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.)