For the last couple years, one of my favorite sources of entertainment has been the wonderful world of YouTube Golf. Intending no disrespect to any other content creators, my favorite channels are the ones by Grant Horvat, the Bryan Bros (not to be confused with the twin tennis duo), Peter Finch, Bryson DeChambeau (of course), and Golf Girl Games (all of them absolutely, positively should have been in the Internet Invitational… but that’s another story for another day).
In a recent Bryan Bros video, my two interests collided. To make a long story short, a golf simulator projected that a tee shot on a par-3 ended 8 feet, 12 inches from the cup.
Co-host Wesley Bryan, to his great credit, immediately saw the computer glitch — this is an unusual way of saying the tee shot ended 9 feet from the cup. Hilarity ensued as the golfers held a stream-of-consciousness debate on the merits of metric and Imperial units. The video is below: the fun begins at the 21:41 mark and ends around 25:30.
A brief clip from Megan Moroney’s video “I’m Not Pretty” correctly uses polynomial long division to establish that is a factor of . Even more amazingly, the fact that the remainder is actually fits artistically with the video.
And while I have her music on my mind, I can’t resist sharing her masterpiece “Tennessee Orange” and its playful commentary on the passion of college football fans.
I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show my series on the mathematical allusions in the novel Shantaram.
Here is one school’s results from a (relatively) recent track and field meet. Never mind the name of the school or the names of the athletes representing the school; this is a math blog and not a sports blog, even though I’m an avid sports fan. Furthermore, I have nothing but respect for young people who are both serious students and serious athletes. While I have no illusions about the global popularity of this blog, and while the information from the meet are in the public domain, I also have no desire to inadvertently subject these student-athletes to online abuse.
With that preamble, here are the school’s results:
The unusual score for jumps caught my attention. Clearly a score of was intended, but this isn’t displayed. (This is also a lesson about using unnecessary precision… unlike the total points field showing 152.33.) Before this unusual decimal expansion, however, I should generally describe how teams are scored at a track and field meet… or at least the high school and college meets that I’ve attended in the United States.
Each meet has multiple events, often categorized as sprints, hurdles, distance races, throwing events, jumping events, relay races, and multi-sport events (like the decathlon). At each event, first place gets 10 points, second place gets 8 points, third place gets 6 points, fourth place gets 5 points, fifth place gets 4 points, sixth place gets 3 points, seventh place gets 2 points, and eighth place gets 1 point.
Let’s explain the last two lines first. At this meet, athletes from this team finished second, sixth, and seventh in the one multi-sport event, earning points for the school. A relay team finished third in the 4×100 meter relay, earning another 6 points for the school.
The third-to-last line — jumps — requires some explanation. No athlete from the school finished in top eight in the long jump or the triple jump (0 points). One athlete won the high jump (10 points). And one athletic finished in a three-way tie for second place in the pole vault. In the case of such a tie, the points for second, third, and fourth place are averaged and given to all three competitors, for points. In total, the school earned points from the jumping events.
In jumping events, it is possible (but rare) for athletes to tie. The table above shows the results of the competition for the top eight finishers. The lingo: P means the competitor passed at that height (to save time and energy), O means a successful attempt, and X means a failed attempt. So, the winner passed at all heights up to and including 2.70 meters, succeeded on the first attempt at 2.85, 3.00, and 3.15 meters. This athlete was the only one who cleared 3.15 meters and thus won the competition. This athlete then failed three times at 3.30 meters: each athlete has three attempts at each height; three failures at one height means elimination from the competition.
The athletes in the next three lines had the exact same performance: success at 2.70 and 2.85 meters on the first attempt, and then three straight failed attempts at 3.00 meters. Because there is nothing to distinguish the three performances, the athletes are deemed to be tied.
The athletes in fifth and sixth place also cleared 2.85 meters, but on their second attempts. Therefore, they are behind the athletes who cleared 2.85 meters on the first attempt. Furthermore, the athlete in fifth place cleared 2.70 meters on the first attempt, while the athlete in sixth place needed two attempts. Similarly, the athletes in seventh and eighth place both cleared 2.70 meters; the tiebreaker is the number of attempts needed at 2.40 meters.
