the power of adhd and the love of pattern-based mathematics

[miskmath]

so i have adhd spurred by the genetic condition mthfr. i often have trouble sleeping due to my brain working its best at night when there are less distractions.

on one such night i wrote out the sequence of x^2 as such

0 1 4 9 16 25 36 49 64 81 100

bored, i subtracted the former from the latter of adjacent numbers, and then repeated because of what i saw. (periods added for spacing)

0 1 4 9 16 25 36 49 64 81 100
. 1 3 5 7 9.. 11 13 15 17. 19
. . 2 2 2 2 . 2 . 2 . 2 . 2 . 2

now i thought to myself, hm, this is interdasting, what about x^3?

0 1 8 27 64 125 216 etc
. 1.7.19 37 61. 91
.. 6 12 18 24 30
. . 6.. 6....6.....6

after graphing it all out for the next several exponential variants, i found as such:

x^1 → 1
x^2 → 2
x^3 → 6
x^4 → 24
x^5 → 120
x^6 → 720

keen observers will note that as it goes up, the result equals the factorial of said number.

adding results in the highest of them keeping its own result:

x^2 * x^3

0 2 12 36 80 150 252
.2 10 24 44 70 102
...8 14 20 26. 32
.....6...6....6.....6

is there a name for this? can i call it void plane mathematics? do you think this will go anywhere? shall i go on a rant about theoretical mathematics? can someone run the numbers on multiplying these? am i just high on caffeine and should go back to bed? is there a way of finding the row above? are there implications in the real world? is there a connection to integrals from calculus? if so why are the transitional numbers varying so greatly? please i need answers!

 

[lookagain]

User miskmath, it is helpful to us for you to write in
sentences. Please capitalize needed words, such as
the first word in each sentence.

 

[Dr.Peterson]

so i have adhd spurred by the genetic condition mthfr. i often have trouble sleeping due to my brain working its best at night when there are less distractions.

on one such night i wrote out the sequence of x^2 as such

0 1 4 9 16 25 36 49 64 81 100

bored, i subtracted the former from the latter of adjacent numbers, and then repeated because of what i saw. (periods added for spacing)

0 1 4 9 16 25 36 49 64 81 100
. 1 3 5 7 9.. 11 13 15 17. 19
. . 2 2 2 2 . 2 . 2 . 2 . 2 . 2

now i thought to myself, hm, this is interdasting, what about x^3?

0 1 8 27 64 125 216 etc
. 1.7.19 37 61. 91
.. 6 12 18 24 30
. . 6.. 6....6.....6

after graphing it all out for the next several exponential variants, i found as such:

x^1 → 1
x^2 → 2
x^3 → 6
x^4 → 24
x^5 → 120
x^6 → 720

keen observers will note that as it goes up, the result equals the factorial of said number.

adding results in the highest of them keeping its own result:

x^2 * x^3

0 2 12 36 80 150 252
.2 10 24 44 70 102
...8 14 20 26. 32
.....6...6....6.....6

is there a name for this? can i call it void plane mathematics? do you think this will go anywhere? shall i go on a rant about theoretical mathematics? can someone run the numbers on multiplying these? am i just high on caffeine and should go back to bed? is there a way of finding the row above? are there implications in the real world? is there a connection to integrals from calculus? if so why are the transitional numbers varying so greatly? please i need answers!

What you have discovered is part of the finite difference method for finding polynomials. (The term "finite difference method" is also used for something very different, so include the word polynomial if you search for more information!) For more, see here. Other names include successive differences and difference table.

 

[blamocur]

If you are familiar with calculus you might also recall that [imath]n[/imath]-th derivative of any [imath]n[/imath]-th degree polynomial is a constant.

 

[Cubist]

Depending on your level of math, you might be interested in also looking at the Lagrange polynomial , which gives a way of finding the polynomial that exactly fits a set of data points. If you use a polynomial to generate several data points, and you want to get exactly the same polynomial back from the interpolation, then you just need to supply enough data points.

are there implications in the real world? is there a connection to integrals from calculus?

Yes indeed.

One use for this kind of math is in "discrete time systems". This is concerned with the mathematics of sampling a signal and then filtering or analysing it in the digital form. This is used extensively nowadays in computer/ mobile phone technology for recording and playing mp3 files, ogg files, etc, etc

 

[Cubist]

is there a way of finding the row above?

Here's a challenge - if you have the first number from the row above, and all the numbers in the current row, then can YOU work out a way to find the whole row above?

 

[Deleted member 4993]

so i have adhd spurred by the genetic condition mthfr. i often have trouble sleeping due to my brain working its best at night when there are less distractions.

on one such night i wrote out the sequence of x^2 as such

0 1 4 9 16 25 36 49 64 81 100

bored, i subtracted the former from the latter of adjacent numbers, and then repeated because of what i saw. (periods added for spacing)

0 1 4 9 16 25 36 49 64 81 100
. 1 3 5 7 9.. 11 13 15 17. 19
. . 2 2 2 2 . 2 . 2 . 2 . 2 . 2

now i thought to myself, hm, this is interdasting, what about x^3?

0 1 8 27 64 125 216 etc
. 1.7.19 37 61. 91
.. 6 12 18 24 30
. . 6.. 6....6.....6

after graphing it all out for the next several exponential variants, i found as such:

x^1 → 1
x^2 → 2
x^3 → 6
x^4 → 24
x^5 → 120
x^6 → 720

keen observers will note that as it goes up, the result equals the factorial of said number.

adding results in the highest of them keeping its own result:

x^2 * x^3

0 2 12 36 80 150 252
.2 10 24 44 70 102
...8 14 20 26. 32
.....6...6....6.....6

is there a name for this? can i call it void plane mathematics? do you think this will go anywhere? shall i go on a rant about theoretical mathematics? can someone run the numbers on multiplying these? am i just high on caffeine and should go back to bed? is there a way of finding the row above? are there implications in the real world? is there a connection to integrals from calculus? if so why are the transitional numbers varying so greatly? please i need answers!

This process is taught in a course of "numerical analysis".