Point of theroretical math?

[rpayne88]

I don't get it. I have to take algebra, trig, and calculus courses for my major (aerospace engineering.) But if these course are anything like what I took in high school, I will be dealing with stuff like "find all real zeros for the equation f(x)=-5X^3+3X^2+59X+12." I've looked in the engineering review book my father (a civil engineer) purchased before he took his general engineering test in college. It is full of formulas. From what I have experienced from my chemistry class in college and honors physics I took in high school, all you do is plug numbers into equations. I'm getting a B in chemistry and I got an A in physics in high school. Yet, in college algebra I'm getting a D-. I need to take algebra and trig before calculus. I know trig has to do with the ratios of a right triangle. But it too is marred with theoretical math (I took it in high school and barely passed.) And I've been told that the gist of calculus is determining how things work as a variable gets exponentially smaller. If all I'm ever going to be doing is plugging numbers into equations, why do I need anything past solving two step equations? I mean, if I can't find a variable through a two step equation, why won't I be able to determine it experimentally. And why do I need to worry about stuff like (3X^2+5)/(4-x). I will never run across a variable without a REAL number in the REAL world. To sum up my question, why do I need to worry about theoretical math I will never use after these classes and not the "practical" math I use on an almost daily basis?

[lajoswinkler]

I will never run across a variable without a REAL number in the REAL world.

Not true.

And with this I rest my case.

[Kerbart]

The easiest answer is that for starters, REAL numbers don't get you very far in a lot of engineering problems. You'll have to resort to complex numbers instead for many problems.

The second answer is that with understanding certain categories of problems will help you to get insight in how certain numbers develop. "Making beam X bigger will not solve the issue as the weight factor increases faster than the extra support it gives," those kind of things. Yes, when the time comes to actually calculate your dimensions it becomes a game of plugging in the numbers. But before you get there you have decisions to make and options to weigh. For those you'll need insight in the processes and that insight comes from, sadly, usually higher math.

Here's a practical example:

• Take a wooden stick of 1m length
  
• Cut it in two random places
  
• What is the probability that you can put the ends together and form a triangle (three substicks of 20, 30 and 50cm can make a triangle. Three of 1, 1 and 99 cannot).
  

A simple "engineering" problem. Yet when you google it you'll see ugly solutions with spreadsheets full of probability analysis. In reality, once you switch from one discipline of math to another and back, you will get a far more accurate answer, without even needing a calculator, in about 5 minutes of time. That's what "theoretical math" can do for you: if gives you the ability to understand a problem and solve it. That will get you solutions that are usually better (and you'll get there faster) than simply plugging in the numbers.

[PakledHostage]

The expression that comes to mind is "We're not in Kansas anymore, Toto"... Your dad's engineering textbook may be full of formulas, but there's more to engineering than plugging numbers into equations. If that's all it was, then we could just let computers do it. Sadly, you need to know where those equations came from, what their physical basis is, what their limitations are (i.e. what assumptions were made deriving them), etc.

Engineering is applied physics, but you also need to be strong in math to get through your engineering degree. It sounds like you may be weak in this area. The best advice I can offer you is to do lots of practice problems and maybe even get a tutor. Don't just work through your exercises mechanically. Really understand what you are doing. Your first year courses are the foundation for all of the harder math and physics that comes later.

[nhnifong]

When I don't understand the point of a mathematical topic, I tend to forget it. So I like to go back and try to understand why the idea was invented and what the context was at the time. Why was it puzzling to the first people who tried to solve it and what was their mathematical background? Doing this has helped to cement many interesting mathematical relationships in my mind. Math is more like history that mathematicians would like to admit. Especially their friggin symbols.

[FanaticalFighter]

You'll be surprised to know that Quantum Mechanics and Fluid Mechanics both make HEAVY uses of complex numbers.

[Depth]

To expand the topic a bit, what is the point of researching n-dimensional mathematical objects, such as the hypercube?

[Nuke]

things used in game programming, like quaternions, would not exist without imaginary numbers.

To expand the topic a bit, what is the point of researching n-dimensional mathematical objects, such as the hypercube?

this also relates to quaternions, for example doing slerp. which is essentially walking across the surface of a hypersphere. i hear you people like games.

Edited November 19, 2013 by Nuke

[falofonos]

[Mr Shifty]

"College" algebra ain't theoretical math. You use algebra when you calculate whether a 16" pizza for $15 is a better deal than a 14" pizza for $12. Theoretical math is more like philosophy, sort of a meta-mathematics where you abstract out as much as possible and discuss what numbers are and what their relationships are in the most general terms. But even some of the theoretical fields have practical applications, e.g. in encryption and other information theory fields.

I'm a working engineer. I can honestly say that I use algebra -- as in the actual manipulation of single and multi-variable equations -- every day that I do design work. Trigonometry is also vital, not because of triangles, but because trig equations describe circles and angles and things that repeat periodically. Calculus I use a couple times a year, but it's also vital to have an intuition about how calculus describes physical systems. I use complex numbers, Fourier analysis and control system analysis regularly. I work at a pure science facility (we exist to do physical sciences research), and the scientists working there use calculus and other complex maths all the time.

