Random Thoughts from a Nonlinear Mind – Page 19

How About Some Tennis Math?

I have been thinking a lot about probability, especially the probabilistic nature of tennis. Why have I been on a probability kick lately? It certainly is a tricky and slippery subject, perhaps the hardest I have ever studied, and I guess I am just trying to keep on top of my game. Hey, you never know when a Golem or a Centaur might show up at my door with a life or death riddle based somehow on probability theory. I guess I just want to be prepared for any contingency.
tt
To begin, I will assume that everyone has a rudimentary knowledge of the game of tennis. I hope you all realize that the server has an advantage, especially as the players get better. As a general rule, this is more true for men than for women, particularly in the professional ranks. The server starts every point, and they know where the serve is going while the receiver has a minimal window of time to figure out the trajectory of the ball. With that in mind, let’s imagine the following scenario. A server wins 60% of their service points against a particular opponent. That means that the returner wins 40% of those points. We can ask and answer a series of questions based only on this information. Let’s get to it. For reasons that will become clear later, we will begin with a game that is already at deuce.

We will start our analysis with a player that wins 60% of all their service points against a random player. Once the game reaches deuce, the probability that the server will win the next two points is .6 x .6 = .36. That means that the server will win the next two points a little over 1/3 of the time, 36% to be exact. What about the receiver winning the next two points to break serve? That would be .4 x .4 = .16. So, 16% of the time the receiver will break serve by winning two points in a row. What is left? The only other possibility is a return to deuce. There are a couple ways we can figure this. We know that all possibilities have to add up to 1 so we can simply solve the following equation:

1- .36 – .16 = .48.

Therefore, 48% of the time the service game will return to deuce. The other way to arrive at that figure is with the following:

.6 x .4 + .4 x .6 = .48.

If you take a minute to study that short equation, you will quickly realize why it makes sense.
tt

Now, this gets a little tricky. With the game at deuce, what is the probability that the server will eventually win the service game? Well, they can win by winning the first two points after the first deuce, which they do at a rate of 36%. The other thing that can happen is that the game can return to deuce, which will happen 48% of the time before the server goes on to win. Therefore we end up with this equation: P = .36 +.48P. Solving for p gives an answer of .692. We now know that the server will win 69.2% of all of their service games against this opponent once the score has reached deuce.
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Notice that this player has a 60% rate of winning individual service points but ends up with a 69.2% success rate when it comes to holding their serve during a service game that has reached deuce. That is pretty interesting, maybe even a little unexpected.
dd
We can now break down other probabilities based on how this player does against other opponents. The first category considered is 50% of service points won. Anything less, i.e., a success rate of less than 50%, and this type of analysis is not very useful. I think we can all agree that the player’s time is better spent on improving their serve than it is doing math. For all other players that are somewhat proficient at serving, the results are as follows.

50% of points won on serve
Server will win the next 2 points 25% of the time.
Returner will win the next 2 points 25% of the time.
Game returns to deuce 50% of the time. Server wins 50% of service games that have reached deuce.

60% of points won on serve

Server will win the next 2 points 36% of the time.

Returner will win the next 2 points 16% of the time.

Game returns to deuce 48% of the time. Server wins 69% of service games that have reached deuce.

67% of points won on serve

Server will win the next 2 points 44.9% of the time.

Returner will win the next 2 points 10.9% of the time.

Game returns to deuce 44.2% of the time. Server wins 80.5% of service games that have reached deuce.

70% of points won on serve

Server will win the next 2 points 49% of the time.

Returner will win the next 2 points 9% of the time.

Game returns to deuce 42% of the time. Server wins 84.5% of service games that have reached deuce.

75% of points won on serve

Server will win the next 2 points 56.3% of the time.

Returner will win the next 2 points 6.3% of the time.

The game returns to deuce 37.4% of the time. Server wins 89.9% of service games that have reached deuce.

80% of points won on serve

Server will win the next 2 points 64% of the time.

Returner will win the next 2 points 4% of the time.

Game returns to deuce 32% of the time. Server wins 94.1% of service games that have reached deuce.

90% of points won on serve

Server will win the next 2 points 81% of the time.

Returner will win the next 2 points 1% of the time.

Game returns to deuce 18% of the time. Server wins 98.7% of service games that have reached deuce.