By contrast, at the 2024 Olympics, the top two competitors in the men’s high jump tied but decided to continue the competition with a jumpoff until there was one winner.
My apologies to any track and field experts if my description of the scoring wasn’t quite perfect.
Back to mathematics… and back to the scores. Why did the computer think that the number of points from jumps was 16.333333492279053 and not ?
There are two parts to the answer: (1) Computers store numbers in binary, and (2) they only store a finite number of binary digits.
Converting 16 into binary is easy: since , its representation in binary is 10000.
Converting into binary is more challenging, and perhaps I’ll write a separate post on this topic. This particular fraction can be found by using the formula for an infinite geometric series:
If we let , then we find
.
Said another way,
Combining the two results,
This is mathematically correct; however, computers use floating-point arithmetic only store a finite number of digits to represent any number. In this case, we can reverse-engineer to figure out how many digits are stored. In this case, after some trial and error, I found that 21 digits were apparently stored after the decimal point:
This is equivalent to the sum ; notice that the last fraction is basically rounding up in binary. Mathematica confirms that this sum matches the sum shown in the school’s team score:
So the computer showed far too many decimal places in the “Jumps” field, and it probably should’ve been programmed to show only two decimal places, like in the “Points” field.
I close by linking to this previous post on the 1991 Gulf War, describing why a similarly small error in approximating in binary tragically led to a bigger computational error that caused the death of 28 soldiers.
Let be the set of all times, and let measure how good day is. Translate the logical statement
where time is today.
This matches the chorus of “Best Days of Your Life” by Kellie Pickler, co-written by and featuring Taylor Swift.
Context: Part of a discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.
In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.
When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.
Let be the set of all times, and let be the statement “I got by on my own at time .” Translate the logical statement
where time is today.
This matches the opening line of the fabulous power ballad “Alone” by Heart.
And while I’ve got this song in mind, here’s the breakout performance by a young unknown Carrie Underwood on American Idol.
Context: Part of a discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.
In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.
When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.
I recently finished the novel Shantaram, by Gregory David Roberts. As I’m not a professional book reviewer, let me instead quote from the Amazon review:
Crime and punishment, passion and loyalty, betrayal and redemption are only a few of the ingredients in Shantaram, a massive, over-the-top, mostly autobiographical novel. Shantaram is the name given Mr. Lindsay, or Linbaba, the larger-than-life hero. It means “man of God’s peace,” which is what the Indian people know of Lin. What they do not know is that prior to his arrival in Bombay he escaped from an Australian prison where he had begun serving a 19-year sentence. He served two years and leaped over the wall. He was imprisoned for a string of armed robberies performed to support his heroin addiction, which started when his marriage fell apart and he lost custody of his daughter. All of that is enough for several lifetimes, but for Greg Roberts, that’s only the beginning.
He arrives in Bombay with little money, an assumed name, false papers, an untellable past, and no plans for the future. Fortunately, he meets Prabaker right away, a sweet, smiling man who is a street guide. He takes to Lin immediately, eventually introducing him to his home village, where they end up living for six months. When they return to Bombay, they take up residence in a sprawling illegal slum of 25,000 people and Linbaba becomes the resident “doctor.” With a prison knowledge of first aid and whatever medicines he can cadge from doing trades with the local Mafia, he sets up a practice and is regarded as heaven-sent by these poor people who have nothing but illness, rat bites, dysentery, and anemia. He also meets Karla, an enigmatic Swiss-American woman, with whom he falls in love. Theirs is a complicated relationship, and Karla’s connections are murky from the outset.
While it was a cracking good read, what struck me particularly were the surprising mathematical allusions that the author used throughout the novel. In this mini-series, I’d like to explore the ones that I found.
In this fourth and final installment, the narrator has a lengthy conversation with his mentor (a mafia don) about his mentor’s philosophy of life.
[The mafia don said,] “I will use the analogy of the way we measure length, because it is very relevant to our time. You will agree, I think, that there is a need to define a common measure of length, yes?”
