It would, without a doubt, be faster for me to walk the 10 miles. Despite being a good distance walker, I am out of shape, and attempting to run the 10 miles would result in extended delays whilst I, lying prostrate on the sidewalk, waited for emergency medical services to arrive and attempt to resuscitate me. They would most likely then bring me to a hospital, or morgue, depending on the relative success, or lack thereof, of their previously stated efforts, which would result in further, possibly infinite, delays. So you see, the wording of the questions on your exams is crucial. Never rush to answer a question without fully reading and understanding every word in the question.

You say the question was unfair because you don’t know me or the status of my health? Well that’s your fault. You should have done your homework!