So, why did we start our analysis with a game that was already at deuce? That is an interesting question. We began there because the math gets a little out of hand if we start at the beginning of a player’s service game. I guess I was just trying to get everyone loose and nimble. Now that we have done our warm-up and taken off our old school polyester tops, it is time to really get down to business. Don’t worry; if the math makes your head spin, you can move on to the table. There is no crime in that.
ss
The probability that a tennis player holds serve (P) is equal to the probability that he holds at love (P @ love) plus the probability that they hold at 15 (P @ 15) plus the probability they hold at 30 (P @ 30) plus the probability they hold when the game goes to deuce or multiple deuces (P @ D(s)). Got it? That makes perfect sense, doesn’t it?

Now consider the following:

The probability that the player holds at love = p⁴

The probability that the player holds at 15 = 4p⁴(1-p)

The probability that the player holds at 30 = 10p⁴ (1-p)

The probability that the player holds in a game that goes to deuce is much more complicated. The equation looks like this:

$20p^{3}\left ( 1-p \right )^{3}D\; where\; D=\frac{P^{2}}{1-2p\left ( 1-p \right )}$

When the algebra is worked out, we end up with this equation:

$\frac{20p^{5}\left (1-p \right )^{3}}{1-2p\left ( 1-p \right )}$

Whew, that was a bit of work. The good news is that we can now determine how often a given player will hold serve based on the percentage of time they will win a service point. The following table contains relevant data for select service point percentages. As you can see, it also includes lots of other information. The mathematics behind the rest of the table becomes a little mind-bending, so I have decided not to add them here. You will just have to trust that the assumptions are logically based on the probability (P) that a given player will win a single service point. The rest of the table is built on that mathematical base. You will also note that I have also included the expected outcome of various matches between players serving at different rates of success.

	serve pts	hold serve	tiebreak	tiebreak set	adv set	3 set match	5 set match
PLAYER A	80.00%	97.80%	52.10%	52.80%	54.80%	54.20%	55.20%
PLAYER B	79.00%	97.40%	47.90%	47.20%	45.20%	45.80%	44.80%
	serve pts	hold serve	tiebreak	tiebreak set	adv set	3 set match	5 set match
PLAYER A	80.00%	97.80%	59.80%	63.90%	71.00%	70.30%	74.70%
PLAYER B	75.00%	94.90%	40.20%	36.10%	29.00%	29.70%	25.30%
	serve pts	hold serve	tiebreak	tiebreak set	adv set	3 set match	5 set match
PLAYER A	75.00%	94.90%	51.90%	52.90%	54.00%	54.40%	55.50%
PLAYER B	74.00%	94.10%	48.10%	47.10%	46.00%	45.60%	44.50%
	serve pts	hold serve	tiebreak	tiebreak set	adv set	3 set match	5 set match
PLAYER A	74.00%	94.10%	51.80%	53.00%	53.90%	54.40%	55.50%
PLAYER B	73.00%	93.20%	48.20%	47.00%	46.10%	45.60%	44.50%
	serve pts	hold serve	tiebreak	tiebreak set	adv set	3 set match	5 set match
PLAYER A	70.00%	90.10%	51.70%	53.10%	53.70%	54.70%	55.80%
PLAYER B	69.00%	88.80%	48.30%	46.90%	46.30%	45.30%	44.20%
	serve pts	hold serve	tiebreak	tiebreak set	adv set	3 set match	5 set match
PLAYER A	70.00%	90.10%	55.20%	59.40%	60.80%	63.90%	67.20%
PLAYER B	67.00%	86.10%	44.80%	40.60%	39.20%	36.10%	32.80%
	serve pts	hold serve	tiebreak	tiebreak set	adv set	3 set match	5 set match
PLAYER A	70.00%	90.10%	58.50%	65.60%	67.50%	72.60%	77.30%
PLAYER B	65.00%	83.00%	41.50%	34.40%	32.50%	27.40%	22.70%
	serve pts	hold serve	tiebreak	tiebreak set	adv set	3 set match	5 set match
PLAYER A	70.00%	90.10%	66.30%	79.50%	81.40%	89.10%	93.80%
PLAYER B	60.00%	73.60%	33.70%	20.50%	18.60%	10.90%	6.20%
	serve pts	hold serve	tiebreak	tiebreak set	adv set	3 set match	5 set match
PLAYER A	67.00%	86.10%	53.40%	56.50%	57.20%	59.70%	62.00%
PLAYER B	65.00%	83.00%	46.60%	43.50%	42.80%	40.30%	38.00%
	serve pts	hold serve	tiebreak	tiebreak set	adv set	3 set match	5 set match
PLAYER A	65.00%	83.00%	51.70%	53.30%	53.60%	54.90%	56.20%
PLAYER B	64.00%	81.30%	48.30%	46.70%	46.40%	45.10%	43.80%
	serve pts	hold serve	tiebreak	tiebreak set	adv set	3 set match	5 set match
PLAYER A	65.00%	83.00%	58.20%	66.30%	67.40%	73.70%	78.50%
PLAYER B	60.00%	73.60%	41.80%	33.70%	32.60%	26.30%	21.50%
	serve pts	hold serve	tiebreak	tiebreak set	adv set	3 set match	5 set match
PLAYER A	62.00%	77.60%	51.60%	53.40%	53.60%	55.10%	56.30%
PLAYER B	61.00%	75.60%	48.40%	46.60%	46.40%	44.90%	43.70%

