Calculus is often used to solve optimization problems. An early famous instance, which is trivial with the simplest calculus but seems very difficult without calculus is “Kepler’s Wine Barrel Problem,” from a true story. Johannes Kepler needed to buy wine for his wedding (look 2/3 way down this webpage. Each cylindrical wine barrel had different dimensions, yet the merchant priced them with a single measurement; Kepler worried he was being cheated. Here’s a page which derives Kepler’s result, using calculus, along with a (rather useless?) interactive Java figure.
In case our OP Lancia missed it, math junkies also sit around arguing about the true value of 0.999…
(Or, more precisely, trying to explain it to the math non-junkies among us.)
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I’m glad you’re so enthusiastic about the subject. My best recommendation would be for you to go to the college library, find the math section, and just randomly pick up books off the shelf until you find ones you like. If you like doing the computations but want more applications and real-motivation, I’d suggest you also take a look at mathematical physics. If you want more motivation, but not necessarily real-world motivations and applications, I’d suggest you take a look at real analysis or topology. You mention being interested in both the theory and applications (which is great!), and the two can be pretty dissimilar at this stage. I’m not necessarily recommending that you take a formal course in these areas or plow through a textbook on them; just read about them, glance through wikipedia, check out a book or two, and see if they’re areas you might find fun and interesting. As for specific textbooks, they all suck. Seriously. I’ve found about half a dozen math textbooks that I’d consider well-written.
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For me, I’ve never wanted to be anything other than a mathematician, and math just sort of clicked in the way other subjects, even ones I did well in, never quite did. Like you mentioned, I’m interested in math for the sake of knowing the beauty of math, and I haven’t found anything else that’s comparable to studying it. Fortunately, if something goes wrong, it’s easy to find a job with a math degree, even one in pure math. I don’t know anything at all about teaching high-school students, but math courses in college are generally taught by college professors (or at least their grad students), and that’s a tricky position to pull off.
I have a two-year college degree (Associate of Science) majoring in Math.
Does that make me employable?
As with anything, it depends on what you sort of job you’re applying for.
Second-order differential equations can be used to work out equations of motion where the forces on an object include gravity, friction braking and a spring. From that, given, say, the mass of a car, the stiffness of its suspension, and how heavily damped it is, you can work out whether the suspension will let you feel every ripple in the road, or will pogo wildly for half a mile if you run over the white line, or will smooth out the bumps unobtrusively. A similar equation will tell you whether a door-closer will slam the thing shut with an annoying bang, take forever to close the damn door, or close it quietly and firmly.
We did something similar on a math course where we had two masses coupled by springs to each other and a frame. Given “ideal” springs of a known spring constant (easily measured with some trial masses and a ruler), the initial conditions predict whether the masses will bounce around chaotically exchanging momentum, or oscillate smoothly either together or in opposition.
Or consider a rocket. The weight of a rocket is largely fuel. The thrust of the motor can be worked out on a test-bed, but how far will the rocket travel and how fast will it be moving when the fuel runs out? Acceleration can normally be determined using Newton’s second law, but here the mass is changing constantly as the fuel is burnt so the math is harder.
It is not coincidental that all the examples of advanced math in “everyday examples” involve physics problem. To the OP, if you are interested in actually learning math for a purpose, switch to become a physics major. Physics is entirely applied math. And quite advanced math too! As a physics major you will continue to learn math and you will never feel like it’s pointless or never feel like it’s not real. And physics majors are extremely employable in many different fields, even if you only have a bachelor’s degree. Being a physics teacher is essentially priceless and you’ll basically be untouchable when everyone else is getting laid off because of budget cuts.
That being said, if you want some advice about becoming a teacher, I’d say only do it if you’re willing to do exactly what your administration tells you and not feel bad about compromising your morals time after time, and if you won’t mind a complete lack of support from everyone you work under and work for. Your only allies will be other teachers and they are always powerless to help you. I got out of teaching high school physics after just 2 years and honestly you couldn’t pay me enough to go back into teaching public HS. College is not better, I hear.
Being 31 and just now deciding to study to be a high school math teacher is rather unusual, but it’s not excruciatingly rare. Being 31 and considering getting a Ph.D. (or even a master’s degree) in math when you don’t even have a bachelor’s degree, is definitely extremely rare. Most people who become mathematicians had already thought about it when they were in high school at least. I remember in elementary school reading about Cantor’s diagonal proof of the uncountability of the rational numbers and thinking that this would be neat stuff to do for a living. That was despite having grown up on a farm with my parents and seven siblings, with a father who was a factory worker and a farmer and a mother who was a housewife, neither of whom went to college. I eventually got master’s degrees in both math and linguistics (which I also got interested in) and have now worked for thirty-one years as a mathematician.