“You mean, in yards and metres, and like that?”
“Precisely. If we have no commonly agreed criterion for measuring length, we will never agree about how much land is yours, and how much is mine, or how to cut lengths of wood when we build a house. There would be chaos. We would fight over the land, and the houses would fall down. Throughout history, we have always tried to agree on a common way to measure length. Are you with me, once more, on this little journey of the mind?”
“I’m still with you,” I replied, laughing, and wondering where the mafia don’s argument was taking me.
“Well, after the revolution in France, the scientists and government officials decided to put some sense into the system of measuring and weighing things. They introduced a decimal system based on a unit of length that they called the metre, from the Greek word metron, which has the meaning of a measure.”
“Okay…”
“And the first way they decided to measure the length of a metre was to make it one ten-millionth of the distance between the equator and the North Pole. But their calculations were based on the idea that the Earth was a perfect sphere, and the Earth, as we now know, is not a perfect sphere. They had to abandon that way of measuring a metre, and they decided, instead, to call it the distance between two very fine lines on a bar of platinum-iridium alloy.”
“Platinum…”
“Iridium. Yes. But platinum-iridium alloy bars decay and shrink, very slowly — even though they are very hard — and the unit of measure was constantly changing. In more recent times, scientists realised that the platinum-iridium bar they had been using as a measure would be a very different size in, say, a thousand years, than it is today.”
“And… that was a problem?”
“Not for the building of houses and bridges,” [the mafia don] said, taking my point more seriously than I’d intended it to be.
“But not nearly accurate enough for the scientists,” I offered, more soberly.
“No. They wanted an unchanging criterion again which to measure all other things. And after a few other attempts, using different techniques, the international standard for a metre was fixed, only last year, as the distance that a photon of light travels in a vacuum during, roughly, one three-hundred-thousandth of a second. Now, of course, this begs the question of how it came to be that a second is agreed upon as a measure of time. It is an equally fascinating story — I can tell it to you, if you would like, before we continue with the point about the metre?”
“I’m… happy to stay with the metre right now,” I demurred, laughing again in spite of myself.
“Very well. I think that you can see my point here — we avoid chaos, in building houses and dividing land and so forth, by having an agreed standard for the measure of a unit of length. We call it a metre and, after many attempts, we decide upon a way to establish the length of that basic unit.”
Shantaram, Chapter 23
After this back-and-forth, the mafia don then described how his philosophy of life can be likened to the need to redefine a basic unit, like the meter, based on our ability to make more accurate measurements with the passage of time.
For the purposes of this blog post, I won’t go into the worldview of a fictional mafia don, but I will discuss the history of the meter, which is accurately described in the above conversation. The definition of the meter has indeed changed over the years with our ability to measure things more accurately.
Initially, in the aftermath of the French revolution, the meter was defined so that the distance between the North Pole and the equator along the longitude through Paris would be exactly 10,000 kilometers. (Since that distance is a quarter-circle, the circumference of the Earth is approximately 40,000 kilometers.)
Later, in 1889, the meter was defined as the length of a certain prototype made of platinum and iridium.
In 1960, the meter was redefined in terms of the wavelength of a certain type of radiation from the krypton-86 atom.
In 1983, the meter was redefined so that the speed of light would be exactly 299,792,458 meters per second. (Incidentally, after 1967, a second was defined to be 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.) Regarding the novel, the above conversation happened in 1984, one year after the meter’s new definition.
These definitions of the meter and second were reiterated in the latest standards, which were released in 2018. This latest revision finally defined the kilogram without the need of a physical prototype.
I recently finished the novel Shantaram, by Gregory David Roberts. As I’m not a professional book reviewer, let me instead quote from the Amazon review:
Crime and punishment, passion and loyalty, betrayal and redemption are only a few of the ingredients in Shantaram, a massive, over-the-top, mostly autobiographical novel. Shantaram is the name given Mr. Lindsay, or Linbaba, the larger-than-life hero. It means “man of God’s peace,” which is what the Indian people know of Lin. What they do not know is that prior to his arrival in Bombay he escaped from an Australian prison where he had begun serving a 19-year sentence. He served two years and leaped over the wall. He was imprisoned for a string of armed robberies performed to support his heroin addiction, which started when his marriage fell apart and he lost custody of his daughter. All of that is enough for several lifetimes, but for Greg Roberts, that’s only the beginning.
