Weight of a Car

I was told this weekend that you can find the weight of a car by multiplying the area of contact the tires make with the ground and the air pressure in the tires. Color me skeptical. I had never heard such a thing.

I was a doubter. I was told the area would be measures in square inches and the pressure in the typical pounds per square inch. Then I thought, area x pressure would then be sq. inches x pounds per square would give you and answer in pounds. So the labels work out, but I was still far from being convinced.

I thought I had a counter proof. What if you put more weight in the car? Then contact area would be greater and I would think the pressure would go up as well. So the car calculations would now be greater. Then I realized I started by making the car heavier, so of course the answer should be greater. So, in trying to disprove this idea, I actually acquire more evidence for it.

I looked online and found that a formula of F = PA or Force = Pressure x Area. (I probably should have remembered that from when I took physics, but that was a while ago.) Since gravity is a force, I'm becoming a little more convinced.

I looked it up online. I literally Googled "tire pressure to find weight of car" and found it. That Google is pretty smart. Anyway, they had a lesson for it. I didn't read it very thoroughly, but it seemed to back up what I had been told. It also said that when done you could check for the actual weight by looking at the sticker inside the driver's door. Well, there is another thing I learned today. I didn't even know there was a sticker there.

It seemed they found the area of a tire and multiplied it by the psi for each individual tire as there could be slight differences from tire to tire. Then they added them altogether.

How does it work out? I don't know. I might try it sometime. Right now the average high temperature is about 35 degrees. If I do spend much time outside, its going to have to be for a better reason than pursuing this. Maybe in the Spring.

How would it work though if I did do it? I know how to find the air pressure. But to find the area? Maybe outline it with chalk? What shape would that be? I'm thinking maybe like a running track with two straight segments and two semicircles. But I'm not sure.

It does seem like an interesting math application and one a class would get into. As a practical matter, probably not. If you really wanted the weight of a care, it seems easier to just check that sticker inside the door.

More Great Videos

Last time I wrote about the group OK Go. It was about one specific video. Well, they actually have a number of amazingly great videos. One you might check out is this one -

Fun, huh?

Following is one that talks about the making of the video. Its doesn't get real detailed on the math, but does allude to the ratios and proportions that must be used to come up with the final product. Granted it might not be an earthshaking important application, but students would get into it and a teacher could make use of it if study ratios at the time. For example ... "The fast portion took 4.2 seconds and the entire video is 4 minutes 12 seconds. If the exploding paint sequence lasted 11 seconds in the long video, then how long..." You get the idea.

There are a lot of hidden gems. It wasn't until I had seen it a few times that I noticed them even throwing in the Doppler effect.

It seems like there is trick photography involved, but there isn't. It takes them weeks to do these videos. Pretty amazing and it shows students the hard work they put in physically and mentally. The videos are cool and addictive - both the videos themselves and their "making of" videos.

Best Music Video Ever

Best video ever. Check it out at okgo.net. It features my now favorite band, OK Go. It's not their only

awesome video either. Did I tell you to check it out? Do so.

The band was approached by a Russian airline about using a plane to do some flights for a video that would involve weightlessness. I know. A little hard to believe. But I read about it in an issue of Smithsonian Magazine. If they aren't a credible source, who would be?

The weightless part of the flight doesn't last long and to do anything substantial, it probably takes several runs at it. You might be familiar with the concept. you can fly a plane or rocket in a parabolic path to achieve weightlessness. It's how the Tom Hanks film, Apollo 13, had its weightless moments shot.

The article (December, 2016, pages 52,53) explained the process.

1. Steady horizontal flight.
2. Hypergravity (1.5 to 1.8 g's) for 20 to 25 seconds at a 47 degree ascent.
3. Microgravity (0 g's) for 20 to 25 seconds.
4. Hypergravity (1.5 to 1.8 g's) for 20 to 25 seconds at a 47 degree descent.
5. Steady horizontal flight.

The weightlessness takes place during that 20 to 25 second sweet spot at the top of the curve. As the authour Jeff MacGregor stated "Then came the math. The song is 3 minutes and 20 seconds long, give or take. Weightlessness during parabolic flight occurs in roughly 25-second increments. That's at the top of each parabola. And for every parabola, it takes five minutes of flight to reset for the next one. To get a single continuous weightless take lasting 3:20 would require eight parabolas - more than 45 minutes of actual flying time. 

This is perhaps not the best ever math application. However, have used it in movies and videos and it has even been used for weddings. Maybe it is the best ever math application.

Book

My book is coming along. My blog is to promote the same idea as my book - applications of mathematics at the high school level. I don't want to be some jerk and use this blog to publicize my book. At least not very often. Let me take this week, though, and tell you where I'm at on it.

I just heard from my peeps at McFarland Publishing. They are pretty much done with what they do. The sent me a PDF of my book - exactly 200 pages. I now have two more jobs.

One is to do a final read through for errors. I'm about 3/4 the way through that. It seems pretty good. The main mistake that has come up is where I use a subscript or superscript in an equation and the next symbol mistakenly gets put into that form as well. No big deal. I think that is easily correctable.

They say I can respond by e-mail if there aren't too many errors (under 30) or run off the offending pages with corrections written in. They let me know it isn't time for general editing. This is just to find errors. I'm not sure I always know the difference, but in most cases I think I do.

Job 2 - Create an index. I've never done that before. I thought I would read it for mistakes and plan out the index at the same time. Nope. That turned out to be way more than my head could handle. So by tomorrow I think I'll be through the initial reading. I don't think I'll have to read every word to do the indexing, so perhaps that will go faster.

I've wondered before how they come up with an index. They sent instructions on how it is done, or at least how they want it done. It's pretty interesting. There are some rules, but at the same time a lot is left to the discretion of the author. Rules just on the correct alphabetize an index is a lot more complex than one would think.

I have to do this and send it in by December 31. It goes to the printer then. I'm not exactly sure when it officially comes out, but shouldn't be too long.