I do know of several other people who grew up in even poorer circumstances and became mathematicians. In each case they discovered how much they liked math by their late teens or early twenties. Incidentally, mathematicians are quite employable. You would probably need a master’s degree. It’s not true that a physics degree is that much more employable.
The problem is that you’re only studying high school algebra at the moment. That’s actually not very similar to higher-level mathematics. To find out if you would like higher-level mathematics, I suggest that you get some of Martin Gardner’s collections of his Mathematical Games articles from Scientific American between 1956 and 1981:
This is a list of those books:
- Hexaflexagons and Other Mathematical Diversions
- The Second Scientific American Book of Mathematical Puzzles and Diversions
- New Mathematical Diversions
- The Unexpected Hanging and Other Mathematical Diversions
- Martin Gardner’s 6th Book of Mathematical Diversions from Scientific American
- Mathematical Carnival
- Mathematical Magic Show
- Mathematical Circus
- The Magic Numbers of Dr. Matrix
- Wheels, Life, and Other Mathematical Amusements
- Knotted Doughnuts and Other Mathematical Entertainers
- Time Travel and Other Mathematical Bewilderments
- Penrose Tiles to Trapdoor Ciphers
- Fractal Music, Hypercards, and more Mathematical Recreations from Scientific American
- The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications
These will give you a feeling for what higher-level mathematics is like while not requiring any knowledge beyond high-school algebra. There’s also a book called The Colosssal Book of Mathematics by Gardner that’s a selection of his best articles. There’s a searchable CD called Martin Gardner’s Mathematical Games with all the articles. There are many books by other authors which give the same feeling for what mathematics is like. What I don’t recommend is going to a college library and looking through college textbooks. Those are always boring if you don’t have the proper background.
I seem to have had a somewhat different experience from most mathematically inclined people. But first let me say that loving mathematics is the most important prerequisite. Finding uses for it requires taking a course in mathematical modeling. The fake problems in texts just don’t give the same flavor.
I always did well in math (although my 7th and 8th grade would never give my an A because I never did her idiotic make-work problems and the fact that my tests were perfect wasn’t enough). I took four years in HS and did very well. But, except for 10th grade geometry, I really didn’t enjoy it at all. In my first year of college, I took no math. In those days, the first year consisted of one term of “college algebra” and one of analytical geometry. I looked at the college algebra text and decided that it was a waste of time. Elementary counting (“How many ways can you arrange a deck of cards?”) and mathematical induction which struck me as utterly obvious. I did take a summer course in analytic geometry and enrolled in calculus in the fall of the second year. Again, it seemed simple–and boring. But I was working in a lab and that had two results. First I discovered that I was not cut out for lab sciences. The second result came when I overheard two graduate students discussing a course one of them had taken and the other was taking. It was called Modern Algebra. It sounded very interesting. (In case anyone is interested, they were discussing–I still remember this–a proposition in Van der Waerden–that is a semigroup is a group if it has a left identity and every element has a left inverse with respect to that left identity. Little did I know how basically useless this is and how poor to put at the beginning of such a course.)
Well, to make a long story short, I fell in love with math. Really with abstract math. The rest is history. I switched my major to math, took that very same modern algebra course the following fall and it changed my life.
There are, of course, jobs for math majors. Aside from HS teaching, there are all those quants in finance, actuarial (although I would imagine a lot of that is now done automatically with computers), … But I might mention that there are an awfully lot of recreational mathematicians out there and there are worse things to do than that. I am now retired (for 13 years) and still do mathematics because I enjoy it.
Just out of curiosity, what is your reaction to the fact that every positive integer is the sum of four (or fewer) squares? Or the irrationality of the square root of 2? Or the unique prime factorization theorem? That’s what loving mathematics is about, not about applications. It is a way of thought.
All right. I have tried three fucking times to respond to this thread and my worthless laptop has crashed or frozen every fucking time. I have to get ready for work, but I will respond to this thread when I’m not tethered to my phone.
Strictly speaking, algebra is the grammar of mathematics, e.g. it explains how you can put the elements together and manipulate them. What is really interesting about algebra is that the modern notation and formulation of elementary algebra (what most students think of when they think of “algebra”) is a relatively recent innovation (circa 800 CE) even though people have been working through complex and sophisticated problems in calculus, differential equations, number theory, and analyltical geometry throughout recorded history. Think of trying to write a symphony in words and you get a notion of just how difficult that must have been. We don’t usually think of algebra as being a technological innovation in the way that the printing press, the steam engine, and the digital computer are, but in fact the evolution of algebra into the current form has allowed for abstracting a particular problem into a statement that can be manipualted and reduced by the application of general rules.