Take some time to study this table. You will find that a 1% advantage in service points won translates into a much bigger chance of winning sets and matches. The table is quite interesting and informative. I will be posting many more essays on this topic in the future. Stay tuned…

An Average Post…(harmonic, that is)

Sometimes I sits and thinks…and sometimes I just sits…
A.A.Milne

Sometimes I sits and thinks…and sometimes I sits and thinks about thinking…
Buford Lister, personal communication

I remember reading somewhere a long time ago, that what made humans unique in the animal world is that we can think about thinking. Not only do we think, but we can take the process to the next level and analyze the thinking itself. This is simply a guess, but I don’t believe any of the pets I have had during my life were capable of such a feat. Actually, I am not sure that most…well, you get the idea.

Over the decades, I have thought a lot about how thinking about thinking works. Back in the 80s, I tried, again and again, to sneak into a great course at Harvard. It was called Thinking about Thinking, and it was taught by Alan Dershowitz, the famous lawyer, Robert Noczik, one of the leading philosophers from the last century, and Stephen Jay Gould, the evolutionary biologist who has influenced me in more ways than I would care to admit. There you have it, three superstar professors from different departments coming together to dazzle a bunch of impressionable minds. The course was designed to show the students how different ways of thinking lead to different approaches to how we view the world and our place in it. At least, that is how I looked at it.

That course was highly popular, and it was near impossible to get anywhere near the classroom. When I was there, the course was taught in the Science Center, a building that resembles an old Polariod camera on its side. There were guards checking registration slips at each entrance. If you were a registered student and happened to forget your slip, you were out of luck, you missed that day’s class. I often thought of trying to crawl through the heating ducts to get in there. After a few minutes, I thought about my thinking and reconsidered, I thought better not to try it.

One day, I was sitting in the small cafe right inside the main entrance to the Science Center. Who do you think sat down near me? It was the three professors. I had no choice; I had to eavesdrop on their conversation. I fell into a bit of a pattern, I made sure I arrived at the cafe around the same time on the days when the class was meeting. I got to hear lots of conversations. I can only recall one topic, it was the same topic the three of them talked about every week. They talked about baseball, baseball, and then more baseball.

Why all this stuff about thinking about thinking, also known as meta-thinking? Easy, we are going to talk about how to take averages, really simple averages. How about this one: if Sally has 40 apples and Billy has 20 apples, what is the average number of apples that the kids have? If I am writing a post about such a thing, you should immediately start doing some meta-thinking, right? If it really was so straightforward, why would I be writing about it? And that is a very good point, and of course, it is true. I wouldn’t be writing about averages if I didn’t have something a little unusual and surprising to say about them.

The average of a and b are calculated in this familiar way: $\large \frac{a+b}{2}$

So, if Sally has 40 apples and Billy has 20 apples, the average number of apples = 30. No problem. The answer is simple and straightforward. Now consider this:

Joe’s car gets 40 miles per gallon, and Steve’s gets 30 miles per gallon. What are the average miles per gallon of the two vehicles?