You can't always determine quantities in advance experimentally. When I'm designing a circuit board for a system that deals with sensitive analog signals, I have to account for all the non-ideal characteristics of the components I'm using. I do this by developing a set of system level equations that describe the behavior of the circuit, including the non-ideal components and compare the result to what I want. I can't do this by experimentation because there are too many variables and my designs have to be solid before they're manufactured.

[konokono]

My advice to you:

If you can't see how the math you're learning is used in practical applications, ask your professor to give some examples. He/she should be happy to oblige. After that, you hopefully won't view it as "plugging in numbers" and instead see the equations for what they mean and what the solutions represent, in the "REAL" world.

[K^2]

If all I'm ever going to be doing is plugging numbers into equations, why do I need anything past solving two step equations?

Because any monkey can plug numbers into equations. In fact, they have computer programs for that now. Any job that requires you to do just that pays about as well as working registers at McDonalds.

Engineer's job isn't to compute some relevant quantity for a specific system. Like I said, there is computer software that will do it for you more often than not, and nobody is going to pay you good money just to type in numbers. An engineer's job is to find an arrangement of parts that gets the desired values within tolerances. That means solving optimization problems more often than not, and that requires understanding of differential equations and optimization methods. If you actually want to be good at it, the later requires you to understand mathematical analysis, which goes way, way beyond calculus.

If you are getting a D- in high school algebra, cut your losses and consider a major in a soft science or liberal arts. You aren't going to be an engineer.

You'll be surprised to know that Quantum Mechanics and Fluid Mechanics both make HEAVY uses of complex numbers.

And statistical mechanics, and electrodynamics, and electrical engineering, and normal modes in classical mechanics, and... Should I continue?

Edited November 19, 2013 by K^2

[Jim DiGriz]

Sounds like you might have hit the issue I hit somewhere 2-3 years into college where all the Math classes were too detached from physical examples for me, and were just pushing symbols around on paper and I had a hard time learning from the Math profs (there was something almost 'linguistic' about their approach to Math and the pure symbol manipulation), while the Physics profs were just unbelievably sloppy at teaching Math and either assumed too much or explained too little and took all kinds of short cuts.

I managed to find some good Applied Math profs and some Physicists who taught some really good Mathematical Physics courses and connected up the Math to the underlying Reality of it.

I'm not sure what to do if you hit this wall at the stage of doing advanced algebra though. At that point it was pretty obvious to me what the applications were. Trig and Geometry are obviously critical. You can't really understand Newtonian Gravity like in KSP without physics and derivatives and integrals and infinitesimal distances. Parts of Algebra are a bit more ponderous, but you need it to get to Calculus.

[Nuke]

schools really suck at teaching math. i went through high school and finished a 2 year degree without any inclination that i would be able to understand advanced concepts in mathematics. then i ended up learning them on my own in a few months of doing projects. the, 'solve 50 completely arbitrary and practically identical math problems every day' way of teaching math needs to die. the endless repetition gives kids this notion that math is boring, the lack of practical exercises gives them the notion that they will never use this, and then when they move into something they dont understand, they get the notion that its way beyond them. and i found out entirely on my own that none of that is even remotely true. math is awesome.

Edited November 19, 2013 by Nuke

[falofonos]

Teach people mathematics properly and some of them may understand compound interest.

Large numbers of people understanding compound interest, especially those who end up in low skill (relatively) jobs, and you get an armed uprising with bankers and politicians swinging from lamp posts.

[K^2]

the, 'solve 50 completely arbitrary and practically identical math problems every day' way of teaching math needs to die.

It does. It's from the wrong era. It's not completely pointless even now, mind, but it should be balanced with things that actually explain what and why.

I mean, take the same complex numbers as an example. Ok, we've seen plenty of examples of where they are used in this thread. But why? Why do they pop up all over the place when every measurable quantity is a real number? You learn the full answer is you study abstract algebra, typically around your 3rd or 4th year of undergraduate degree in pure mathematics. Field of complex numbers is an extension of the algebraic field over rational numbers. In simple terms, and dropping some nuances, complex numbers are important because some of them are roots of polynomials with real coefficients.

Well, that's simple enough, polynomial f(x) = x² + 1 obviously has no real roots, but it has two complex roots, which is nice, but why should anyone care? Honestly, if it was just about the roots of polynomials, nobody would care. This is where group theory and analysis come in. In particular, Lie groups make a grand entrance. Long story short, because this would take up half a text book, any time you have infinitesimal transformations of any kind, the total transformation can be written as e^ix for some x. This is the reason why you see complex exponents show up in so many solutions of differential equations, and consequently in every branch of physics from mechanics to relativistic quantum field theory.

Naturally, it's not necessary to learn all of the details of why it works this way, unless you are getting really deep into theory, but it helps when there is some natural progression to things. When you are presented with some examples of where things are used as you go along, and maybe learn some fundamentals of higher maths along with it, rather than just treat it as something too complex to grasp.

