Explanation?

Ma123 · 2013-09-27 18:07:22

Hi. I'm not well educated in math, just a high school student interested in math. I was playing around with numbers and noticed a pattern. If I lined up consecutive perfect squares like {0,1,4,9,16,25,36...} and then took the difference between each pair of consecutive numbers, I got {1,3,5,7,9,11,13...}. Then I took the difference between each consecutive pair of numbers and got {2,2,2,2,2...}. Taking the difference between those gives {0,0,0,0,0...}. Interesting.

I then thought about using other powers, say cubes. I take the sequence of perfect cubes {1,8,27,64,125...} and make a sequence of the differences like before {7,19,37,61,91...}. I continue the process of taking the differences between consecutive numbers (12,18,24,30...} then {6,6,6,6...} then {0,0,0...}.

I then try the fourth power. {1,16,81,256,625,1296...} --> {15,65,175,369,671,1105...} --> {50,110,194,302,434...} --> {60,84,108,132...} --> {24,24,24,24...} --> {0,0,0...}

So that's the process.. I wrote (x+1)^n-x^n where n is the exponent.

Then I noticed that the number of times you have to take the difference in order to reach a difference of 0 is (n+1).
Also, the final sequence before {0,0,0...} will be a sequence of {n!,n!,n!...}

I think this is really cool! But I have no idea what it means. I was hoping one of you brilliant minds would be able to expand on this pattern for me. So do these things have a relationship? Is it just coincidence? Does it have a use? I appreciate your help!

Agnishom · 2013-09-27 19:43:57

Ma123 wrote:

So do these things have a relationship? Is it just coincidence? Does it have a use? I appreciate your help!

These are no coincidence, just inherent property of numbers. You could easily try and prove them.

It has uses. Using it you can find formulas to sum different series.
For example, the sum of the first n odd numbers is n^2. This is a formula that can be derived from your first observation.

Finally, Welcome to the forum! Cool Observation! Well Done!

Bob · 2013-09-27 19:47:02

hi Ma123,

Welcome to the forum.

You have indeed found out a true mathematical result. Well done! But you are not going to become world famous for it, I'm afraid, because others got there first.

Any quadratic formula will do the same.

eg

will generate a similar set of differences.

Each time the highest power goes up by 1, it takes one more set of differences before you get {0,0,0....}

Is it useful?

Well yes. If you have investigated a pattern of numbers and you find the differences, it will tell you what the highest power of n is and thus can help you to find the formula for the pattern. There's a bit about it here:

http://www.mathsisfun.com/algebra/seque … -rule.html

It's great that you found this out for yourself. It's a sign of a true mathematician Welcome to the forum.

Bob

ps. One way to prove why it works involves using this:

http://www.mathsisfun.com/algebra/binomial-theorem.html

bobbym · 2013-09-27 21:45:13

Hi Ma123;

Welcome to the forum. You have discovered the difference calculus. Not as popular as it once was but still important. You can interpolate, and approximate derivatives with them. You can do sums and a whole lot more.

Did you know you can multiply polynomials with them?

Agnishom · 2013-09-28 00:05:09

bobbym wrote:

Hi Ma123;
Welcome to the forum. You have discovered the difference calculus. Not as popular as it once was but still important. You can interpolate, and approximate derivatives with them. You can do sums and a whole lot more.
Did you know you can multiply polynomials with them?

Please explain

bobbym · 2013-09-28 03:17:07

Which part? The multiplication?

Ma123 · 2013-09-28 07:37:09

Could you explain both the approximation of derivatives and the multiplication? That sounds cool

bobbym · 2013-09-28 07:59:54

Hi;

This technique was used by our group on small handheld programmable calculators to do symbolic multiplication.

We want to expand

Form a table of differences, starting with the first difference.

4th difference is 384 so divide by 4!
384 / 4! = 16
Coefficient of x^4 is 16, store that.

Subtract that from the original expression:

Form a table of differences of that:

3rd difference is 576 so divide it by 3!
Coefficient of x^3 is 576/6 = 96, store that.

Subtract that.

Form a table of differences of that:

2nd difference is 432 so divide it by 2!
Coefficient of x^2 is 432/2 = 216, store that.

Subtract.

Form a table of differences of that:

First difference is 216 so divide it by 1!
Coefficient of x is 216, store that.

Subtract.

Form an x y table of that:

Zeroth difference is 81 so that is the constant term.

Put it all together:

Multiplication of polynomials by differences. Wunderbar!

Did you know it was because of work like this that Charles Babbage designed his difference engine and later his analytical engine. For that reason he is called the father of computers.

Bob · 2013-09-28 19:51:32

hi bobbym,

You've lost me. Where did these numbers come from?

Bob

bobbym · 2013-09-28 21:57:01

Hi Bob;

I have left out the first 2 lines for brevity instead it caused confusion.

You first form a table of x and y values of y=(2x+3)^4. It is shown below.