$\large \frac{30+40}{2}\neq 35\: MPG$

And, no surprise, that answer is wrong. Why? Let’s suppose that both Joe and Steve drive for 120 miles. Joe would use 3 gallons of gas, and Steve would use 4 gallons. Now, we can add everything up. A total of 7 gallons of gas was used to travel 240 miles. Therefore, the average is 34.28, and that is the correct answer. 34.28 is the harmonic mean or harmonic average, it is quite different from the simple averages we are used to calculating.

This is the equation for harmonic averages:

$\large \left (\frac{\frac{1}{a}+\frac{1}{b}}{2} \right )^{-1}$

That equation can be reduced to the following:

$\large \frac{2}{\frac{1}{a}+\frac{1}{b}}$

The important thing to think about is that you want to get a common denominator, not a common numerator. That creates a lot of confusion when it comes to computing harmonic averages.

So, where do we now stand? We all now know that some averages are more simple to compute than others. As always, the trick is to know when to use a harmonic average instead of a simple one. I will be posting more about this topic in the future. Simple averages and harmonic averages are not alone in their “average” universe, check back in and you will see what I “mean.”

The Package

This is a piece of Flash Fiction. The topic: A person arrives home to find a package on their doorstep.

Simeon Langdon felt an unnaturally cold chill as he unbuckled his radiation suit and removed his propulsion pack. Damn, the hairs on my arms are standing on end. Why are they doing that? What is going on with me? He touched the proper sequence of buttons on the control panel, and the pad lifted him to his apartment.

He instantly saw the package as the platform rotated and then locked into place. Huh, I don’t remember ordering anything. I wonder what it could be… He picked up the box and opened his door. He set it down, injected himself with an aqua fluid, and went to the mandatory decontamination chamber. He tried to relax as the toxins were slowly removed from his body.

After about 20 minutes, he made his way out of the chamber and examined the box. He found perfectly symmetrical block letters on all six sides.

BROCKTON LANGDON
256 JOHNSTON COMPLEX
TUSCON, REPUBLIC OF ARIZONA
(TO BE DELIVERED ON FRIDAY, AUGUST THIRTEENTH IN THE YEAR 2652)

Simeon took a deep breath, followed by a long pause. Brockton was his grandfather, and he had been dead for at least 50 years. How is this possible? I didn’t live at this address, there wasn’t a Republic of Arizona when he was alive, how and why did I get this package? He closely examined the box and the material used to seal it. No return address, no other clues, nothing.

Simeon cautiously opened the package. He jumped back as an orb floated up out of the box and settled near the ceiling. A flash of energy engulfed the room, and then a hologram of his grandfather appeared. Not the grandfather he knew, not the aged, mysterious figure who showed no emotion and kept to himself. The hologram seemed to be the 20-year-old version of a man he barely recognized.

Out of the eyes of the hologram came a shot of laser light targeting the data port in Simeon’s left shoulder. He was overwhelmed by the encoded information, all ones and zeros, the binary language of computers. Simeon heard the words the data stream was speaking to him, but he couldn’t understand how and why.

“The Langdon family is directly descended from the beings who seeded this planet with life billions of years ago. The DNA animating this planet is ours. It was me, in consultation with The Superiors, who set the wheels in motion all those years ago to destroy much of the life on Earth. It is now your job to eliminate what is left of it in North America. Your cousins will take care of the rest. You will initially fight and struggle within yourself, but you will do your ancestor’s bidding. Moreover, after you fully understand, you will want to do it, you will be compelled to do it. The orb will deliver the devices, all you have to do is deploy them. You will know where to go, what to do, and when to do it.”

The hologram disappeared into the orb, and then the orb disappeared into Simeon. It felt warm and energizing. He smiled as he basked in the epiphany that revealed his life’s true purpose. As it was, gentleness and compassion never were words in his lexicon. He now felt emboldened, fully realized, complete. He put his head down and started to analyze the Orb-driven algorithms running through his matrix, his concentration only momentarily broken by the faint screams of “HELP!” coming from the kidnapped women he had locked in his bedroom closet.

02.02.2020

The First Rule of Palindrome Club: Name no one, man.
The Second Rule of Palindrome Club: See above.
Anonymous (personal communication)

Today is February 2, 2020. Any idea what makes this day so special? Well, I just watched The Australian Open men’s final. Novak Djokovic won…again. It is just a matter of time before he and Raphael Nadal both pass Roger Federer for the most Grand Slam titles. Can you still be the GOAT if you are third on that list? I doubt it.