He arrives in Bombay with little money, an assumed name, false papers, an untellable past, and no plans for the future. Fortunately, he meets Prabaker right away, a sweet, smiling man who is a street guide. He takes to Lin immediately, eventually introducing him to his home village, where they end up living for six months. When they return to Bombay, they take up residence in a sprawling illegal slum of 25,000 people and Linbaba becomes the resident “doctor.” With a prison knowledge of first aid and whatever medicines he can cadge from doing trades with the local Mafia, he sets up a practice and is regarded as heaven-sent by these poor people who have nothing but illness, rat bites, dysentery, and anemia. He also meets Karla, an enigmatic Swiss-American woman, with whom he falls in love. Theirs is a complicated relationship, and Karla’s connections are murky from the outset.
While it was a cracking good read, what struck me particularly were the surprising mathematical allusions that the author used throughout the novel. In this mini-series, I’d like to explore the ones that I found.
In this third installment, the narrator a sudden realization that he had.
I put all of my focus on the beating of my heart, trying by force of will to slow its too-rapid pace. It worked, after a time. I closed around a single, still thought. That thought was of [a mafia don], and the formula he’d made me repeat so often: “The wrong thing, for the right reasons.” And I knew, as I repeated the words in the fearing dark, that the fight with [another mafia don], the war, the struggle for power, was always the same, everywhere, and it was always wrong.
[My mafia don], no less than [other mafia dons] and all the rest of them, were pretending that their little kingdoms made them kings; that their power struggles made them powerful. And they didn’t. They couldn’t. I saw that then so clearly that it was like understanding a mathematical theorem for the first time. The only kingdom that makes any man a king is the kingdom of his own soul. The only power that has any real meaning is the power to better the world. And only men like [my noble friends, not in the mafia] were such kings and had such power.
Shantaram, Chapter 41
The author’s choice of language is music to my ears: “I saw that then so clearly that it was like understanding a mathematical theorem for the first time.” There have been many, many times throughout my education and career that I struggled to understand some theorem. But the moment that I figured it out, I couldn’t believe what had taken me so long to finally get it. That’s the type of epiphany that the author seems to be describing.
I again quote at length from Richard P. Feynman, who did a far better job of explaining the emotions of such a sudden realization after being stuck in a rut than I ever could:
Then I had another thought: Physics disgusts me a little bit now, but I used to enjoy doing physics. Why did I enjoy it? I used to play with it. I used to do whatever I felt like doing–it didn’t have to do with whether it was important for the development of nuclear physics, but whether it was interesting and amusing for me to play with. When I was in high school, I’d see water running out of a faucet growing narrower, and wonder if I could figure out what determines that curve. I found it was rather easy to do. I didn’t have to do it; it wasn’t important for the future of science; somebody else had already done it. That didn’t make any difference: I’d invent things and play with things for my own entertainment.
So I got this new attitude. Now that I am burned out and I’ll never accomplish anything, I’ve got this nice position at the university teaching classes which I rather enjoy, and just like I read the Arabian Nights for pleasure, I’m going to play with physics, whenever I want to, without worrying about any importance whatsoever.
Within a week I was in the cafeteria and some guy, fooling around, throws a plate in the air. As the plate went up in the air I saw it wobble, and I noticed the red medallion of Cornell on the plate going around. It was pretty obvious to me that the medallion went around faster than the wobbling.
I had nothing to do, so I start to figure out the motion of the rotating plate. I discover that when the angle is very slight, the medallion rotates twice as fast as the wobble rate–two to one. It came out of a complicated equation! Then I thought, “Is there some way I can see in a more fundamental way, by looking at the forces or the dynamics, why it’s two to one?”
I don’t remember how I did it, but I ultimately worked out what the motion of the mass particles is, and how all the accelerations balance to make it come out two to one.