[Smidge204]

Well, one way to look at it is such: Without a theoretical groundwork, you won't even be able to conceive of the experiments - let alone design them - to "determine it experimentally." You think the guys at CERN just decide to blow nearly US$10B on the LHC just to see what would happen, without any consideration for what the machine would need to be capable of? They did the math.

If you honestly feel that all you're doing is "plugging equations" then I fear you are failing yourself in the whole point of higher education. You're not in high school anymore. While you might still get that diploma if all you do is pass your tests, if you don't develop a deeper understanding of the subject then you've pretty much wasted your time and money. Without that understanding you won't even know where to start when addressing "real life" problems, because real problems are not clearly spelled out in textbooks.

Now, I say this from someone who has had more or less the same experience... as a mechanical engineer (who ended up taking some aerospace classes due to small class sizes ) math classes were far too abstract for me and I had to take a lot of summer classes to catch up. It's rough. (Quite literally the only think I remember from my class on multivariable differential math is the word "Eigenvalue" and I still have no idea what that even is. "D" is for "Done" I suppose!)

Hell, I still write "SOHCAHTOA" on the corner of my notepad whenever trig comes up...

As for "real numbers in the real world" I assume you aren't referring to complex numbers vs. real numbers. So there are lots of examples of where this kind of math comes in handy, and being able to manipulate the formulas is more vital than simply pluggin'-n-chuggin' into a calculator. The classic spring-mass-damper problem comes to mind, as well as anything to do with frequency analysis (either analog signal processing or mechanical resonance, your pick!).

If plugging numbers into a calculator is all you can do, I can guarantee that someone who DID learn the math will be building a machine to do your job very shortly.

=Smidge=

[Vat]

Complex numbers are an important component in AC electrical theory.

[magnemoe]

Now if you want to have real fun you can begin with software development and make your own equations.

document.getElementById(maintrenodeid+s1+'e').style.pixelLeft=x1+x2-10+x3;

Business software however is pretty straight forward, game development is harder, imagine the math behind KSP

Even an skyrim mod I made about farming used plenty of math.

[Seret]

I've looked in the engineering review book my father (a civil engineer) purchased before he took his general engineering test in college. It is full of formulas. From what I have experienced from my chemistry class in college and honors physics I took in high school, all you do is plug numbers into equations.

Yep, to be fair plugging numbers into equations does suffice most of the time. However, you need to understand the relationship between the terms of the equation, and the theoretical framework that often underpins that. Sometimes there is no theoretical basis, the equations will have been derived empirically, but usually there is a mathematical relationship there.

You can solve simple engineering problems by simply rearranging formulas and plugging in whatever numbers you've got. Engineering only takes from science what ever has a practical use, all most engineers want from scientists is a formula they can use to make predictions. But that doesn't mean that purely theoretical pursuits aren't productive, lots of very practical discoveries have come out of blue sky research. And competence with some of the more advanced mathematical techniques is required for some engineering specialities. If you wanted to get into something like fluid dynamics or writing software for FEA then you'll need a maths background that goes beyond rearranging formulae.

[Seret]

If you are getting a D- in high school algebra, cut your losses and consider a major in a soft science or liberal arts. You aren't going to be an engineer.

That's a bit harsh. Getting that score could be about lack of motivation as much as aptitude. That's a solvable problem.

Engineering is an exceptionally broad field. Some parts of it will require very good maths skills, some require little or none. The level of maths required to get through an engineering degree in something like civils isn't as high as something like electrical, for example (although they will expect a certain level of competence, obvious).

Unfortunately for the OP though they are just going to have to suck it up and lift their grade. Just see it as an annoying but necessary hoop you have to jump through while still grinding through the academic system.

[AeroEngy]

You are going to need to get comfortable with higher level math especially if you are going into Aerospace Engineering (that's what I do). There are lots of different areas you can get into being an aerospace engineer but they all pretty much involve some kind of higher math. Compressible fluid dynamics, thermodynamics, aircraft/spacecraft controls, flight dynamics, etc. require complicated math partial differential equations, Laplace & inverse Laplace transforms, etc.

Once you get out of school and into industry there are software packages where you don't really need to fully understand all the underlying math. However, you should really understand how it all works ... and you won't get out of school without getting the underlying principles.

[Mr Tegu]

I got a D in A level (~high school) maths and I now have mechanical engineering degree so you can do it, the maths will kick your arse though. And, unfortunately, you really need to get on it now cause you will be using it in a year or 2 when you start subjects like control systems and vibrations etc. If they're teaching it to you as part of your degree its because you are going to need to understand it.

[Lexif]

It can be demotivating to first learn the tools, then the application. In my first year studying physics on a german university, I had to take the same algebra and calculus courses as the math students. I didn't understand why I'd need such a deep and very abstract understanding of math as our physics lectures hardly had a differential equation here and there. Then, in the second year, we got theoretical physics courses (mechanics, quantum mechanics, thermodynamics and statistical mechanics). It was very interesting, but I didn't have the background to keep up with the speed and depth of the courses, struggled hard and had to repeat a course.

Don't know about engineering, but don't assume you know better what you'll need later on.

[PakledHostage]

Something to lighten up the tone of this thread (ref: XKCD):