Today is also Groundhog Day. I hear that Phil is about 50% correct with his shadowy predictions. By the way, how do they tell if he really saw his shadow? Does The Old Farmer’s Almanac offer instructions or some kind of insight into groundhog vision?

In a few hours, I hear that the Super Bowl will be televised. I haven’t watched a football game since the Browns packed up and left Cleveland for Baltimore. For me, that was the end. Football no longer warrants any of my time or attention.

So, what is special about 02.02.2020? I am sure that you have already figured it out. That series of numbers form a palindrome. Notice that is doesn’t matter what order you put the day or month in, you still are golden. It is also a palindrome in ISO format.

Here are a few more interesting points about 02.02.2020. That day is the 33rd day of the year (a palindrome), and there are 333 (a palindrome) days left in the year. This particular confluence of circumstances will never happen again.

In a somewhat shocking and surprising (though totally predictable) turn of events, I am going to tell you something about mathematical palindromes that I find astonishing. If I told you that any positive integer can be written as the sum of three palindrome numbers, would you believe me? As incredible as it sounds, it is true.

Usually, I would go through the basics of the argument and then give a couple examples. I can’t do that this time, the paper this idea is based on contains 40 pages of dense mathematics. A number of algorithms are required to solve for any and every case. In total, everything is quite complicated. Instead, I suggest you click on this link.

On the website, you will find that all the math has been coded. Just type in any number you can think of and the special numbers that form it will appear.

I have just one final thought as I get ready to head to the gym. A person from Finland who deals in soap (a saippuakuppinippukauppias) has to be losing their mind today.

A Most Curious Text

I have a bunch of relatives that live down South. Many of them live in M i crooked letter crooked letter i crooked letter crooked letter i humpback humpback i. At least that is how I was taught to spell Mississippi by my cousin Christy when she was around 5 years old.

Christy’s sister Tammy is one of those Mississippians. When she was young, she lived in Ohio, then moved to Mississippi. She went to college in Georgia and…wait…what’s that? Did I hear someone say “fascinating, tell me more ” (If you don’t recognize that little squiggle, check out my post from 1/14/20 ).

Well, I am going through Tammy’s history because I am trying to figure out where and when she learned French. Have I heard her speak French? No. Did she write me an email in French? She didn’t do that either. So, what evidence do I have that she knows French? I offer the following text she sent after she read the last post, the one about the mathematical exploits of her uncle.

Do you notice anything about her text? Anything unusual at all? I did. It appears to be written by a French person or someone who knows how the French like to punctuate their sentences. See the extra space between “that” and the question mark? That is the giveaway. It is one of those things that strikes Americans as unusual. It just doesn’t look right.

Speaking of punctuation that does not sit well, I have a friend who insists on punctuating her texts in the British style. Any and all quotation marks are inside of all punctuation. She does this even though I am reasonably sure she has not been further east than Ohio. It is just one of those things. That said, I do believe that Tammy’s text is unique in my experience, I can not recall ever getting a message with French punctuation.

So, I had to do some research to get to the bottom of this. It didn’t take me long to discover that autocorrect has been known to add an extra space before punctuation. In fact, I read that many people have noticed this and are not fans of this quirk. I have never been a fan of autocorrect, it sometimes has a mind of its own and that intelligence doesn’t necessarily mesh with mine.

There is another possibility, the predictive text function that shows up when texting. If you choose a word from the available selections, the program puts a space after it. It requires a little extra work to get rid of it, and I stand with those who can’t be bothered by such nonsense.

This is the end of this short post about a text Tammy sent me. Do I think that she knows French, or is the extra space a function of her texting app? The fiction writer in me wants to believe that she is a spy, planted here by the French government at a very young age for some nefarious purpose. The scientist in me…well, who cares what that guy thinks, the spy theory makes for a much better story.

Do Me a Solid…

Do Me a Solid…and Read my Post about a Platonic Solid.

Something extraordinary happened a couple days ago. I woke up and went upstairs to my library, sat down and turned on my computer; after that, I got some lunch and then went to the gym. Nothing too exciting except for the fact that by the time I got out of the shower, my Dad solved a math problem that no one knew the answer to. You read that right, at the age of 83, my Dad made an original contribution to mathematics by answering a question about a Platonic Solid. How cool is that?