I still remember going to Hans Bethe and saying, “Hey, Hans! I noticed something interesting. Here the plate goes around so, and the reason it’s two to one is . . .” and I showed him the accelerations.
He says, “Feynman, that’s pretty interesting, but what’s the importance of it? Why are you doing it?”
“Hah!” I say. “There’s no importance whatsoever. I’m just doing it for the fun of it.” His reaction didn’t discourage me; I had made up my mind I was going to enjoy physics and do whatever I liked.
I went on to work out equations of wobbles. Then I thought about how electron orbits start to move in relativity. Then there’s the Dirac Equation in electrodynamics. And then quantum electrodynamics. And before I knew it (it was a very short time) I was “playing”–working, really — with the same old problem that I loved so much, that I had stopped working on when I went to Los Alamos: my thesis-type problems; all those old-fashioned, wonderful things.
It was effortless. It was easy to play with these things. It was like uncorking a bottle: Everything flowed out effortlessly. I almost tried to resist it! There was no importance to what I was doing, but ultimately there was. The diagrams and the whole business that I got the Nobel Prize for came from that piddling around with the wobbling plate.
Richard P. Feynman, “The Dignified Professor,” from Surely You’re Joking, Mr. Feynman!
I recently finished the novel Shantaram, by Gregory David Roberts. As I’m not a professional book reviewer, let me instead quote from the Amazon review:
Crime and punishment, passion and loyalty, betrayal and redemption are only a few of the ingredients in Shantaram, a massive, over-the-top, mostly autobiographical novel. Shantaram is the name given Mr. Lindsay, or Linbaba, the larger-than-life hero. It means “man of God’s peace,” which is what the Indian people know of Lin. What they do not know is that prior to his arrival in Bombay he escaped from an Australian prison where he had begun serving a 19-year sentence. He served two years and leaped over the wall. He was imprisoned for a string of armed robberies performed to support his heroin addiction, which started when his marriage fell apart and he lost custody of his daughter. All of that is enough for several lifetimes, but for Greg Roberts, that’s only the beginning.
He arrives in Bombay with little money, an assumed name, false papers, an untellable past, and no plans for the future. Fortunately, he meets Prabaker right away, a sweet, smiling man who is a street guide. He takes to Lin immediately, eventually introducing him to his home village, where they end up living for six months. When they return to Bombay, they take up residence in a sprawling illegal slum of 25,000 people and Linbaba becomes the resident “doctor.” With a prison knowledge of first aid and whatever medicines he can cadge from doing trades with the local Mafia, he sets up a practice and is regarded as heaven-sent by these poor people who have nothing but illness, rat bites, dysentery, and anemia. He also meets Karla, an enigmatic Swiss-American woman, with whom he falls in love. Theirs is a complicated relationship, and Karla’s connections are murky from the outset.
While it was a cracking good read, what struck me particularly were the surprising mathematical allusions that the author used throughout the novel. In this mini-series, I’d like to explore the ones that I found.
In this second installment, the narrator describes a conversation with a new acquaintance.
“My father was a teacher of chemistry and mathematics… My father was a stubborn man — it is a kind of stubbornness that permits one to become a mathematician, isn’t it? Perhaps mathematics is itself a kind of stubbornness, do you think?”
“Maybe,” I replied, smiling. “I never thought about it that way, but maybe you’re right.”
Shantaram, Chapter 26
When I think of stubbornness, I think of the determination of a marathon runner to push through fatigue to keep running hour after hour to complete all 26.2 miles of the course. I don’t usually think of a mathematician.
Nevertheless, the author certainly hit on something with this allusion. Mathematicians certainly need a healthy dose of stubbornness when staring a conjecture and trying to figure out its proof; it’s normal for that frustration to last for weeks, months, or even years. That said, I wouldn’t say that this is unique to mathematicians — researchers in just about any field of study need to be persistent to discover something that nobody else has figure out before.
Nobel Prize laureate Richard P. Feynman had a couple of vivid descriptions about the frustration of getting stuck on a research project and the stubbornness that was necessary to break out of that rut.