I hope this wasn't boring, but I've never done this before and going through the process has been interesting.

Israeli Elevations

I had mentioned last week that I had just been in Israel. Finding math applications in a foreign land is not easy. I have enough trouble doing so in the US.

Last week I looked at converting temperatures in your head from Celsius to Fahrenheit. That made me thing about negative temperatures. That made me think about negatives in general.

When encountering negative numbers for the first time, many students reject the notion that numbers can be less than zero. But of course they can. While you can't have a negative number of apples, you can have a negative amount of money, degrees, elevations, etc.

Specifically, let's look at subtraction in finding a difference of elevation. The Dead Sea is at 1,414 feet below sea level. Jerusalem is at 2,582 feet above sea level. What is the difference in elevation? 2,582 - (-1,414) = 3,996 feet.

What if we subtract in the other order? -1,414 - 2,582 = -3,996 feet. What are we to make of the negative answer? That is what the absolute value is for. If a and b are elevations, the difference in elevation is |a-b|. This is easier than writing something cumbersome like: a-b if a > b and b-a if a < b.

This shows students a practical example of integer subtraction and the absolute value concept.

One more example of this: The Sea of Galilee flows into the Dead Sea by way of the Jordan River. All of these are below sea level. The Sea of Galilee is 696 feet below sea level. Here we are subtracting two negatives. |-696 - (-1,414)| = |-696 + 1,414)| = 712 feet.

Israel Temperatures

I haven't blogged for a while because I've been in Israel. It is and will be the longest trip of my life. We went through ten time zones to get there. Not a lot of math applications on this trip, however there was one I came up with.

Most things are translated. Many speak English - at least at the touristy spots. Most of the signs are in Hebrew, English, Arabic, and might even include a picture.

However, you don't get any help on the metric system. Speeds signs were in kilometers per hour. Temperatures were done in Celsius. At the Dead Sea there was a thermometer. On television we could see weather forecasts. All were in degrees Celsius.

It was pretty warm there. It is a little closer to the Equator than I am used to. Also, were were in places below sea level. Those along with the fact that they said they were experiencing a warmer than usual November made for pretty warm days. The temperatures were usually in the 20's. My wife would see these and ask me how hot it really was.

Converting can be done with the formula F = (9/5)C+32. However, most people aren't going to be in the mood for this formula with or without a piece of paper, and usually we were without paper.

I came up with a passable method. My method was round off to the nearest multiple of five. Then divide by five, multiply by nine, then add 32. As I thought about it, I kept refining my method. My goal was to come up with something easy to use and would give a pretty good approximation for the degrees Fahrenheit.

• The above method - round off to the nearest multiple of five, divide by five, multiply by nine, then add 32.
• Multiply by two (close to 9/5) and add thirty-two.
• Multiply by two and add thirty.

I thought the last was pretty good. Adding thirty is easier than thirty-two. And it might compensate for using the larger value of two rather than nine-fifths.

So how does that work? Lets try it out for 20, 25, and 30 degrees Celsius.

• 20 degrees C:  Real temperature is 68 degrees F and with my method 72 degrees F.
• 25 degrees C:  Real temperature is 77 degrees F and with my method 82 degrees F.
• 30 degrees C:  Real temperature is 86 degrees F and with my method 92 degrees F.

I felt somewhat good about myself and my new method. Some further expansion on this application:

Graph y = (9/5)x+32 and y = 2x+30.

How close do these match up?

Are there temperature ranges this approximation works for and doesn't work for?

Are the better conversion formulas that approximate the temperature?

What would be a good approximation formula for an Israeli visiting the U.S.?

Golden Gate Bridge

I'm going to be out for a couple of weeks, so I thought I should squeeze in one more post before I take off.

Last week I ran a half marathon. I know, foolish. But the main reason I did it is because it was in San Francisco and the course crossed the Golden Gate Bridge twice. You can see the bridge from a distance in downtown San Francisco. You can also see it driving across in a car, but it goes by pretty quick. Neither of those are the same experience as crossing it on foot. The cable droops down almost all the way to the road. It was fun to get right up close to it. Its just about head level at its lowest point.

An interesting thing I noticed is that the roadway is curved. You definitely run a bit uphill then down. The highest point, though, is not in the middle. I'm sure engineers had a reason for that, but that would be beyond me.

You can get many statistics regarding the bridge on-line. I was going to cut and paste them here, but they are easy to find. A teacher could fashion math applications for anything from arithmetic to calculus. A cool one is to find the equation of a parabola approximating the cable.

Since the cable is about head-height, using the roadway as the x-axis, we can take the center of the cable to be the point (0,6). The distance between the two towers is 4,200 feet. The height of the towers above the roadway is 500 feet. So, two other points on the cable could then be (2,100, 500) and (-2,100, 500). Using a system of equations with those three points could give you an equation of a parabola.

Also, the distance from the mean high water mark to the road is 220 feet. So a parabola could be found with the Pacific Ocean represented by the x-axis.

Or it could be found using meters rather than feet.

You could probably spend a week just studying the bridge. And it would make for a cool field trip.

Election Day

This just happens to be election day. Let's follow up on my posting last week about polls. Today there won't be polls, there will be projections. They change the name, but they're really the same thing. It turns out they can be wrong.

• I read an article that said there was a primary several months ago in which pollsters took data to say that Clinton had a 99% chance of winning. Sanders ended up with a narrow victory. I guess you can't say he was wrong. Some things that are predicted to happen one percent of the time, do happen. Still probably embarrassing for those pollsters, though. 
• In 1948, Harry Truman famously held up a newspaper declaring that "Dewey Defeats Truman". He didn't. Dewey had such a lock on it. As George Gallup Jr. said about this, "We quit polling a few weeks too soon." That'll do it.
• In 1936, Literary Digest conducted a survey of its readership. It picked Alf Landon over Franklin Roosevelt. It turns out that Literary Digest (which has since gone out of business for obvious reasons) mostly appealed to a higher-income type person. That skewed Republican, thus predicting a President Landon.
• In 2000 the television networks declared Al Gore the winner of Florida. That was all he needed to be president. They had to retract that, declaring the race "too close to call". Overnight the networks declared George Bush the winner. Later, back to "too close to call". 

