Tag Archives: math

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.

A Pythagorean Pun

There were once 3 kingdoms that bordered the same lake. In the middle of the lake there was an island, and the 3 kingdoms had been fighting over it for years. No one seemed to be able to keep the upper hand for very long and no one had been victorious. The wars over this little island were very costly, but all 3 kingdoms wanted it because of its great beauty and resources.

Finally, the monarchs agreed to a way to settle the matter permanently. Each would send their knights and squires to the island and they would
fight it out. Whoever’s knights and squires won the day would keep the island forever.

One kingdom sent many knights and each knight had a few squires. The night before the battle, the knights polished their armor while the squires readied the weapons. When the armor was finished, the knights sat around the fire drinking.

The second kingdom sent more knights than the first and each knight had several squires. The night before the battle, the knights drank around the fire while the squires scurried about polishing armor and readying weapons.

The third kingdom only sent one knight and he had only one squire. While the squire polished armor and readied the weapons, the knight hung a single pot from the tallest branch of the tree and tied a rope with a loop at the end from another branch. Then the knight sat by the fire and drank while the squire kept working.

The fateful day came and all the squires came out to the battlefield. (The knights had stayed up too long drinking.) The battle was fierce. In the
end, only the lone squire from the third kingdom was left standing. Proving once again, the age old theorem:

The squire of the high pot and noose is equal to the sum of the squires of the other two sides.

The punch line of this joke is a pun on the Pythagorean Theorem, often stated as: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. You may recall from our venture into trigonometry humor that a right triangle has one ninety-degree angle, and that the hypotenuse is the side opposite the right angle (also the longest side in the triangle).

Never argue with this triangle; it is always right.

Never argue with this triangle; it is always right.

The Pythagorean Theorem (named after Pythagoras of Samos, but known to people long before his time) states a curious relationship between the lengths of the sides of a right triangle. Imagine extruding the sides of a right triangle outward to form three squares, like this:


According to the Pythagorean Theorem, the area of the square formed from the hypotenuse is equal to the combined areas of the other two squares. In other words, if you could dismantle the squares formed from sides a and b and recombine them into one square, it would be the same size as the square formed from side c.

Mathematically, the Pythagorean Theorem can be expressed as a2 + b2 = c2. It holds true for any right triangle (as long as the triangle is flat; it doesn’t work for triangles printed on curved surfaces). Here’s an animation demonstrating how the Theorem works, courtesy of Wikipedia.

Now about that joke: I dig the pun at the end, but number-wise, it doesn’t quite work. If the third kingdom represents the hypotenuse (or high pot and noose), it should be the largest force of the three kingdoms. Assuming the first kingdom had, say, 300 squires and the second kingdom had 400 squires, the third kingdom would need 500 squires to be their equal, Pythagorealistically speaking. 3002 + 4002 = 5002. I know, I know…it’s just a stupid pun, but I want this blog to be an avenue for learning as well as laughing, so I would be remiss not to mention it.

Cuckoo for Cross Products

Q: What do you get when you cross a mosquito with a mountain climber?

A: You can’t cross a vector and a scalar!

Man, that is one nerdy joke. To be honest, I had to do a little review so I could confidently discuss it.

See, it’s a pun! Actually, it’s a triple pun, because it relies on the multiple meanings of three words: vector, scalar, and cross. Prepare for MAXIMUM NERDAGE as we fearlessly leap across disciplines!

What is a vector?

Biologically speaking, a vector is an organism that transmits diseases from one animal or plant to another; a mosquito, e.g. Mosquitos can transmit malaria, West Nile virus, dengue fever, yellow fever, and a bunch of other germs you really don’t want.

In mathematics (and its beautiful child, physics) a vector is a quantity that has both a magnitude and a direction. If I tell you that I am driving east at 80 kilometers per hour, I have given you a vector: specifically, my velocity. Velocity is a vector quantity because it has both a magnitude (80 km/hr) and a direction (east). Other examples of vectors are acceleration, force, and momentum.

What is a scalar?

To scale a mountain is to climb it, so a mountain climber might be called a scalar (or would it be spelled scaler? I’m not sure, but the joke doesn’t work if you spell it with an e, so I’ll stick with scalar).