Back to the query of the o.p., at the age of 31 and at a starting level of education in higher mathematics a career in research mathematics is probably not a reasonable expectation, but studying applied math is certainly feasible, and as others have noted, the vocations using this span from education to statistics and actuarial to computer science and engineering.
However, before you jump in with both feet, it is probably best to get a firm grounding on the more abstract elements of mathematics and logic (which are inextricably intertwined). It is, as the o.p. noted, one thing to work through the mechanics of reducing a problem, but another to understand the fundamental logic behind it. In that vein, I would actually recommend taking a course in basic symbolic logic before advancing into calculus or analytical geometry so as to have a firm grounding on how to develop and present general proofs. Being able to understand the sometimes arcane and jargon-dependent language of a proof is key to undetstanding concepts like the fundamental theorem of calculus or the mean value theorem, and therefore understand the simple beauty and power of calculus before getting wound up in the details of solving particular problems. I personally think Velleman’s How to Prove It is actually one of the best applied introductions to symbolic logic and wish that it had been published before I went through my calculus and differential equations sequence.
And even if you don’t end up pursuing a career in mathematics, it can still be an intertaining diversion and hobby. Check out Martin Gardner’s Mathematical Games, a collection of columns that he wrote for Scientific American introducing various problems, puzzlers, and apparent paradoxes, and his successor Douglas Hoftstadter’s Metamagical Themas which are generally more abstract and philosophical, but all the more fascinating for being so.
Good luck to you.
Stranger
Hmmm…maybe the person who programmed your laptop needs a remedial math course. 
I use algebra and something I didn’t learn about until nursing school called “Dimensional Analysis” for medication administration and also some simple geometry for doublechecking the work of the folks (generally surgeons and physical therapists working together) who use traction devices which need to be at particular angles and levels of force on patients post-op. Not rocket scientist level math, to be sure, but more than I thought I’d ever use as a grown up.
When making herbal medicines (which I did for while as part of a job, as well as taught it), it’s useful to know dilution and ratios - if I have 151 proof Everclear and need 50% alcohol at a 1:4 and I want 16 ounces of this herb tincture, how much water, alcohol and herb do I need? (Although in all honesty, I figured it out *once * for various strengths and ratios and wrote a chart. Still made my students figure it out once though. Never know when you’re going to spill something on your chart and render it unreadable.) I assume compounding pharmacists need to be able to do the same for their jobs.
I’m also called in when they want to put in a new garden where I camp and they only have so many feet of fence to do it with. Yep, that literal garden variety College Algebra example of functions, and I really have used it a dozen times! 
Hari Seldon, you’re even older than me, so perhaps it was easier back in your day to put off studying calculus till your sophomore year of college and still be able to major in math. I graduated from high school in 1970, and my high school (which was several steps below contemptible) didn’t offer calculus. It took some work for me to teach myself the first and second years of calculus in the summer before college and during my freshman year. Most math majors even then had probably already taken a year of calculus before starting college. Nowdays calculus is even more typically a high school course. My high school still doesn’t offer calculus, according to the online course catalogue I checked. That puts it in the bottom 10% of all high schools in the U.S. I congratulate you, Hari Seldon, on only deciding to major in math in your junior year of college and yet still making it through to a Ph.D.
This thread has got me tempted to start a thread on a project I had a few years ago. My skill as an all wood bow and arrow maker got me recruited to build a giant bow for a Da Vinci designed catapult the discovery channel was building. Some of you might have seen it. I had about 10 days to secure materials and design and build a giant bow that actually worked. It was really a confidence builder for me that I was able to learn and formulate a lot of math that I had never been exposed to. I was really disappointed that the show chose to concentrate on conflicts rather than all the science that was going into this project. I really fell in love with math all over again after this project.
Ok, lets try this again… fingers crossed…
I certainly can see some real world applications for this… say I have to mow x acres of grass and I have y gallons of fuel and I know I can mow z square feet on a gallon of gas… I could chart the two lines and figure out how much grass I can cut on the available fuel (right…? I’m having a hard time figuring out the actual equation, but it seems like that is an example of a real-word application for this sort of algebra).
I have no clue what any of these things mean, but now my brain has decided it just has to know and is planning on thoroughly abusing Google in an attempt to discover some answers.