Last time we looked at how the polls work. A newscaster might say, "Candidate X is at 57% with a 3% margin of error". They usually don't mention the level of confidence. I thought they use a 90% confidence level. I saw something lately that said it is usually 95%. Regardless, they're pretty confident. But they aren't certain. 

Let's take that example and use a 95% confidence level. 

Candidate X is at 57% with a 3% margin of error translates to:

We are 95% sure that he is somewhere between 54% and 60%.

There is actually more to it than that. Consider a bell-shaped curve peaking at 57%. Of all the possible outcomes, 57% is most likely. Then 56%, then 55%, then 54%. Even 53% or below is not out of the question. Very unlikely, but not out of the question.

So if Candidate Y is at 45%, he/she is probably going to lose. However, it won't be because the 3% margin of error says he has to.

Pretty confusing. No wonder the pollsters get embarrassed every once in a while.

Political Polls and margin of error

At the time of my writing this, it is about a week until the election. It's Trump vs. Clinton - Duel of the Century. I thought I would look into political polls as a mathematics application this week. Specifically, let's look at what is called the margin of error.

I looked at a couple of what I think are reputable websites. However, they seemed to not quite get this concept. For example, something like this was stated by a couple of sites:

A poll states that candidate A is at 52% with a margin of error of +/- 3%. This means the candidate could actually be polling anywhere from 49% to 55%.

Unless I've been lied to in my past math classes, I believe this is wrong information. This is a common misconception, but I didn't think I would find news agencies writing this.

He is what I believe is the correct scoop. Polls usually have a confidence level. Part of the confusion is when CNN, NBC, etc mention their polls, they don't talk about this. Anyway, for most polls it is 90%. So, in actuality, a much truer fact is that there is a 90% chance that that candidate A is between 49% and 55%. She (or he - I'll stick with "she" the rest of the way so I don't have to mention both genders each time. Why "she" rather than "he"? I flipped a coin. Seriously.) is probably in that range, by she can't be certain of that.

You can never be certain of polls. Common sense tells you that you can't have absolute certainty. If there are millions of voters in the U.S., and your survey covers a few thousand, how do you know you didn't just happen to survey only ones that are against candidate A. Yes, unlikely, but it could happen. So if a poll states A is ahead of B, 57% to 42% with a margin of error of 5%, it's all over, right? No, it isn't. It's not looking good for B, but it's not all over.

We see surveys during election years a lot, but we see them often at other times without knowing it. The government's unemployment reports, bestselling books, the top TV shows for the week are all done by random sampling of a relatively small sample.

Students could figure out the margin of error. It goes like this:

Margin of error = z x squareroot(p(1-p)/n). The z-value is based on how accurate you want your poll result to be. You would have to look that up. The value of p is your polling result and n is the number in your sample. (Oddly, the number in your total group, whether it is the entire U.S., the state of Oregon, or your bowling league, has nothing to do with the answer.)

Common sense tells us that there is a trade-off. The more exact you want to be, the wider your interval is going to end up being. I might be able to state, from a recent survey of adult males, that I am 90% certain the average height of all adult males is between 5'7" and 5'11". One the other hand, if I want to be 99.99% certain, I might only be able to state that the average height is between 3' and 8'. You gain in certainty and you lose in precision.

Let's try one out.

We polled 1,000 people. Of those, 560 said they would vote for Candidate A. So, she is polling at 56%. We want to be 90% certain of the range her number would actually land in. Looking up the 90%, we find a z-value of 1.645.

1.645 x squareroot(.56(1-.56)/1,000) = .0258. If we round it to 2.5%, she is 90% sure of her actual number being between 53.5% and 58.5%.

Just for fun, here are some other possibilities.

Suppose we chose a confidence level of 95%:

95% corresponds to z = 1.96, so
1.96 x squareroot(.56(1-.56)/1,000) = 3.1%, giving a range of 52.9% to 59.1%

Suppose we take our original example and assume we surveyed twice as many people:
1.645 x squareroot(.56(1-.56)/2,000) = 1.8%, giving a range of 54.2% to 57.8%

I was right. That was fun.

Standard Deviations and Baseball

Its World Series time and I feel compelled to stick with a baseball theme this week. I've considered this application since I was not much more than a child. I wasn't sure how the math on it would work, and I'm still not certain, but I thought it would be worth exploring.

Batting averages are the ratio of hits to times at bat. So getting one hit in four times up to bat gives a batting average of .250.

It would make sense that the overall batting average in baseball might vary over the years. Things have changed since it started in 1869. There used to be no night games. That is mostly because the electric light hadn't been invented yet. Night games have made it harder for hitters. Although, they've outlawed spit balls. That has made it easier.

Does it all even out? Apparently not. There used to be quite a few batters that hit .400 or better for a season. No one has done that in the past few decades, though. I've wondered if there a way to even things out mathematical. I've seen some attempts at this.

I found a person's website that has the major league batting average for each season. Over a century worth, it is at .263. The highest year ever was 1894 when it was .309. So maybe a player that year could have their batting average dropped by .046 (.309 - .263 = .046). Similar adjustments could be made for players of each year.

Not a bad idea. I've seen other similar methods. However, I've thought that some measure of variance should come into play. I've had a theory that the standard deviation of the batting average statistics have been going decreasing over the years. So, there were more .400 hitters in the past, far above the league average, but I would guess that back then there were also more hitters far below the league average.

Why might that be? Now there are scouts going to colleges, high schools, Japan, Dominican Republic, etc. looking for possible talent. In the early days, they took what they could get. It wasn't necessarily the best baseball talent. Someone might come in from the coal mines, look pretty good, and you sign him to a contract. Over the years the process has improved.