Now you just make a difference table from that.

625 - 81 - 544,

2401 - 625 = 1776

6561 - 2401 = 4160 etc.

repeat for each line until you come to constant differences.

The first line are called the first differences. The second line is differences of the differences and is called second differences.
I left out the operator notation so as to not confuse the OP.

Bob · 2013-09-28 22:14:10

OK, got it now. Thanks.

Bob

Ma123 · 2013-09-30 12:04:04

That's really cool! So how can you approximate derivatives with that?

bobbym · 2013-09-30 20:19:31

Hi;

One of the simplest ideas uses the zeroth difference ( first line in a difference table ) to compute the derivative at a specific value.

Supposing we wanted to find the value at x = 2 of the derivative of y = √x.

Start by making a table of values:

This last line is the zeroth difference, the first line of the difference table.

Now just take that line and plug in to a formula:

This agrees with the exact answer of 0.3535533... very well

There is an even more direct way if you want to see that to.

A thought:

It is important to remember that numerical mathematics and classical school taught math are very different. Things like integration are hard in math but easy in numerical math. In the same way differentiation is comparatively easy in math but difficult in numerical math. Difficult in the sense that numerical differentiation using differences is highly unstable and subject to roundoff error.

Ma123 · 2013-10-02 13:06:29

Thank you! That is really cool!

Using those differences and the fact that the last difference before 0 is n!, i have found a formula for n!

n! = nC0(x+n)^n - nC1(x+n-1)^n + nC2(x+n-2)^n - nC3(x+n-3)^n + ... +- nCn(x)^n for any x

the +- is at the end because if n is even it will be +, and if n is odd it will be -

Not sure if that's useful at all. Probably not, but it was fun anyway

Ma123 · 2013-10-02 14:38:14

I converted it into sigma notation, but I'm not sure how to write that on the computer.

In maple, it is sum((-1)^p*nCp*(n-p)^n, p=0..n)

Last edited by Ma123 (2013-10-02 17:01:33)

bobbym · 2013-10-02 19:09:47

Hi;

You know how to use Maple?

Ma123 · 2013-10-02 19:12:29

Somewhat. Is that summation useful at all? I thought it was pretty cool how it followed coefficients from Pascal's triangle.

bobbym · 2013-10-02 19:18:16

Hi;

Could you show a small example with numbers?

Ma123 · 2013-10-02 19:33:44

Sure, I will find 4! using the summation:

Substuting 4 in for n in sum((-1)^p*nCp*(n-p)^n, p=0..n) gives sum((-1)^p*4Cp*(4-p)^4, p=0..4).

Taking the sum gives (-1)^0*4C0*(4-0)^4 + (-1)^1*4C1*(4-1)^4 + (-1)^2*4C2*(4-2)^4 + (-1)^3*4C3*(4-3)^4 + (-1)^4*4C4*(4-4)^4

= 1*1*256 + (-1)*4*81 + 1*6*16 + (-1)*4*1 + 1*1*0

= 256 - 324 +96 -4 +0

=24

bobbym · 2013-10-02 19:41:23

Hi;

It does work on all the examples I gave it. Here is what it looks like in latex.

Ma123 · 2013-10-02 19:42:37

Yeah! That's what I was looking for. Is this not a known formula?

bobbym · 2013-10-02 19:48:19

Hi;

I would say yes but so what. If you discovered it on your own then it is unique and new for you.

Ma123 · 2013-10-02 19:51:09

Well that is cool

Could it have any practical uses?

bobbym · 2013-10-02 19:52:33

When we speak of practical uses then we get into efficiency.

Is your formula more efficient for a computer then the normal one for say 100! ?

Ma123 · 2013-10-02 19:55:24

I couldn't imagine it would be. Oh well, it was still fun to derive!

Math Is Fun Forum

#1 2013-09-27 18:07:22

Explanation?

#2 2013-09-27 19:43:57

Re: Explanation?

#3 2013-09-27 19:47:02

Re: Explanation?

#4 2013-09-27 21:45:13

Re: Explanation?

#5 2013-09-28 00:05:09

Re: Explanation?

#6 2013-09-28 03:17:07

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#7 2013-09-28 07:37:09

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#8 2013-09-28 07:59:54

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#9 2013-09-28 19:51:32

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#10 2013-09-28 21:57:01

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#11 2013-09-28 22:14:10

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#12 2013-09-30 12:04:04

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#13 2013-09-30 20:19:31

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#14 2013-10-02 13:06:29

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#15 2013-10-02 14:38:14

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#16 2013-10-02 19:09:47

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#17 2013-10-02 19:12:29

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#18 2013-10-02 19:18:16

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#19 2013-10-02 19:33:44

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#20 2013-10-02 19:41:23

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#21 2013-10-02 19:42:37

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#22 2013-10-02 19:48:19

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#23 2013-10-02 19:51:09

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#24 2013-10-02 19:52:33

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#25 2013-10-02 19:55:24

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