I seem to recall having this discussion with you before… I would love to sit down and talk with you about your experiences in education… it’s pretty clear it left a sour taste with you and I’m curious to hear more. You’re from my neck of the woods, next time your in town maybe we can do lunch.
From talking with my math instructors, a masters degree in applied mathematics is the most useful. Basically, one earns a bachelors degree in mathematics then focuses on the ‘applied’ part during their masters program – basically, taking a bunch of engineering classes. I’ve not heard that physics majors have a leg up over applied mathematics majors.
Thanks for the information, I’ll look into finding copies before school resumes inn January. Considering I’m just returning to school, no matter what field I choose I suspect I’ll be at a disadvantage because of my age.
Bolding mine.. I confess that I’ll have to go to Google for most of the above. But now I’m curious, which I think is a good sign. Most people would just say fuck it and walk away. Not me. I want to learn this stuff. I want it to make sense to me.
Let me share a story… a friend of mine and I were sitting in lecture hall waiting for a geology class to start. The professor was chronically late, so we began to discuss the previous day’s lecture, which had to do with S- and P-waves. My friend wondered if such things could be graphed using sine and cosine. Now, my trig is non-existent, but I figured you could based on what little I know. I theorized that one would have to know the different chemical and molecular compounds of the different soils and rocks that were present in the earth’s crust that the waves passed through - obviously the crust is not a uniform mass. Therefore, it would be a collection of different charts giving the geologist a visual representation of the wave (I was totally blue-skying it, this may be total bunk).
She then theorized that the geology instructor should chart such waves and allow those of us with functioning curiosity to see just such a representation. She said something to the effect “That would make everyone’s head explode”. We then began trying to figure out what kind of formula best represented such an explosion.
Ten minutes later we were both huddled over pieces of graph paper arguing the best way to represent flying brain matter – both with and without a vacuum present. One of us looked up and realized that the instructor had arrived, had the power point ready to go, and was watching with unconstrained amusement – along with the entire lecture hall (maybe 50 people)-- our little spat. (Someone told me later it sounded like a full-on husband and wife fight). We were informed that this was neither trigonometry nor anatomy class, and would we be so kind as to save such conversations for our own time?
So yeah, I think I have it pretty bad.
Linear programming. This is one of those unfairly neglected corners of mathematics, because it doesn’t fall into any of the big common courses, like calculus or statistics or trigonometry. I only learned about it when I had to teach it in a class I myself was teaching, but when I did figure it out, I found the simplex (matrix) method of solving linear porgramming problems to be both very interesting from a pure mathematical point of view, and potentially very useful in real-world applications.
It has been said that you learn arithmatic by studying algebra, you learn algebra by studying trig, and you learn trig by taking calculus, etc. It’s a never ending sequence.
Linear programming is the bread and butter of airline operations. It doesn’t get much more applied than that.
Nitpick - are you absolutely sure you were taught that*? Thisis what I understand that term to mean, and tis a much more advanced concept than the level you seem to be at. Or are you talking about completing the square, which is probably about the right level, highly relevant towards solving quadratics, and much more likely? And regarding one reason it is useful see:
(*Won’t be your fault if it’s wrong term of course, blame yer teacher!)
That’s where completing the square is useful where you are - it’ll help you sketch the chart more easily. When the term is y = (ax + b)^2 + c, -b is the x coordinate and c is the y coordinate of the minimum point of the function, while a is the steepness. Further definitions of minimum point and steepness will come when you do some calculus btw, which you are not far off.
I haven’t the foggiest idea about education, especially in the US, but I can say the following: If it’s like the UK, then while it’s very easy to find a good media studies or other guffy subject (I accept that media studies is a legitimate academic discipline but not for 9 year olds for gawds sake, and it’s certainly not equivilant to maths) graduate who wants to teach, schools are massively crying out for “hard science” graduates, for four main reasons. Firstly, there are less of them now because so many who would have done them have done softer pointless subjects. Secondly, almost anyone with a hard science degree can get a “better” job than teaching, in a financial sense (and also career progression if one doesn’t want to stay in education). Thirdly, in general for all kinds of reasons men aren’t going into teaching any more compared to women, it’s like 4:1 or something which is possibly related to point four: In the best schools teaching is probably a joy. But in possibly a majority of schools, even though most of the pupils are there to learn (even if they won’t admit it) a few shits spoil things for everyone and it’s impossible to discipline them.
I touched on the briefer job point slightly, but to reiterate - you can get metric shitloads of jobs with a good maths degree. It signifies a top notch brain and employers know this. Almost any field would be interested. Of course the best paid straight out of university are the quants, although I think they have to have Phds etc mostly.