# Useless math

For the experienced scientists and educators, what are the most useless math topics from a practical standpoint. I would list my favorite:

1. Polynomial long division. If the numerator and denominator don't factor, what is the use of performing the long division?

I know of only one -- to find the leading order behavior of a rational polynomial. But who would care about that other than a theoretical physicist or chemist?

## Answers and Replies

I dunno. Sometimes if you have a polynomial you can spot one factor real easy, but the others are hard to find. It becomes easier if you divide through by the factor you've found.

i think it applies more readily when the numerator is of signifigantly higher order and a simple factor is hard to remove visually. also, checking work for removing a factor from a larger order polynomial.

younger kids are taught to write out the whole part seperate and not to have a higher order on top. also, i believe, that some kids cant complete squares easily, so they need to use polynomial long division to get the +d-d part as a -d remainder. but i dont do that, so i cant remember why they taught it to me specifically.

basically its just used to convert improper rational expressions as a proper rational expression plus a polynomial. id suppose its hand for integration using ln x and u'/u formations, since the top is reduced to an order less than the denominator. i still like trying to find factors and completing the square just by looking, but i can see how some kids dont.

Rather than a specific subject, I'd like to nominate the generic set of all problem solving techniques where students are taught something incredibly overspecialized simply to solve one certain class of problem.

For instance, "mixture" problems. (John has two containers of punch. One is 5% juice, the other is 10% juice. How much of each should John mix together to get a 10-liter solution of 7% juice?) Rather than using these problems as general examples of how to describe and solve mathematical relationships involving percentages, students are frequently taught to set up specialized grids that include the given information, and then compute the missing information using the pattern of the grids. Essentially, rather than being taught problem solving skills, students are taught yet another step-by-step process to follow, without understanding why it works or how they could develop a similar process on their own for a different sort of problem.

I can understand why this would be appealing to math teachers - anyone can learn a process through repetition and reproduce it on demand, given sufficient practice, but teaching someone a concept is harder. Still, I think it's a disservice to students to do it that way.

Also - unrelated to the previous point - quaternions are a subject that will be of no use to most people, even in the sciences, although they do produce elegant solutions to certain problems.

matt grime
Homework Helper
the geometry of the quaternions (and related divisiona algebras over C) looks like it will be of use in theoretical physics.

jcsd
Gold Member
Is:

$$\frac{x^2 +12x +30}{x+7}$$

analytic at x = -7 ?

can the singularity be removed?

I rest my case.

matt grime
Homework Helper
you probably ought to have chosen a better example where -7 was a root of both top and bottom with unknown multiplicity.

HallsofIvy
Homework Helper
A very practical application: if P(x)/(x-a)= Q(x)+ r (r is the constant remainder) then
P(x)= Q(x)(x-a)+ r so P(a)= r. Any computer scientist knows that calculating r by (synthetic) division is faster than evaluating P(a).

Good catch, Halls. I had forgotten some of these applications (although I think few students will ultimately need to know them).

RE: " Essentially, rather than being taught problem solving skills, students are taught yet another step-by-step process to follow, without understanding why it works or how they could develop a similar process on their own for a different sort of problem."

Here! Here! I noticed these techniques cropped up with the onset of new math, and I find them abominable. Why not just reason it out? Sure it's hard, but math is hard.

Polynomial division is good for curve sketching too. Try finding a slant asymptote (if necessary) without long division.

Functions and relations have got to be down there. Calling a function a subset of a relation is all that was about IIRC. Pretty stupid if you ask me. But i guess math people like formalizing things.

I wasn't impressed with at least half of the discrete math i've taken thus far.

edit: ooh, i gotta go back to calculus two for this one. Rotating things about an axis to calculate surface area and volumes. It's much faster and easier, and works for all conditions, if you do it via triple integration and everything.

edit2: Now that i remember way back to calculus 2, i remember 1/2 of it was trivial integration techniques. Trivial in the sense that they were very limited in uses. Most of the time nowadays i try to use maple/matlab to calculate things. I seriously doubt anyone still remembers those ways of integrating those nasty partial fractions (or something like that) using triangles and trig.

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Math Is Hard
Staff Emeritus
Gold Member
JohnDubYa said:
Sure it's hard, but math is hard.

Hey! Leave me outta this! :rofl:

matt grime
Homework Helper
Although it pains me to say it Goalie ca, many mathematicians do not think of relations in terms of subsets ("orbits" would be my view of them), but that notion you dislike is useful for computer science, certainly for some langauges. You define the relation as ntuples, then evaluate expressions on entries in the elements. So is.father.to( , ) is a list of father child pairs, and one can call functions such as father.of( ).

But then there is the other reason for defining functions in terms of sets. If you don't, what are they? Letting one of the many secrets out of the bag, they way you're taught about functions for most of your life is mathematically unsound. One needs only look at the number of questions I see written by supposedly good mathematicians that ask something like is the function 1/x from R to R continuous everywhere. It isn't even a function from R to R.

