Tag Archives: mathematics

Integrating Humor Into Your Daily Life

Two mathematics professors, Bob and Steve, are talking in a bar.  Bob laments to Steve that the average person just doesn’t understand math anymore.  Steve thinks Bob is being too hard on people.  When Bob excuses himself to use the restroom, Steve walks over to a pretty blonde sitting by herself and says “When my friend comes back from the bathroom, I want you to make some pretense of walking by us.  I’m going to stop you and ask you a question, and I want you to say “x squared”.  Got that?”

The blonde looks puzzled: “x…squared?”

“Yes, that’s right.”

The blonde repeats: “x squared…okay, I think I got it.”

A moment later, Bob returns from the bathroom.  Steve says “Hey, I bet I can prove that the average person knows math.  Take this woman, for example,” he says, motioning to the blonde who, right on cue, is walking past them.  “Excuse me miss, so sorry to bother you, but I wonder if you can tell me: what’s the integral of 2x?”

“x squared” says the blonde, and she starts to walk away.  Bob is flabbergasted.

Then the blonde turns around and yells “PLUS THE CONSTANT OF INTEGRATION!


First, I know it’s been a long time since I’ve updated this blog.  When I started writing it, I thought I would be able to dig up enough nerdy jokes to fill a football stadium,  but I lost momentum as it became more difficult to find jokes that dealt with topics about which I had not already written.

Still, I’m not ready to call “Nerdy Jokes” a done deal just yet.  I’d like to continue to update this blog whenever possible, if for no other reason than for my own education and for the entertainment my two or three loyal readers.  Thanks: I love you guys!

Now then…integration is a mathematical process from calculus with applications in physics and other sciences.  To teach you how to do integration would be far beyond the scope of this blog, so here are some helpful links.  Take a few minutes to study these websites, and you should be up to speed on calculus in no time at all!

Integration is the opposite of differentiation, and it’s much easier to understand integration in that context.  Plus, understanding what differentiation is will make it easier to comprehend the blonde’s triumphant rejoinder vis-à-vis the constant of integration.  So let’s start there.

Consider the function f(x) = x2, whose graph is shown below.

graph of x squared

This graph is a parabola; in fact, you might say it’s the archetypal parabola.  Every other parabola you can imagine (and I don’t know about you, but I can imagine quite a few parabolas!) is merely a variation on this theme.  Some are wider, some are skinnier, some are upside down, but all parabolas have in common a certain set of characteristics; chief among them is that the height of any point on a parabola is directly proportional to the square of the horizontal distance of that point from the parabola’s axis of symmetry.

But pardon me; I digress.  Could you find the slope, or slantiness, of this parabola?  The answer seems to be no; after all, since the parabola is not a straight line, it does not have a constant slope.  But what if I asked you to find the slope of a single point on the parabola?  Again, this seems like an impossible task; the slope of any line segment can be expressed as rise over run; a single point, by definition, has neither rise nor run, and so it cannot have a slope.

Fine, fine, but what if I told you to imagine a straight line lying tangent to the parabola at a certain point, say, x = 1.  Surely you could find the slope of that line, couldn’t you?

slope and climber

And the answer is: yes, you can find the slope of a straight line lying tangent to a curve at a specific point.  That’s where differentiation comes in.  When you differentiate a function, or find the derivative, you come up with an expression that gives the slantiness of a line that glances off of a function at a certain point.  In the case of f(x) = x2, the derivative is f'(x) = 2x.  (You can read the first part as “f-prime of x”.  There are many different ways to indicate a derivative.)  So if you want to know the slope of a line lying tangent to the graph of f(x) at the point x = 1, you simply substitute 1 for x in the derivative equation, and that gives you the slope of that particular tangent line:

f'(x) = 2x

f'(1) = 2(1) = 2

The slope of line lying tangent to f(x) at x = 1 is 2, which means that this line rises 2 units for every 1 unit it runs.  A line tangent to any other point of the parabola would, of course, have a different slope.

If you’re still struggling with the concept of differentiation, here’s a real-world analogy that might help.  Say you’re sitting at an intersection when the light turns green.  You step on the gas pedal, causing your car to accelerate from rest to, say, 100 km/h (roughly 60 mph, for those not inclined to use metric units).  If you were to look at the speedometer at any time while speeding up, the reading of the speedometer would be similar to the derivative of a function.  It would tell you how fast you were going right that moment, not ten seconds ago or three minutes from now.

Integration is the opposite of differentiation.  Imagine that I gave you the derivative of an equation, and asked you to suss out the original equation.  You could use integration for that purpose.  If 2x is the derivative of x2, then x2 is the integral of 2x.  Right?  Riiiiiiight?

Well, almost.  See, there are infinitely many functions that have the derivative 2x.  Besides f(x) = x2, there’s also:

  • f(x) = x2 + 1
  • f(x) = x2 – 1
  • f(x) = x2 + 8
  • f(x) = x2 – 148
  • And so on…

Basically, you can add or subtract any number – called a constant – from x2 and you’ll get a new function that has the exact same derivative.