I don’t believe I can really do without teaching. The reason is, I have to have something so that when I don’t have any ideas and I’m not getting anywhere I can say to myself, “At least I’m living; at least I’m doing something; I’m making some contribution”–it’s just psychological.
When I was at Princeton in the 1940s I could see what happened to those great minds at the Institute for Advanced Study, who had been specially selected for their tremendous brains and were now given this opportunity to sit in this lovely house by the woods there, with no classes to teach, with no obligations whatsoever. These poor bastards could now sit and think clearly all by themselves, OK? So they don’t get any ideas for a while: They have every opportunity to do something, and they’re not getting any ideas. I believe that in a situation like this a kind of guilt or depression worms inside of you, and you begin to worry about not getting any ideas. And nothing happens. Still no ideas come.
Nothing happens because there’s not enough real activity and challenge: You’re not in contact with the experimental guys. You don’t have to think how to answer questions from the students. Nothing!
In any thinking process there are moments when everything is going good and you’ve got wonderful ideas. Teaching is an interruption, and so it’s the greatest pain in the neck in the world. And then there are the longer periods of time when not much is coming to you. You’re not getting any ideas, and if you’re doing nothing at all, it drives you nuts! You can’t even say “I’m teaching my class.”
Richard P. Feynman, “The Dignified Professor,” from Surely You’re Joking, Mr. Feynman!
Also from Feynman:
The problem was to find the right laws of beta decay. There appeared to be two particles, which were called a tan and a theta. They seemed to have almost exactly the same mass, but one disintegrated into two pions, and the other into three pions. Not only did they seem to have the same mass, but they also had the same lifetime, which is a funny coincidence. So everybody was concerned about this…
At that particular time I was not really quite up to things: I was always a little behind. Everybody seemed to be smart, and I didn’t feel I was keeping up…
Anyway, the discovery of parity law violation was made, experimentally, by Wu, and this opened up a whole bunch of new possibilities for beta decay theory, It also unleashed a whole host of experiments immediately after that. Some showed electrons coming out of the nuclei spun to the left, and some to the right, and there were all kinds of experiments, all kinds of interesting discoveries about parity. But the data were so confusing that nobody could put things together.
At one point there was a meeting in Rochester–the yearly Rochester Conference. I was still always behind, and Lee was giving his paper on the violation of parity. He and Yang had come to the conclusion that parity was violated, and flow he was giving the theory for it.
During the conference I was staying with my sister in Syracuse. I brought the paper home and said to her, “I can’t understand these things that Lee and Yang are saying. It’s all so complicated.”
“No,” she said, “what you mean is not that you can’t understand it, but that you didn’t invent it. You didn’t figure it out your own way, from hearing the clue. What you should do is imagine you’re a student again, and take this paper upstairs, read every line of it, and check the equations. Then you’ll understand it very easily.”
I took her advice, and checked through the whole thing, and found it to be very obvious and simple. I had been afraid to read it, thinking it was too difficult.
Richard P. Feynman, “The 7 Percent Solution,” from Surely You’re Joking, Mr. Feynman!
is a factor of 
. Even more amazingly, the fact that the remainder is 
actually fits artistically with the video.
was intended, but this isn’t displayed. (This is also a lesson about using unnecessary precision… unlike the total points field showing 152.33.) Before this unusual decimal expansion, however, I should generally describe how teams are scored at a track and field meet… or at least the high school and college meets that I’ve attended in the United States.
points for the school. A relay team finished third in the 4×100 meter relay, earning another 6 points for the school.
points. In total, the school earned 
?
, its representation in binary is 10000. 
into binary is more challenging, and perhaps I’ll write a separate post on this topic. This particular fraction can be found by using the formula for an 
, then we find
.
; notice that the last fraction is basically rounding up in binary. Mathematica confirms that this sum matches the sum shown in the school’s team score:
in binary tragically led to a bigger computational error that caused the death of 28 soldiers.
be the set of all times, and let 
measure how good day 
is. Translate the logical statement
Mean Green Math 