# Is all math essentially: Solve for x?

1. Aug 15, 2012

### Tyrion101

Is all math essentially: "Solve for x?"

I am still learning basic math, and in one form or another it all seems basically: "Find x, or in some case some other letter." Or does it become different when you get to the higher maths?

2. Aug 15, 2012

### DonAntonio

Re: Is all math essentially: "Solve for x?"

$${}$$
Not even close

Very, very different...fortunately enough: much more challenging, interesting, beautiful, astonishing...although, of course, now and then we still have to tackle a "find x such that so and so" question.

DonAntonio

3. Aug 15, 2012

### Mentallic

Re: Is all math essentially: "Solve for x?"

I'm in uni and still finding x's or some other kind of value. Although, the x's I'm typically finding aren't numbers, they're matrices, vectors, functions etc.
But that's just the start of it. There are so many other problems that'll you'll also need to tackle - and likely a lot of times these techniques you've learnt will be used in order to help you find x, but also, they might not be.

I'm not sure how basic the basic math you're talking about is, but have you learnt about expanding factors yet? Such as a(x+y)=ax+ay? This kind of thing is pretty much as big in maths as finding x is.

4. Aug 15, 2012

### voko

Re: Is all math essentially: "Solve for x?"

Mathematics is useful because it helps us solve some problems. In such a situation, yes, it is very frequently about "finding x that satisfies these conditions". But this is mostly known as "applied mathematics'. Pure mathematics is about building theories about properties and relations of things that may or may not be useful in applications. Note, however, that there is no clear-cut division between pure and applied. Often, it is a matter of attitude. Plus what is known as pure and even considered not ever likely to become applied, can become applied one day. For example, modern cryptography is almost entirely based on (once) purest branch of mathematics: number theory.

5. Aug 15, 2012

### Staff: Mentor

Re: Is all math essentially: "Solve for x?"

Proof that for n > 2 there are no integers a, b and c that satisfy the equation an+bn=cn is hardly of the kind "solve for x".

6. Aug 15, 2012

### xodin

Re: Is all math essentially: "Solve for x?"

Similarly to what others have said, higher math often times doesn't even use numbers--they keep it all in variables. For example, in basic calculus you might be given a function of x, and your job is to solve for its derivative or integral, which is still just a function of x. Sure you could plug numbers into any one of those functions, but that's not always the goal.

7. Aug 15, 2012

### phinds

Re: Is all math essentially: "Solve for x?"

I had to laugh at that 'cause it's EXACTLY what my reaction was when I saw the question. I guess it would be for most folks who've had much exposure to math but it still tickled me to see you took the words right out of my mouth.

8. Aug 15, 2012

### voko

Re: Is all math essentially: "Solve for x?"

It's a qualified answer that for n > 2 "I cannot solve for x".

9. Aug 15, 2012

### phinds

Re: Is all math essentially: "Solve for x?"

That seems like quite an odd way of looking at it.

10. Aug 15, 2012

### voko

Re: Is all math essentially: "Solve for x?"

As I wrote earlier, applied vs pure is a matter of attitude.

11. Aug 15, 2012

### I like Serena

Re: Is all math essentially: "Solve for x?"

The problem can be rephrased as: find integers a, b, c, and n > 2 such that an+bn=cn.

12. Aug 15, 2012

### Tyrion101

Re: Is all math essentially: "Solve for x?"

Interesting, I guess I'll continue slugging through the basics, and hopefully get to where I don't need to use "find x" as much as I am right now. Thanks for the replies, it helped a lot. I think that's why I keep zoning out in math when I keep trying to go back to it, is it seems mostly the same to me.

13. Aug 15, 2012

### voko

Re: Is all math essentially: "Solve for x?"

Any theorem is essentially either $$(\exists x \in X' \subset X) \textbf{P}(x)$$ or $$(\forall x \in X' \subset X) \textbf{P}(x)$$ which can be regarded as solutions to a problem "find x in X such that P(x) holds".

14. Aug 15, 2012

### xodin

Re: Is all math essentially: "Solve for x?"

Math is about as cumulative as it gets. Best to learn to enjoy the repetition and master the subject at hand, as you will use every single aspect of it down the road. Maybe not in your next class, or the class after that, or the class after that, but then you'll run into classes that expect you to remember exactly how to do everything you covered in algebra, geometry, trig, etc; and in many cases, expect you to know or learn on the fly the stuff that wasn't necessarily covered in detail before, such as obscure trigonometric identities, etc. Plus, as an aside, don't expect to be using calculators in advanced math, or to be able to use formula sheets on tests, but if you pay attention, do the coursework, and learn the material as you go, you won't need them.

15. Aug 15, 2012

### DonAntonio

Re: Is all math essentially: "Solve for x?"

Psst, that's a trivial question with a trivial answer:
$$a=0\,\,or\,\,b=0\,\,or\,\, c=0\,\,,\,\,\forall\,n\in\Bbb N$$
are (infinite) solutions.

DonAntonio

Ps. Pheew! Good it wasn't required $\,abc\neq 0 \,$ , otherwise I'd have to Wiles-Taylor this question out!

16. Aug 15, 2012

### chiro

Re: Is all math essentially: "Solve for x?"

Mathematics is about three things: transformations, constraints, and representation.

Representation is how you define something. Transformation is how you change something and a constraint is basically a kind of assumption that you use to make problems tractable and also for actually defining something (if you can't define something or it's too unconstrained, you won't be able to analyze it).

The above hold for every single area of mathematics without exception, and this is pretty much what it's all about.

With your problems of "finding x" your goal is to "find x" and in the process you start off with all your initial conditions and then you are "transforming your way" to a solution. You may make extra assumptions along the way or approximations but the idea is the same.

We do this in proofs as well: again it's invariant with respect to the field of mathematics.

17. Aug 17, 2012

### I like Serena

Re: Is all math essentially: "Solve for x?"

But... a,b, c are integers... didn't I mention that they are elements of the (apparent) same $\Bbb N$ that you are referring to? :shy:

18. Aug 17, 2012

### DonAntonio

Re: Is all math essentially: "Solve for x?"

Nop. You wrote "The problem can be rephrased as: find integers a, b, c, and n > 2 such that $\,a^n+b^n=c^n\,.$"

And then I wrote what I wrote. :)

DonAntonio

19. Aug 17, 2012

### SteveL27

Re: Is all math essentially: "Solve for x?"

For sake of discussion I'll take the opposite side of that.

Find integers n > 2, a, b, c such that a^n + b^n = c^n or else show that no such integers can exist

That's a math problem in which we're asked to find four unknowns that satisfy a condition. In that respect it's no different than being asked to find x given that 2x = 5 (or show that no such x exists. Maybe we're in the integers.)

We state a condition or relationship among some variables; then we're asked to find particular numbers that satisfy that relationship, or prove that none exist.

You could do the same thing for the Riemann hypothesis. Find x such that x is a zero off the critical line, etc.

Last edited: Aug 17, 2012
20. Aug 17, 2012

### chill_factor

Re: Is all math essentially: "Solve for x?"

no. "solve for X" is science. Math is much more.