Zurtex
Homework Helper
Goalie_Ca said:
edit2: Now that i remember way back to calculus 2, i remember 1/2 of it was trivial integration techniques. Trivial in the sense that they were very limited in uses. Most of the time nowadays i try to use maple/matlab to calculate things. I seriously doubt anyone still remembers those ways of integrating those nasty partial fractions (or something like that) using triangles and trig.
Surely the writers of such programs as matlab do...

I'm only very much at the start of my mathematical life and am at the moment revising such integration techniques. But the way I always see it is rather than having a set problem and a set solution is that you have a logical problem and you need to try and find a logical solution, thus getting in the mind frame for solving any particular problem.

Hurkyl
Staff Emeritus
Gold Member
I dunno, I can do a lot of those integrals from Calc 2 faster than it would take to fire up a computer and run Mathematica.

And, sometimes, knowing a technique can be important even if you never use it; for instance, I've seen theorems in my Complex Analysis and Algebra courses whose proofs would be totally mystifying if you didn't understand partial fractions.

HallsofIvy
Homework Helper
matt grime said:
Although it pains me to say it Goalie ca, many mathematicians do not think of relations in terms of subsets ("orbits" would be my view of them), but that notion you dislike is useful for computer science, certainly for some langauges. You define the relation as ntuples, then evaluate expressions on entries in the elements. So is.father.to( , ) is a list of father child pairs, and one can call functions such as father.of( ).

On the contrary, many mathematicians do. Mathematicians prefer to have a variety of ways of thinking about the same concept.

krab
JohnDubYa said:
.... But who would care about that other than a theoretical physicist or chemist?
... and theoretical physicists and chemists do what? Useless stuff?

Chances are its useless math if 5 years from now you'll never use it again.

I mean, all math is usefull, in some way or another, but who the hell cares about most of that stuff. Maybe its the engineer in me, but I mostly just wanna learn the math i can need and use and get the idea where it came from so i can derive things and understand it. I'm not one for pointless details, particularly techniques for solving stuff that my calculator can do!

Janitor
When I was in high school, multi-function hand-held calculators were becoming available for something less than a king's ransom. My math teacher, Mrs. W., must have been well past the age at which a public school teacher can retire, but she hung in there for some reason. She diligently taught us how to take square roots using some process that I have forgotten, other than it was laid out on paper kind of like the way you do long division. I have never had a moment when I cursed myself out for having forgotten that technique.

Hurkyl
Staff Emeritus
Gold Member
I've had a few occasions where I was very annoyed I couldn't remember how to do a square root by hand.

Goalie_Ca said:
I mean, all math is usefull, in some way or another, but who the hell cares about most of that stuff. Maybe its the engineer in me, but I mostly just wanna learn the math i can need and use and get the idea where it came from so i can derive things and understand it. I'm not one for pointless details, particularly techniques for solving stuff that my calculator can do!
Some details may be pointless to you at this stage of the game, but remember that these details needed to be understood in order to develop the calculator.

The attitude here seems to be: Why do I need to learn how prepare meals from scratch where I can just go to the store, buy a TV dinner and be done with it?

If there is a more efficient technique to solve a problem, then by all means use it. If you can evaluate definite integrals by hand faster than with Mathematica, then do it by hand. If you find that Mathematica is faster, then use Mathematica. Whatever floats your boat.

Janitor
I've had a few occasions where I was very annoyed I couldn't remember how to do a square root by hand.- Hurkyl

Ah, but if you just remember the Newton-Raphson method, then you can always do it that way. Maybe Mrs. W's way was faster, but if it is not something that is easy to remember how to do...

hurkyl>

you can just use newtons method

or this iteration form...
i think thats the guess, divide, average, guess again.

like sqrt(12) guess is 3
12/3 = 4
(3+4)/2=3.5
12/3.5=3 3/7 (3.428571428571)
(48/14 + 49/14)/2 = 97/28

you can check by squaring the 97/28 and subtracting the original value, and thats the square of your error.
(97/28)*(97/28) = 12.0013 which is pretty damn close.

the longer way is here

it uses a little guessing, but can be done very precisely without many repetitions.

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Hurkyl
Staff Emeritus
Gold Member
Anyways, one of the biggest dangers of not learning a technique because your calculator can do it is, quite simply, that you won't know the technique.

This can cause a myriad of problems including:

Being unable to solve a problem that requires multiple techniques because you'd have to apply the technique you never learned.

Being unable to recognize part of a problem can be attacked via the technique you never learned.

Being unable to recognize that a problem can be transformed into one that can be solved with the technique you never learned.

Being unable to apply the technique you never learned to transform into another one that cannot be solved, but is still easier to analyze.

Being unable to learn a more complicated technique that uses, or generalizes, the technique you never learned.

HallsofIvy