To take a shot at that proving my theory, I used a website that showed the league average for each year. I then entered twenty years worth of yearly batting averages and found the standard deviation. Its not perfect, but I think it kind of backs me up. Here we go:

1871-1900  Standard deviation = 15.91
1901-1920  Standard deviation = 10.66
1921-1940  Standard deviation = 7.38
1941-1960  Standard deviation = 3.76
1961-1980  Standard deviation = 7.91
1981-2000  Standard deviation = 5.84
2001-2012  Standard deviation = 5.15

So to really do this right, I probably should find the standard deviations of each individual year using each player, rather than using the year as a whole. However, that seemed like a lot of work, so I settle for this. I bet there is some data base that has all the averages and the capability of adjusting the mean averages and the standard deviations for each year and adjusting each player's batting average accordingly. It won't be me, but somebody should take that on.

No hitters

Sorry, but I can't help but go back to baseball stats for the next couple weeks. It is playoff time for baseball, so I can't really help it.

Clayton Kershaw had a no hitter going for a while a couple days ago. No hitters are pretty rare. I got to thinking that you could maybe estimate the chances of a no hitter. Let's say a team would normally bat 0.250 against you. That is, they would get a hit every four times at bat. What are your chances of a no-hitter? You need to retire 27 batters (3 in each of the 9 innings). The probability you retire the first batter is .75. The probability of retiring the second batter is 0.75 x 0.75. The probability of having a no hitter in just the first inning is 0.75 x 0.75 x 0.75 or 42.2%.

For the whole game, the probability of a no-hitter would be (0.75)^27 = 0.000423. Unlikely.

You can give up walks or have batters reach on errors and still have it count as a no-hitter. I don't think we need to take that into account, though, as they do not count as official at-bats anyway. I had to think about that a bit, but I'm pretty sure I'm right on that.

Then I thought about estimating how many there should be in a season or any given period of time. I went back to 1998 because that is the last year major league baseball added teams. Since then, to the present day, there have been 30 major league baseball teams. With 162 games for each team, from 1998 to 2016 (19 years) there have been 162 x 30 x 19 = 92,340 save opportunities. If we use the above probability, the number of no-hitters during that time would be: 92,340 x 0.000423 = 39.1 saves.

How many have there actually been? 49. Keep that number in mind. We'll compare other outcomes to that.

So, not bad. In fact, a lot closer than I thought it would be.

The big question mark in all this, I think, is the batting average. The overall major league average is a little higher than this, maybe 0.260. Doing the math again would give an estimate of 27.2 (lower than the aforementioned 39.1).

But maybe we shouldn't be talking about the league average. You would figure the type to get a no-hitter is a better than average pitcher. And in fact, looking at the list of those that have thrown no-hitters shows some of the best pitchers of the past 20 years - Jake Arrieta, Max Scherzer, Cole Hamels, Clayton Kershaw, and Justin Verlander. (And there have been some pitchers that had some talent, but also a good amount of luck on their side that day of their no hitter.)

So maybe the correct batting average would be 0.240 -- 55.9 no-hitters.

Or maybe a batting average would be 0.230 -- 79.6 no--hitters.

Anyway, for those somewhat interested in the topic of baseball this was an interesting math application on baseball and probabilities.

Running Pace

I'm running in a race in about a month. Its a half marathon, which is 13.1 miles. I think I can make it, but I'm not absolutely certain of that. Being in shape for a race is not really enough of a challenge for me. The only reason I'm doing it is for the scenery. It goes across the Golden Gate Bridge. Twice, in fact. So that will be an adventure in itself. I have run two other races in the past that are highlights in terms of the races themselves. One was in Knoxville, Tennessee that finished on the 50 yard line of the University of Tennessee stadium. The other was a half marathon in Indianapolis whose course included one lap (2.5 miles) on the Indy 500 track.

Anyway, on to math. Since I'm not 100% sure I can even finish, I'm definitely not sure what pace I should try to run. There was a predictor in my latest copy of Runner's World Magazine. They gave a way to predict your pace in various races by looking at times for shorter distances. I thought - good application.

The had predictors for the 5K, 10K, half marathon, and marathon. Since it applies to my situation, I'll just use the one for the half marathon.

• The Workout - "Race" a 10K at 80 percent effort. 
• The Formula - Take your 10K time in minutes (for example, a 55:30 is 55.5) and add 0.93. Multiply the result by 2.11.
• When - Three to five weeks before race day.
• Why - A 10K is great because it has that endurance aspect of a half marathon but doesn't require you to run too much so close to race day,

Yes, they could have condensed things quite a bit by using an equation rather than an explanation.

So the "formula" is f(x) =  2.11(x+0.93), 

To take their example of 55:30, you would have a half marathon time of f(55.5) = 2.11(55.5+0.93) = 119.067 minutes or 1 hour 59 minutes 4 seconds.

To put a little more algebra into this, we could say that we are hoping to run the half marathon in one hour 50 minutes. What kind of 10K would predict that kind of time?

Answer:  110 = 2.11(x+0.93), so  x = 51.203 or 51 minutes 12 seconds.

The other three races; 5K, 10K, and marathon; have different, but similar formulas, and would be great for Algebra I classes.

Morse Code

I saw something about Morse code and thought it might be an interesting topic as a mathematics application.

First, some background.

Samuel Morse was born in 1791. He attended Yale, graduating in 1810. He aspired to be a painter. I didn't realize he did of this other career until I read about his paintings in David McCollough's book, The Greater Journey: Americans in Paris. Here is his portrait of President James Monroe.

He lost his wife and both parents in a three year span. As an escape, he went to Europe. During this time he made some contacts that led to led to the invention of Morse Code.

It didn't catch on for a few years. A U.S. congressman showed interest and a test was done with a wire stretching from Washington D.C. to Baltimore. He successfully asked, "What hath God wrought" and the rest is history.