A scalar is also a quantity that has a magnitude but not a direction. Speed is an example of a scalar quantity. If I tell you that I am driving at 80 kilometers per hour – but neglect to mention which way I am driving – I have told you my speed. That’s the difference between speed and velocity: velocity has a direction; speed does not. Other examples of scalar quantities are: mass, distance, and energy.

What does it mean to cross two things?

There are lots of jokes that start with “What do you get when you cross X with Y?” I’ve always assumed that this crossing was some sort of ethically questionable breeding program being conducted in a secret laboratory deep beneath a mad scientist’s mansion. Think about that the next time somebody asks you a joke like this one. It’ll make you cringe a little and not feel so sure you want to know the answer.

In math/physics, crossing is a specific mathematical operation that can be performed on two vectors in three-dimensional space. That doesn’t make a bit of sense, I know, so let’s back up. Imagine that vectors are arrows…that’s how most scientists think of them anyway. Now imagine two arrows pointing outward from a single location, sort of like this:

Two vectors diverged in a yellow wood...

Two vectors diverged in a yellow wood…

We’ll call the green vector A and the red vector B. The cross product of A and B is written as A x B (read as “A cross B”) and is calculated using the following formula:

A x B = |A| |B| sinθ n

In this formula, |A| is the magnitude (or length) of vector A. |B| is the length of vector B, θ (theta) is the measure of the angle between A and B, and n is a unit vector, which helps you figure out which way the cross product points in three-dimensional space. And what exactly is the cross product of A and B, I hear you asking? It’s a third vector (really, a pseudovector…don’t ask) that is perpendicular to both A and B. The blue arrow below represents the cross product of the green and red vectors.

Hot cross vectors

Hot cross vectors

You’re probably thinking “Great…so what?” Well, the cross product has a lot of practical uses, particularly in the field of engineering. Cross products are used in some calculations involving torque, a force that causes things to spin. Yes, despite what sounds like a lot of made-up mathematical nonsense, it does have real world value.

Now then, on to the heart of this joke: why can’t you cross a vector with a scalar? Because you specifically need two vectors – two quantities that have directions. Since a scalar has no direction, you cannot cross a vector with a scalar. Insert uproarious laughter.

Now that you thoroughly understand vectors and cross products, here’s a follow-up joke to send you on your way. Enjoy!

Q: What do you get when you cross an elephant with a banana?

A: |elephant| |banana| sinθ n

Mandelbrot Mirth

Q: What does the B stand for in Benoit B Mandelbrot?

A: Benoit B Mandelbrot

Benoit Mandelbrot was born in 1924 in Warsaw, Poland. He rose to prominence as a mathematician who developed fractal geometry, which is all about measuring the “roughness” of nature. It also produces some interesting shapes, called fractals, which are known for being self-similar.

A self-similar fractal looks the same on the small scale as it does on the large scale. If you could examine the fractal pattern from far away, then zoom in to one point in particular…you’d get the same image. What’s that you say? You want to experience zooming in on a fractal pattern while electronic dance music plays in the background? As you wish!

These patterns are generated by a formula that performs a series of calculations on each point in the pattern and assigns colors based on the outcome of the calculations. Although these calculations may be performed by hand, the process is slow (very slow, in fact) and laborious. The development of computers really helped mathematicians explore Mandelbrot patterns in arbitrarily high detail.

I know what you’re thinking: this is all very pretty but what practical value does it have? Well, as I said before, Mandelbrot originally developed fractal geometry as a means of describing the roughness of things found in nature, like, say, coastlines or mountain ranges. Fractal geometry has also led to the development of cell phone antenna that maximize reception while minimizing space.

Now, about that B: According to his obituary in The Guardian, Benoit Mandelbrot added the B himself in order to make his name more distinctive. It didn’t stand for anything in particular (much like the S in Harry S Truman). However, if you assume that the B stands for Benoit B Mandelbrot, then it works beautifully as an analogy of his life’s work. Zoom in on the B and you get…Benoit B Mandelbrot. Zoom in again and you get…Benoit B Mandelbrot. Zoom in again and…well, you get the point.

Not an actual fractal.

Not an actual fractal.

Whether Mandelbrot had a humorous motive in choosing the letter B as his middle initial, we may never know. I will say this much: it’s hard to imagine that a man as clever as Mandelbrot would have completely missed the connection.