This is my story…

The other day I happened upon a Numberphile video about the mighty Dodecahedron, famous the world over for being one of the five Platonic Solids. A Platonic Solid is a regular polygon, meaning that the same number of identical shapes meet at each vertex, or corner. And that’s right, there are only five of them. Try as you might, you won’t find another one. Here is a fantastic video about Platonic Solids. While this is not the video that inspired this sequence of events, I include it because we must have an understanding of these forms before I can get to the heart of my tale.

The solid we are interested in is the Dodecahedron, and yes, I still have a lot of trouble trying to spell it. As I was doing some research on Platonic Solids, I came across this GIF. I remember that scene from The Simpsons, and I am happy I found it. It adds that little extra something to the post, don’t you think? Anyway, I couldn’t possibly write a post about the Dodecahedron without including it.

As the Numberphile video demonstrates, the five Platonic Solids are the Tetrahedron, Cube, Octahedron, Dodecahedron, and Icosahedron. The only thing we need to know about those shapes is that for four of the solids, it has been proven that you can not start at a vertex, head out in a straight line and return to your place of beginning without running into another vertex. This has been proven for all the solids except for the Dodecahedron. For that shape, the answer was unknown. People had speculated that it was possible, but no one had found such a path or proven that such a trajectory would be impossible. Until now.

In their impressively short paper, which I have included below, Jayadev S. Athreya and David Aulicino show such a path. They took a Dodecahedron and unfolded it into the form of a net. A net, in this case, is a flattened out two-dimensional version of the three-dimensional object in question. In fact, their proof of the theorem is a little unusual and not mathematical at all. Grab some scissors, cut out the net, and then tape the Dodecahedron together. You will see that the line is straight.

Download (PDF, 2.48MB)

After I watched the Numberphile video about the Dodecahedron, which I have included at the end of this post, I downloaded the paper. After reading it, I remembered something that Professor Athreya said near the end of his presentation. He stated that it wasn’t yet known if the special transecting line cut the Dodecahedron in half or not. I looked at Figure 1 in their paper, said, “hmmm… to myself,” and then went to find my Dad at the office.

I handed him the paper, and he quickly read it. After I told him that the areas of the shapes above and below the red trajectory line were unknown, he sat down at this computer. It was apparent that he knew how to solve the problem.

Here is my Dad’s solution to the area problem. First, he had to compute the coordinates of every vertex for each of the 12 Pentagons shown in the figure. He used Coordinate Geometry to do this. Once this was done, he was able to calculate the length and angle of the line transecting the Dodecahedron. Then it was straightforward to calculate the areas above and below the red trajectory line. How cool is that?

Download (PDF, 88KB)

I emailed my Dad’s solution to the professor. He got back to me the same day. He called my Dad’s approach to the problem “Awesome!” and thanked him for doing the calculation. How cool is that?

One final thought, my Dad was most likely the first human being to ever calculate the areas created by the special transecting line. He was the first person to know that the area above the line gets 47.7% of the original total while the bottom gets 52.3%. All I have to say is…How cool is that?

NOTES:

Here is the Numberphile video about Platonic Solids. Professor Athreya talks about the problem my Dad solved at the 18:00 mark. The video is very good, it is worth the investment of time to watch the whole thing.

My Dad’s name is Jerry Slay. His email address is surveyor5298@yahoo.com. If you wish, send him a message letting him know how cool this story is.

We’re Number…Huh?

If I gave you five guesses, could you identify the most famous equation in mathematics? Perhaps this would be your first guess:

$\huge a^{2}+b^{2}=c^{2}$

I must admit that is not a bad choice. I think that one might be my personal favorite, but I do believe there is a general consensus about the most beautiful and elegant equation math has to offer. It is known as Euler’s Identity, it looks like this:

$\huge e^{i\pi }=-1$

If we do just a little manipulation we get this:

$\huge e^{i\pi }+1=0$

That is the version I prefer. Within one simple equation, we get the five most important numbers in mathematics. And as an added bonus, we get to see exponents. Pretty cool.

There are lots of excellent videos on the internet on the topic of Euler’s Identity. The equation can be derived in lots of different ways, each more fascinating than the next. There is no need for me to include even a single derivation here. I am going to do something much more interesting.