It relies on a series of dots and dashes. They can be communicated with electronic impulses or light impulses. It was very important, but began to fall out of favor with the invention of Bell's telephone in which actual words could be used instead of a code for spelling out words. It is still used in various areas, including signal lamps by the coast guard. Those without speech can use the tapping of Morse code to communicate. But for the most part, it is found in history books.

SOS, for example, is ...---... How many are combinations of dots and dashes are needed to cover the alphabet? This could be found use the fundamental counting principal (If there are "m" ways to do one thing, and "n" ways to do another, there are "m x n" ways to do both.)

• Using one symbol means a dot or a dash could be used - two choices.
• Two symbols means there are 2 x 2 = 4 ways.
• Three symbols means there are 2 x 2 x 2 = 8 ways.
• Four symbols means there are 2 x 2 x 2 x 2 = 16 ways.

Since there are 26 letters in our alphabet, this still isn't enough. We could use five symbols, but that makes it more cumbersome. I can be done, though, by using one, two, three, or four symbols. Since 2+4+8+16 = 30. That is plenty to cover the whole alphabet.

If we need more that just words - digits, or symbols like ? and ;, we are going to need more. So for them, we need to use 5 symbols. How many possibilities would that give us?

Two to the fifth power is 32, and that means we have 2+4+8+16+32 = 62 possibilities. That gives us enough for 26 letters, 10 digits, and 26 more symbols beside.

Where Are We Going to Use This?

I was browsing through and found one of those site where someone asks a question and others respond with their thoughts. This one asked about where advanced math gets used in real life. I thought, "Hey, that is right up my alley", so I checked it out.

I thought I would include it here in my blog. The responses are quite interesting. I just pasted them in as is, so there might be some grammatical or logical errors. That is just what makes it interesting. The original website was at https://www.physicsforums.com/threads/what-is-advanced-level-mathematics-used-for.500513/

So, here is what people had to say:

I barely understand the bare basics of algebra, my math skills are abysmal. But what applications do advanced mathematics such as stochastic calculus and linear algebra have? Other than in physics, science, and engineering, what other things can advanced-level mathematics be used for? How about in daily life?

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It can enhance your abstract thinking ability. Not directly useful for anything in daily life.

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I fully disagree, how do you think barcodes were invented? A few uses of Linear Algebra: Codabar system Digital image compression Calculating life expectancy Modelling population growth Profit maximization Universal Product Code Lots more. If you want a thorough discussion of exactly how they are used, then just open up some linear algebra books, or do a Google search. Higher math isn't just solving puzzles (In fact, that's not really what math is). People don't just do mathematics to improve their thinking abilities; it certainly helps, but it has many applications. Keep in mind that mathematics need not be applied to anything. Just because you can't use a result of mathematics (at first) for anything practical does not make it useless.

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There is great deal of uses in physics (also era involves some Chemistry) and engineering. In general relativity a lot of linear algebra and calculus is required. In engineering, mostly calculating some basic mechanics problem. Other than these, Economics uses great deal of calculus to model the market which is very important. You can search more on financial mathematics (not accounting~~boring).

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Also, I've heard that matrices can be used to balance out chemical reactions in chemistry. I haven't actually done that myself, but I was happy to know that there was an easier way than what I did in my first year chemistry class!

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Oh sure it's useful in all sorts of professions. But not daily life, and many well paid jobs don't need it either. But the trouble is, when you're 16 you don't quite know what you'll be doing in 10 years time, and by then it's too late to learn so easily. 

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Everything has its applications. Of course, applications of some fields are more obvious than others.

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Mathematics models the natural world. So your question is nearly identical to ...... What use is it to learn English ? Furthermore, Logic, one of the foundations of Mathematics, is the link between The Arts & The Sciences. All art forms (nearly) seek to communicate. How better to make your case than with clear precise easy to understand logic ? Be it painting, screenplay, poem, courtroom summation or a novel. Some of the very the best lawyers were good at Math. That is one of the reasons they excel at the Law. Mathematics underlies nearly everything you see around you. But it will not guarantee a good life. That is the province of religion and moral philosophy.

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I think balancing equations is simply simultaneous equations, I mean for rather complicated equations. Matrices are just simple forms of simultaneous equations, they just save your paper and ink. Of course there are many ways of balancing equations, but many of them might not work for all cases. Also, matrices are very useful in doing statistics, though I haven't learn much of statistics, I heard of something called covariant matrix that is used for complicated systems. And statistics can be applied to many areas. May be you can look more on that.

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How does that even matter? Just because some random CEO doesn't use his knowledge of basket-weaving doesn't make basket-weaving useless. That's a really poor argument. Why does everything have to be immediately useful in daily life, and by daily life, I assume you mean eating, breathing, sleeping, and no more. It seems to me that you think that if you don't use something every day, or can't use it to make lots of money, then it's useless. As said many times in this thread, mathematics is all about logical and abstract thinking; it's basically a form of creativity. Now tell me, how useless are the former?

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When people ask me what the use is for mathematics, I always respond with the following poem by Morris Bishop: There's a tiresome young man in Bay Shore. When his fiancee cried, "I adore the beautiful sea". He replied, "I agree, it's pretty, but what is it for?"

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He never said higher math was useless in the general context, simply useless in daily life. There's nothing wrong with the validity of his statement since we take the meaning of "useless" in every day conversation as "generally useless" rather than "completely useless". However, there are few professions that are useful in daily life, such as cooking, etc. so the statement, although basically true, is misleading.

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Cryptography is a pretty big one.

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I prefer this: A math professor, a native Texan, was asked by one of his students: "What is mathematics good for?" He replied: "This question makes me sick! If you show someone the Grand Canyon for the first time, and he asks you `What's it good for?' What would you do? Well, you kick that guy off the cliff!"