Binary Befuddlement

There are 10 kinds of people: those who understand binary and those who don’t.

This joke really only works in writing. I’ll explain why later.

Binary (also called base 2) is a number system; a way of counting. It’s really not that different from the decimal number system (base 10), which is the system that most of us know and love from our days in elementary school. Still, binary feels unintuitive because we’re used to thinking in powers of ten, not powers of two. But maybe I’m getting ahead of myself.

Before I jump right into binary, it might be educational to review base 10 first. As you know, the base 10 number system uses – wait for it – ten digits. You’ve probably heard of them: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The first nine counting numbers are each represented by a single digit, but when you get to ten, you have to switch to two digits. Why, you ask? Because there’s no digit in base 10 that represents the number ten. Instead we use a new “place”…a place where the digits 1 through 9 represent multiples of ten instead of one. So 10 can be read as “1 ten and 0 ones” and 11 as “1 ten and 1 one”. Whenever we accrue ten ones, we roll it over and bump up the tens column by one. Easy peasy so far, right?

And so it goes until we reach 99 (9 tens and 9 ones). Adding one more causes the digits to spill over into a third place, which we call the hundreds place. The number 100 could be read as “1 hundred, 0 tens, and 0 ones”, although we normally forego mentioning the tens and ones if there aren’t any to speak of. Up through the three-digit numbers we climb until we reach “9 hundreds, 9 tens, and 9 ones”, or 999. Adding one now will cause the digits to spill over into the thousands place, and four-digit numbers are born.

If you’re particularly math-savvy, you probably noticed that each of the “spill-over” numbers is a power of ten. The first two-digit number is 10, or 101. The first three-digit number is 100, or 102. The first four-digit number is 1000, or 103. That’s no accident; our number system is built up from multiples of powers of ten, hence the name “base 10”.

Perhaps the following illustration will help. The number one hundred seventy-five may be thought of as 1 hundred, 7 tens, and 5 ones.

A picture is worth 103 words.

A picture is worth 103 words.

Okay, okay, I hear you object, I know how to count. Indeed you do, gentle reader, perhaps too well. Most of the time when we think of numbers, we think of them in sort of a holistic sense; rarely do we stop to consider how they are put together. But now that you have considered the construction of numbers, you’re ready to tackle binary.

See, binary uses the same rules as decimal, except you get only two digits: 0 and 1. Instead of being based on powers of ten, the binary number system is based on powers of two. You don’t have a tens place, a hundreds place, a thousands place, etc; you have a twos place, a fours place, an eights place, a sixteens place, and so on. The first counting number is written as 1, just as in decimal, but when you add 1 to that, you don’t get 2. Instead, the numbers spill over into the twos place, and you get 10. Don’t read that as “ten”, because it isn’t. It’s “1 two and 0 ones” in binary. The next counting number is 11 (1 two and 1 one), which we would write as 3 in decimal. Next comes 100, or “1 four, 0 twos, and 0 ones”. Hopefully you get the drift.

For your edification, here are some numbers of interest in both the decimal and binary number systems.

Description Decimal Binary
The loneliest number 1 1
The loneliest number since the number one 2 10
Blind mice, wishes, bears, etc 3 11
Pi, approximately 3.14159265359 11.0010010000
Number of seasons 4 100
Bo Derek 10 1010
Candles 16 10000
Days of summer vacation 104 1101000

And just to utterly drive home the point, here’s a visual representation of the same number as above (one hundred seventy-five) but in its binary form: 10101111.

A picture is worth 1111101000 words.

A picture is worth 1111101000 words.

Now I said before that the joke really only works in writing. If you try to say it out loud, you run into a problem or 10. First, you wouldn’t read it as “There are ten kinds of people” because then the rest of the joke doesn’t make sense; your listener would be justified in expecting you to describe eight more types of people. You could read it as “There are one-zero kinds of people”, but that makes the joke awkward and telegraphs the punch line. You could just say “There are two kinds of people” but then it sort of becomes an anti-joke, and not a very funny one at that. Sorry, aspiring math-based stand-up comics; you’ll have to leave this one out of your repertoire.

In closing, I’d like to offer this follow-up funny, which I’m sure will leave you in stitches.

There are 10 kinds of people: those who understand binary, those who don’t, and those who didn’t realize that this joke is actually in base 3.