  • If f(x) = x2, then f'(x) = 2x
  • If f(x) = x2 + 1, then f'(x) = 2x
  • If f(x) = x2 – 148, then f'(x) = 2x
  • And you get the point…
two graphs

Both f(x) = x2 and f(x) = x2 – 5 have the same derivative, because their tangent lines have the same slope at any point x.

It’s impossible, given only the derivative of an equation, to integrate and figure out which constant to add or subtract.  You need additional information.  Neophytes learning about integration for the first time are often told to include a constant of integration – a variable that stands for any conceivable number that might be added to or subtracted from the function.  When asked to integrate 2x, the most appropriate response is:

∫ 2x dx = x2 + C

Where C is the constant of integration.  It could stand for 4, or -5, or it could even be zero.  But we don’t know for sure, so we use C as a stand-in.  Unless we’re given further information (say, the coordinate at which the function intersects the y-axis), we cannot determine what the value of C is.

The blonde in our joke proved that she was calculus-savvy enough to run with the mathematics professors – perhaps even more savvy, since Bob failed to mention the constant of integration in his instructions, and she still managed to stick the landing.  Good for her.


Paradoxical Playfulness

Q: What’s a good anagram for Banach-Tarski?

A: Banach-Tarski Banach-Tarski.


 

If you’re scratching your head about this one, don’t worry.  I was too at first.  I had to do some research about the Banach-Tarski Paradox before I understood the joke.  Now that I get it, I think it’s quite clever; sadly, it has very limited re-tell value.  (See what I did there?)

Let’s start with the easy part: an anagram is a rearrangement of the letters in a word or phrase, usually in such a way that they make readable new words.  For example, you can rearrange the letters in Nerdy Jokes to form Yo, Send Jerk!  I’m not sure what that means, but there you go.

You normally expect an anagram to have the same total number of letters as the original word or phrase, and that’s sort of where the humor of this jokes comes in.  Let’s talk about the Banach-Tarski Paradox, and maybe you’ll see why this joke is funny.

The Banach-Tarski Paradox comes from something called Set Theory: a mathematical theory dealing with the ancient Egyptian storm god, Set (I assume).  I’m not really sure how Egyptian mythology ties into the Banach-Tarski Paradox, but hey, I’m not a mathematician!

The Banach-Tarski Paradox starts off like this:  Imagine a three-dimensional solid ball.  Are you imagining it?  See how round it is?  Good.  Now imagine breaking the ball into a finite number of pieces.  Next, rearrange the pieces to form two solid daughter balls from the broken pieces of the original.  You can’t stretch or add pieces, you can only move and rotate the existing pieces.

Now…imagine that each daughter ball has the exact same volume as the parent.

If that doesn’t make you cry “Balderdash!” then perhaps you haven’t been paying attention.  I’ve just said that you can take the limited pieces of an object and rearrange them in such a way as to make two exactly identical copies.  That flies in the face of everything we think we understand about the relative permanence of volume.  You simply shouldn’t be able to do that.

But mathematically, you can.  And furthermore, you can do it endlessly, meaning that a single ball can be replicated endlessly until the universe is lousy with balls.

If you’re the enterprising sort, you’re probably already scheming about a way to get your hands on a gold bar and a hacksaw, perhaps thinking that you can replicate your wealth ad infinitum.  Well, here comes the part where I rain on your parade by telling you the caveats I didn’t mention before.

This only works on mathematical objects.  A mathematical object is different from a physical object in several important ways: first, a mathematical object is a collection of points (not particles), which, in the case of a ball, lie within well-defined parameters.  Second, since mathematical objects are not made of atoms, there’s no limit to how often or how finely they can be divided.  Third, and perhaps most importantly, the non-granular nature of mathematical objects means that when you do divide them into smaller pieces, the facets of the cut can be arbitrarily, even infinitely, small.

This potential for infinitely rough cuts makes the Banach-Tarski rearrangement possible.  A piece with an infinitely rough surface has an undefinable volume; consequently, the normal rules for adding volumes do not apply.  In fact, you only need to divide a mathematical sphere into five pieces to create two clones, which is somewhat suprising to me.  Of course, the whole idea is surprising to me, so I guess more surprises shouldn’t be surprising, and I think I’ve just talked myself into another paradox.

So let’s revisit the joke: a normal anagram of Banach-Tarski would yield a word or combination of words with a sum of twelve letters, since there were only 12 letters to start with (Example: Satan Hack Bit).  But a mathematical rearrangement of Banach-Tarski, in the manner of Banach-Tarski, would yield two identical copies of the original (assuming, of course, that Banach-Tarski were a mathematical collection of points instead of a real-world physical object.)  I have been assured that the mathematicians who first described the paradox, Stefan Banach and Alfred Tarski, were in fact physical objects, meaning they were not subject to the rearrangement that bears their name.  (Nerdy Jokes would like to strongly discourage its readers from attempting to perform Banach-Tarski rearrangements on living organisms.)