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Yes, I was one of those people who said that. But I think the OP was concerned about direct applications. Like when you would want to sit down with a pen and paper to write an integral or perform a matrix operation. For most people the answer would be never in their life. Us mathy types would think about it all the time when we hear news stories or write on internet forums, but that's not normal people. If you aren't inclined to analyze things for fun, then knowing how to integrate won't make you do it.

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*Science *Physics *Engineering *Computer programming *Genetics and other fields of biology *Chemistry *Business accounting/finance and economics But people other than physicists, scientists, and engineers wouldn't have any real use for any advanced maths. 

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I believe that advanced-level mathematics (especially pure maths) is for fun and for appreciating the beauty. I am currently at high school and love learning math (college-level math), but find the maths at high school very dull and boring. The current education system just turns maths into a very systematic work. For example, when we learn Pythagoras Theorem, after teaching the theorem itself, we are told how to (1) Find the length of the hypotenuse if the sides are given (2) Find the length of one side when the hypotenuse and one of the sides are given while the teacher can just let us find the way of doing it ourselves with the original theorem. Even with the exercises provided, the questions are divided into parts about the first type of problem and second type of problem. It is just plain stupid (sorry for being a bit too rude).

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Finance, cryptography...

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There is a method to this madness. True you can derive these different formula's from the original theorems each and every time you need to use them. But, you will need to apply these things from time to time as you move into higher level work and it's helps if you have a lot of these very basic things like trig formulas memorized (at least somewhat memorized) such that you don]t have to go back and derive them each and every time you need them. Of course there is nothing wrong with learning how to use the theorems and understand their meanings to derive the formulas or, given a formula prove it's validity based on the theorem(s). When i was in high school (and freshman college) many of the more fundamental courses omitted the proofs or simply glossed over them. But I never felt comfortable, I always preferred working through the proofs and, thinking of other approaches I could take to them.

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You must distinguish between these two questions: "What do people often use advanced mathematics for" and "What can advanced mathematics be used for". People who have training in advanced mathematics, and a certain knack for applying it, can apply it to almost any subject, even art and literature. Statistically, the people who know advanced mathematics tend to be engineers, physicists etc. so that's where you most often see advanced mathematics applied. If an artist or literary historian happened to be an expert in differential equations, they might well be able to apply it to their field of study. However, they might not find many of their peers able to understand or appreciate their work.

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Because of it's sometimes mind-boggling complexity and difficulty, it is difficult to wrap your mind around that level of math without having to have an IQ of 140+ Math is despised by most high school and college students. Only the people with high IQ's (a tiny percentage of the population) tend to excel in, and take further interest in it. 

Baseball Distances

I stumbled onto an interesting website. It had baseball statistics and had some stats I didn't know they even kept track of.

It gives numbers on average flights of batted balls for each player. It's interesting to look at as a math application. I tried it out using formulas and didn't get the quite the same answer. However, the trajectory equations don't account for the air resistance encountered. And, of course, I might have just done the math wrong. More on this after I let you know how far off I was.

The categories were "Average Launch Speed", "Average Distance", "Average Velocity", "Average Launch Angle", and "Average Height". For example Evan Longoria (no relation to the actress) had:

• Average Launch Speed: 92.17 miles per hour
• Average Distance: 248.2 feet
• Average Launch Angle: 14.39 degrees
• Average Height: 46.06 feet

I assume Launch Speed could be found with a radar gun. Launch Angle perhaps by camera, although it seems like that would depend on where the camera is in relation to the camera. Ideally, the camera would be pointed perpendicularly to the ball's trajectory, I would think, but that wouldn't always be the case.

So, I wondered if I could compute what they had for Average Distance. I'm guessing that is how far the ball before it hits the ground. But what if Evan hits a line drive and is caught? It went a certain distance, but would have gone farther without the fielder there? Anyway, here we go.

First I figured I need to get its average speed into feet per second to match with the other categories.

92.17 miles per hour = 286,657.6 feet per hour = 135.183 feet per second

I then used the formula:  y(t) = h + (vsinA)t -16t^2.

I'll assume an height of the ball when making contact with the bat to be 5.5 feet. I want to see how long it takes to hit the ground (y(t) = 0).

0 = 5.5 + 135.183sin(14.39)t - 16t^2

Using the quadratic formula, this game me two answers, the positive one being 2.25 seconds. 

Then I used this to find how far it went with x(t) = v(cosA)t = 135.183(cos(14.39))2.25 = 294.62 feet

According to that website (http://m.mlb.com/player/446334/evan-longoria) the distance is only 248.2 feet. 

I was ready to call this a big old fail. But, perhaps not. Like I mentioned before, I'm not sure how they figure balls that are caught before they land or balls that bounce off the outfield fence. And are those distances found by observation of where the ball seems to land? Air resistance slows down the ball quite a bit. They say that the Colorado Rockies in mile-high Denver is the easiest place to hit home runs because of its thin atmosphere. The math equations assume a vacuum, so the formula would give a greater distance. 

So, maybe my math is all right. Regardless, it's a nice math application.

 

Trapezoids

I always had a bit of a tough time finding examples of trapezoid applications out there in the real world. There are a few, but certainly not as easy as finding shapes such as circles, rectangles, triangles, squares, ... I thought I would go looking and here are a few I found.

The trapezoidal rule is actually from calculus. A little above the geometry level, the basic idea of it would be quite understandable to a geometry student.

Another interesting one is the Mars Rover which contain the Rover's solar panels. I believe they are in the shape of a trapezoid because the panels are initially folded up against the Rover. The trapezoid shape is best for that unfolding transition.

I don't have specific info on most of the others. They're just trapezoids out in the real world.

Snowflakes

"No two snowflakes are alike." You've undoubtedly heard that a time or two. Seemingly, not an important math application. I got to thinking about it and that statement does bring up some important points. 

First it brings up some lessons in basic logic. If we are trying to prove there are not two snowflakes alike, how would be prove or disprove a statement like this? Disproving it could be easy. Find two that are the same. If we could do this, we could put this issue to rest. 