If there are any sports fans out there, I have a suggestion for you. I don’t believe I have ever seen such a sign, but the world is undoubtedly in need of one. The next time you head out to a game, you might want to take the time to get some poster board and markers, clear off a desktop, and get to work. It won’t take long.

I suggest the following text:

$\huge WE'RE\; NUMBER\; -e^{i\pi}$

If you really are proud and excited about your sign, feel free to add an exclamation point at the end, after all, 1! does indeed equal 1. If a cameraman catches you, you just might make it onto ESPN. If that happens, you can bask in the thought that 5 or 10 people across the nation chuckled when they saw your handiwork.

0.20787…

Check this out…

$\huge i^{2}=-1\; or\; i=\sqrt{-1}$

That looks familiar, right? I think most of us have come across both of those definitions at some point. What you probably didn’t learn is this…

$\huge i^{i}=e^{\frac{-\pi }{2}}$

Take a close look at that…and then look at this…

$\huge i^{i}=0.20787...$

Truly astonishing, an imaginary number raised to the power of an imaginary number gives us a never-ending decimal. As always, if you are interested in deriving the answer, do a Google search. You will be amazed at what you will find.

How About a New Punctuation Mark?

Do you have any idea what this is?

How about a hint? Oh yeah, that is a really good idea

Can I have another one, that last one was so helpful

OK, you can have a couple more.

You look great Where are you going, gravedigging

Nice ketchup stain, it really pulls your entire ensemble together

Yes, I present the mighty Sarcmark. Douglas Sak, a man in need of a punctuation mark to better denote sarcasm in his emails, founded Sarcasm, Inc. in 2006. Shortly thereafter, the Sarcmark was offered for sale. Not really a bad idea, is it?

I am always on board when someone is offering up ways to better clarify ideas, especially those of the written variety. The website claims that the symbol is gaining traction but I must admit I have not seen it being used anywhere. Except here. Except now. Where else on the entire internet could you get such valuable information this easily?

If you really feel you need this Punctuation Mark in your grammatical arsenal, then I urge you to get over to the Sarcmark website ASAP. When I came across the site years ago, the Sarcmark cost real money, now it is free. I suggest you head on over, I would hate for you to send an email to a friend only to have them not realize you were insulting them.

Inspired by a Yahoo

Inspired by a Yahoo: Inspiration is in Short Supply, we all need to take it Wherever and Whenever we can get it.

I am bursting at the seams. Writing about Edwin Goodwin (the clown who “discovered” that π = 3.2) has elevated me; my ambitions now include worldwide fame. I want riches, texts from supermodels (OK, Danica Patrick would do just fine), and a driver named Jeeves. How am I going to get all these things? Easy, I stayed up late last night and came up with the following rock-solid mathematical proof that 2 = 1. Read on.

$\! \! \! \! \! \! a=b \; \; (easy\; enough)\\\\ a^{2}=ab \; \; (multiply\; both\; sides\; by \; a)\\\\ a^{2}+a^{2}-2ab=ab+a^{2}-2ab \; \; (add\; a^{2}-2ab\; to\; both\; sides)\\\\ 2(a^{2}-ab)=a^{2}-ab \; (simplify)\\\\ \frac{2(a^{2}-ab)}{a^{2}-ab}= \frac{a^{2}-ab}{a^{2}-ab}\\\\ \therefore 2=1$

Why would I give you only one proof when I have two in the bag? Behold the following:

$\! \! \! \! \! a=b \; \; (easy\; enough)\\\\ a^{2}=ab \; \; (multiply\; both\; sides\; by \; a)\\\\ a^{2}-ab=ab-b^{2}\; \; (ab=b^{2})\\\\ (a+b)(a-b)=b(a-b)\\\\ a+b=b\\\\ b+b=b\\\\ 2b=b\\\\ \therefore 2=1$

I am basking in my own genius. I am going to call my congressman and get this thing written into law ASAP. I wonder which supermodel will text me first? My mind is racing, good grief I might even hear from Athena (you will be hearing much more about her later this year). Wow, I’ll probably have to set up some kind of a rotating schedule so that all these women don’t show up at my house at the same time. I don’t want them fighting over little old me. So, do you think this will be a big problem or have I missed something? Do I need to buy some new socks and shirts or should I just chill?