Creating Functions?

therussequilibrium · 2012-12-21 11:04:39

Okay hopefully this doesn't make me look incredibly stupid, but this is my problem. I'm wondering how mathematicians come up with their formulas. For an example, I ran into one of those pictures where it asks you how many squares are inside the other square - and instead of counting them I wanted to find out how to create an equation that would solve the problem. So say the square is a 4x4 square(one big square made up of 16 smaller squares. I attempted to try to figure an equation...well what I came up with was a function that would solve the problem, but only if you knew what the previous function was, as in:

f(x)=f(x-1)+f(x)^2
f(4)= f(3)+16

Well, yeah I created a function, but I can only use it if I know the f(x-1), so I haven't really solved anything. I ended up looking up the formula and the formula to actually solve the equation is:

f(x)=x*(x+1) * (2x+1) / 6
f(4)=4*(5) * (9) / 6 = 180/6 = 30

And I can't figure out for the life of me how you would come up with this equation.
I seem to try to create equations for almost anything, and I feel like I have really good ideas, but what I always come up with gets me close to something but it never gets me to that something which is extremely frustrating.

Another example of me trying to do something like this was when I was extremely bored at work so I decided to try to figure out how to convert Fahrenheit to Celsius, in which well I had no idea how to go about doing this but I came up with something that get's you fairly close to the correct answer, as long as the numbers don't get really big.(Which makes me feel like Newton compared to Einstein, which probably means it's because I don't have a large enough understanding of Mathematics.)

C= (F/2) - 16 (Which is obviously rather simple, but gets you somewhat close to the correct answer. And yes I do know the correct formula now...)

But anyways, any help/advice/ or even links to something that could help me in understanding how Mathematicians find these functions would be greatly appreciated.

noelevans · 2012-12-21 16:57:38

One approach: Work out lots of cases and examples and look for patterns that can be turned into
formulas. Nowadays it is lots easier to do with the aid of computers that can run out cases by the
bucket load.

Anybody else care to contribute your favorite methods?

bobbym · 2012-12-21 17:08:49

Hi therussequilibrium;

Welcome to the forum.

There are several ways to come up with formulas. There are also several ways to play when you do not have them.

Definitely, STP, or Spot The Pattern is one of the best. It was used by all the mathematicians of the past and still is used. You can usually prove the pattern by inductiom.

Sometimes you can solve the problem by forming a generating function or a recurrence. Sometimes you just thrash around with the symbols algebraically until you get what you want.

Did you mean,

therussequilibrium · 2012-12-21 17:45:21

bobbym -

Thank you for the welcome, glad to be here.

But yes the function you last stated was the one I came to, but it doesn't really solve the f(x) unless you know the f(x-1).

And, those are the processes I used in attempting to figure out how to solve the square problem. And yes, there are a lot of patterns in the problem because it deals with squaring numbers and adding them. So, I guess my real question, what would help me the most is: how did whoever came up with the function to solve how many squares are in a larger square, get to their function?

That function being, f(x)=x*(x+1)*(2x+1)/6

There has to be some sort of understanding, maybe something to do with Geometry, that I don't have for someone to come up with this function.

Last edited by therussequilibrium (2012-12-21 17:47:24)

bobbym · 2012-12-21 17:48:39

Hi;

We can start with the answer you got rather than his.

Did you mean,

for your answer?

therussequilibrium · 2012-12-21 17:50:03

Yes, that's what I meant, not sure why I put f(x)^2 at the end, I just mean x^2.

bobbym · 2012-12-21 18:02:53

Hi;

Yours is generating the sequence

Did you check that against the other formula?

therussequilibrium · 2012-12-21 18:03:56

Just noticed the math function that we could use, so sorry about the ugly looking equations before.

Yes, that's correct.

therussequilibrium · 2012-12-21 18:05:49

Yes, they are the same pattern, they both solve the square question - but to use my equation you have to know what

is equal to, so it seems incomplete.

bobbym · 2012-12-21 18:08:31

That is not correct, yours is a recurrence. You only needed to know f[1] to prime it all the rest come from that. That is an excellent way to solve the problem!

All you need now is to turn the recurrence into an equation. Want to know how?

therussequilibrium · 2012-12-21 18:09:45

I would love to, because the only way I can possibly see how to is go through and manually figure them all out, which could take days if the square is a really large one lol.

bobbym · 2012-12-21 18:13:01

Actually a recurrence is quite fast to do with a graphing calculator or a computer and is somethimes the preferred way.

Hold on and I will show how to solve the recurrence so you get a formula.

therussequilibrium · 2012-12-21 18:15:26

Thank you, I really appreciate your help in this.

bobbym · 2012-12-21 18:29:07

Do not appreciate it yet. This will be tough to understand at first.

First way avoids the recurence altogether and uses the summation calculus.

So we have our first formula:

therussequilibrium · 2012-12-21 18:49:40

Yep, a little confused. I see that

I'm just not sure where that equation came from, it looks a lot like the equation they found to solve the square problem.

bobbym · 2012-12-21 18:53:50

That is the equation just in a different form. The summation calculus is like the integral calculus. Only it is for the discrete not the continuous.

Let's look at the second way.

We collocate and hope for the best.

therussequilibrium · 2012-12-21 18:58:39

Ahh, okay, I see it now - just not completely sure what happened on the second step, but I do see that the final equation equates to the original.

bobbym · 2012-12-21 19:09:55

Hi;

Do not worry about that now, we are just looking at ways to solve your recurrence.

Let's look at the second way.

We collocate and hope for the best.

We guess that the form is

We form the collocation equations.

We know from the recurrence that

n = 1, f(n) = 1
n = 2, f(n) = 5
n = 3, f(3) = 14
n = 4, f(4) = 30

So just plug in to the avove equations and solve for a,b,c,d

we solve that set:

therussequilibrium · 2012-12-21 19:18:29

Okay, now that's awesome, but how did you guess the pattern was in the form of

? Is this a well known equation that I don't know?

bobbym · 2012-12-21 19:22:09

The third method, tells us the the answer is a cubic. Also even if you can not do the summation in the first method you know that the integral ( almost like the sum ) of an x^2 involves an x^3.

Supposing you did not know any of that you would try different forms until you found one you like. Here we do not have to do that.

When you are ready we will look at the third method. Tell me when.

therussequilibrium · 2012-12-21 19:28:08

I'm ready, I'm writing this all down so I can study it and get a better grasp of it all.

therussequilibrium · 2012-12-21 20:00:51

Alright, no worries, I appreciate you taking your time out to help me - so take all the time you need.

bobbym · 2012-12-21 20:04:45

I want to point out that I am just showing ways to solve your recurrence. We have not examined if there was an easier way to solve the original problem besides the recurrence.

The third idea gives us all the answers to previous questions.

therussequilibrium · 2012-12-21 20:17:13

I understand, this information is still very helpful to me. It's obviously showing me there is a lot I need to learn; I'm a freshman in college and I really want major in mathematics, but from your general knowledge and understanding it would seem that I'm extremely far behind and I'm not sure I can catch up. So, this helps a bit...

bobbym · 2012-12-21 20:20:37

Everyone can catch up. In my country there is a growing feeling that to learn math you need special talents. Other countries I am being told believe that it is just a matter of hard work and desire.

To show who is right, my country has gone from number one in mathematics to number twenty eight since following this policy.

The third idea gives us all the answers to previous questions.

This one uses differences and a formula.

Math Is Fun Forum

#1 2012-12-21 11:04:39

Creating Functions?

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Re: Creating Functions?

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