How about proving it to be true. We've all seen lots of pictures of snowflakes. None alike so far. The pioneer in this seems to be a Wilson Bentley from Vermont. He was born in 1865, a time when there weren't a lot of pictures being taken of anything. He had a collection of over 5,000 photographs of snowflakes. He was single (not a surprise) and had plenty of time to devote to his work. None of his matched. The fact that none of them match would not constitute a proof. This would be a good example of inductive reasoning. Here is an opportunity to discuss inductive verses deductive reasoning, and the benefits and drawbacks of each. 

Can deductive reasoning be used here? Let me state that I'm well out of my area here. My lowest grade in high school was a C and that was in Chemistry. I was fine with that since I probably deserved lower. I really did try. Chemistry and I just do not click. Regardless, here we go.

I did some reading to try to figure this out. A water droplet might freeze onto a dust particle. They freeze in a hexagon shape. Most of what I saw kind of glossed over why that is. One statement explaining the snowflake pattern went like this. And I quote:

Hexagonal ice ([1969], ice Ih ⁱ see Phase Diagram), is in Space group P6₃/mmc, 194; symmetry D_6h, Laue class symmetry 6/mmm; analogous to β-tridymite silica or lonsdaleite, having a a six fold screw axis (rotation around an axis in addition to a translation along the axis).

Curse you Chemistry. Anyway a hexagon is formed. Other water molecules latch onto the vertices of the hexagon, growing the snowflake as it falls through the air. Different shapes come about based on the temperature and humidity of the surrounding air. These flakes all take different paths to the ground, thus slightly changing its weather conditions, thus slightly changing the shapes as they grow. These different paths cause different shapes. At any one moment in time, the forming shape has the same weather conditions, giving the snowflake its symmetry. 

One article states that there are 10,000,000,000,000,000,000 molecules of water in a snow flake and they can be rearranged in many different ways. A couple of articles likened this to factorials. The ways to arranged 6 books is 6! (= 720), 7 books is 7! (5,040), and 8 books is 8! is (40,320). I'm not sure finding the number of snowflake designs is as simple taking the factorial of the number of molecules, but I guess their point is that the number of patterns must be huge.

However, I'm still not convinced. Someone estimated there have been approximately 1,000,000,000,000,000,000,000,000,000,000,000,000 snowflakes. Really? No two alike in that bunch. We haven't looked at them all. Even if we did, even with global warming, I'm sure there will be a bunch more. 

In fact, some of the scientists think there might have been duplicates. If the snowflake doesn't have far to fall, and thus doesn't have the chance do grow very much, that greatly increases the chance that two of them could be similar. 

Actually, one scientist claimed she has found a pair. In 1998, Nancy Knight claims she found two alike. I saw a picture and they look pretty convincing. Nerdy scientists, though. have balked at this. Some of the hydrogen atoms (approximately 1 in 3,000) could be deuterium. (Hydrogen usually has just one proton in the nucleus. Deuterium has a proton and a neutron.) This would likely make snowflakes that looked the same, still not be identical.

Some people just can't admit defeat. 

I hope this was helpful. I did my best to be as accurate and thorough as I could. (I'm sorry Mr. Gustafson. I really did try in Chemistry.) 

Fractal Video

I saw an interesting video on fractals. It was produced by NOVA called Fractals: Hunting the Hidden Dimension. Being for the general public, it, of course, didn't get too hard core with the mathematics, but it didn't completely back away from it each.

Because of that, a few of the applications left one with some questions. Such a case was when someone in the video said, "Fractals are important in code breaking." Then they leave it there because to try to explain it would cause most people's heads to explode. In a lot of cases, to have to skip over the math is somewhat unsatisfying, but probably pretty much unavoidable.

Interesting to me was how Benoit Mandlebrot first got involved with applications of fractals. As computers were just starting to communicate, there were problems. Computer data was being sent over telephone lines. However, it often wasn't getting through as intended.

Benoit B. Mandelbrot, then an employee of IBM, noticed a certain pattern of interference over, say, a ten minute span. He then noticed that same pattern would appear if he looked at maybe a five minute span, then a two and a half minute span, etc.

The video called it, "self-similarity". The fact that this self-similarity was taking place, told him this situation could be modeled with fractals.

It is a good video. It is from 2011, so not too out of date. Students no doubt will be chagrined at how excited the math nerds in the video get over these fractals. Even with that - a good video.

Statis Pro Baseball One Last Time

I know my blog has been a little heavy with the baseball applications. Specifically with regards to the best game ever made - Statis Pro Baseball. One more week, then I'll move on. This and other older games are great, though, for math applications because its right there in front of you. All the computer games have the statistics / mathematics hidden away in the computer program running it.

This is application is actually from a different game that I played once with a friend of mine. At the time I thought it was kind of ingenious, although I'm not sure I put a lot of thought into how they did it. Each baseball player had a card with spinner which represented statistically what you could expect from him in an at-bat.

If the first batter up was Ty Cobb, I would take his card, spin the spinner and see what he did. It might land on a colored section of card marked "Out". How did they come up with the colors on the cards anyway? Let's make Ty's situation real simple and divide it into sectors for hits and outs. For his career he 4,189 hits in 11,434 times at bat. That makes a batting average of 0.366. This means of course that he gets a hit 36.6 percent of the time which would be a sector of 36.6% of 360 degrees. This is a sector of 131.76 degrees. His chance of going out would be a different colored sector of 360 - 131.76 = 228.24 degrees.

There were more divisions than just hits and outs. Although it has been a while, I'm sure there were singles, doubles, triples, home runs, outs, and walks at least. To build the circle would mean finding percentages, changing them to degrees of a circle, dividing up the circle, and coloring and labeling the sectors. (The picture is not what the spinner looked like, obviously, but that's the idea.)

Whoever came up with it, I thought it was a pretty good game. It also didn't last, but that's the way it goes, I guess.

So, this isn't a high level math application, obviously, but I think an interesting one, and it reviews, protractor use, percentages, and circles.

Computer Baseball

Last week I wrote about Statis Pro Baseball - a game I played growing up. Drawing cards numbered from 11 to 88 determined how a player did in a particular at-bat. A computerized version of this can be used making use of the same randomness as drawing from the deck of shuffled cards.

Let's take one random baseball player. How about Babe Ruth? I was recently in his boyhood home / museum in Baltimore. It is really cool. Anyway, let's take his 1927 season.

 That year, Ruth in 540 at-bats had 95 singles, 29 doubles, 8 triples, and 60 home runs. The percent of each of these types of hits out of 540 at-bats is as follows:

Singles:         17.6%
Doubles:         5.4%
Triples:            1.5%
Home Runs:   11.1%

Now, we could use a random number generator, available on many calculators or on-line and simulate any number of at-bats. Adding together the above percentages and change them to numbers from zero to a thousand, we get the following:

Singles from 0 to 176
Doubles from 177 to 230
Triples from 231 to 245
Home Runs 246 to 356
Everything 357 and above will be an out

Running the random generator for twenty numbers and giving the results:

791 - Out
365 - Out
258 - Home Run
320 - Home Run
494 - Out
75   - Single
929 - Out
842 - Out
461 - Out
536 - Out
210 - Double
388 - Out
836 - Out
914 - Out
214 - Double
812 - Out
509 - Out
978 - Out
955 - Out

20 At bats, 1 Single, 2 Doubles, 0 Triples, 2 Home Runs

Actually, a little sub-par for the Babe. I'll spare you all the numbers, but I did it two more times just for fun, and here are the results.

20 At bats, 0 Singles, 3 Doubles, 0 Triples, 4 Home Runs

20 At bats, 3 Singles, 3 Doubles, 0 Triples, 2 Home Runs

I think this is a pretty good Algebra I application covering ratios and to get them thinking about the law of large numbers

Statis Pro Baseball

As a kid/adult I had what was called a Statis Pro Baseball game. It came in a box and you could recreate games with actual player statistics. It has since gone out of business, but some addicts still make the player cards and put them on-line. I'm reminded of this game because I recently saw sets for the 55 World Series (Yankees vs. Mets) and the 1919 World Series (Reds vs. White Sox (a.k.a. the Black Sox)) You just take the players' statistics and convert them to numbers the game uses. It used the numbers 11 to 88. I'm not certain why those numbers specifically, but that is what they did.

This is a fun (for fans of this kind of thing) math application involving ratios. So for example, say you have a batter, Johnny Baseball, that was up to bat 428 times and he had 101 singles. Of the 78 numbers used in the game (11 to 88), how many would be used to represent the singles?

The proportion 101/428 = x/78 gives a value of x = 18.4. So we round that off to 18 and for the Statis Pro game that is represented by the first 18 numbers - 11 to 28.

Suppose Johnny had 13 doubles. So, use the proportion 18/428 = x/78. This gives x = 3.3. So another three units is represented by the next three numbers, which would be 29, 30, and 31. Continuing in that manner, you could keep going and compute the numbers for triples, home runs, strike outs, walks, being hit by a pitch, or making an out. All that info would be on his player card.

You could divide things differently, of course. The "hit by pitch" category could be combined with walks. The game actually divides singles by singles to left, center, or right field. Depending on where the ball is hit helps to determine how far base runners can advance. (If you don't get that, don't worry about it.)

So how were those numbers used anyway? Besides the player cards there was another groups of cards marked randomly 11 to 88. Suppose Johnny Baseball is up to bat and his number drawn is a 17. Since 17 is between 11 and 28, he hit a single. If he was up and drew a 31, he hits a double.

It gets a little more complicated than than, but not too bad. You have to figure in not just the batter's card, but the pitcher his is facing has a card that describe how well he pitches. That has to be taken into account as well. Despite all this complexity, a 9 inning game can be played in an hour. So individual at-bats take well under a minute. That is much better than the eternity they take in real games.

So, I've probably done enough reminiscing right now, but I think the idea behind this is pretty valuable. Computer games today depend on this kind of randomness. We'll check that out next time.

Space Equation

I try to focus on high school math applications in this blog. Therefore, this picture probably doesn't quite fit. However, it is such a cool picture that I wanted to put it in. I saw it, or at least a portion of it in the September, 2016 issue of Reader's Digest. I went looking for the picture and found an expanded view of it on-line at http://rarehistoricalphotos.com/nasa-scientists-board-calculations-1961/. The Reader's Digest piece said that it was taken on October 10, 1957. These are equations related to satellite orbits. The picture was taken six days after the launch of Sputnik, putting the USSR up 1-0 in the space race. That seemed to get things going in the United States. NASA was created the next month and two months later, the U.S. had launched its own satellite.

Initially, I thought the photographer did some kind of time-lapse photography and these were all the same guy. Although, one person did the writing - astronomer Samuel Herrick - these are all different scientists. Its been my observation that everyone from the 1950s looked more or less the same. I think that is the reason for my confusion.

At the above website, I got some more information about the photo. The point was made that there are no calculations here - just equations that they might use. That makes sense being at the start of the space race and smack dab in the middle of the cold war. So no top level information was being given away in this photo.

Usually I think its a poor idea to present applications that are over the heads of students, but I think an exception could be made here. There is virtually no calculus here and concepts in trigonometry, "e", etc would be recognizable to many high school students.

The article ended with this:  For a complex equation that deals with time-steps and feeds back on itself, the prominent scientists of NASA would have “math parties”!!! [exclamation points, mine]. Everyone would master one part of the equation. Then the first guy would do his part and hand it off to the next guy and so on. Eventually the final guy would go back to the first person and give him the new inputs for 1ms [microsecond?] further in time. After a few hours you could have a nice neat graph of everything over a 1-2 second period. That is how the first nuclear reactors, nuclear bombs and a lot of aerospace calculations were done.