So I'm reviewing my rules of radicals prior to teaching it to students, and found out I'm a little rusty on them.
Suppose you hit an answer that ends with a prime number as your radicand. Provided you used mathematically valid reasoning to get there, does this prime-number radicand now suggest that you arrived at the most simplified form, or are there "dead ends" distinct from the right answer?
]]>Much as before, the idea is that in your post, you surround equations with special characters, and MathJax will convert the contained text into an equation for you. There's two types of equation that you can typeset:
\( y=x^2 \)
. Note that we do not support $ signs as most LaTeX users would be familiar with, since this occurs too frequently in text.
\[ y = \int f(x) dx \]
, which we note is exactly what one would type in a usual LaTeX document.
For reference, the old guide is still available and has a number of useful examples for those getting started.
Finally, please note that for legacy posts, the old [math] [/math]
tags will still continue to work and these will display equations as inline. However it's likely that older posts may look different to the way that they did before.
]]>
Firstly, I noticed that the surface area looks like it's the derivative of volume with respect to radius... which come to think of it makes sense as the rate of change in volume at a point in time is that outer spherical shell being added times its thickness.
But secondly I also noticed that the ratio of the two is r/3. As in, as if the average particle in a sphere were only a 1/3 of the way to the outside.
More generally, V/A is in length units. Am I figuring this right? Does V/A represent average distance, root mean square distance, or whatever other measure of central tendency from center to outside?
More generally than that, how is this extrapolated to other shapes? Does V/A represent anything in particular more generally or is its dimensionality usually meaningless?
]]>Urgent!
Hackers have accessed and corrupted this project multiple times despite it being saved only locally on my iPad with myself being the only legitimate user.
I fear that they may be exploiting the method for factoring large semi-primes to crack security.
I need the help of a reputable mathematician to alert CERT about this threat as they are ignoring my attempts to contact them.
https://www.fema.gov/community-emergency-response-teams-cert-action
Figure 1: The Sieve of Erasthonenes
By noting that prime factors occur at regular intervals. ie multiples of 2 reoccur every other number, multiples of 3 reoccur at every third number, etc. we can leverage this periodicity of prime factors to identify all non-prime positions within a predifined large range of natural numbers arranged in an array.
This periodicity of prime factors means that we can apply the concepts of Standing Wave Harmonics to find all composite numbers with a given range based on these wave patterns. Therefore we will also know the relative positions of all prime numbers within the same range.
Figure 2: Patterns of Standing Wave Harmonics
These positions can be stored as a "0" for non-prime or "1" for prime rather than needing to store the entire number, thus alleviating the computational issues with large numbers.
The method
The key is to arrange numbers into rows of N numbers which is defined by the product of the first n primes.
N = P₁x P₂...x Pn
The standing wave like effect of those first prime numbers will cause their prime factors allign into their periodic columns which, in turn causes the primes to allign themselves within the remaining collumns though they will be intermingled with other composite that are defined by primes greater than Pn.
for N = 2x3 = 6
multiples of 2:
xx, 02, xx, 04, xx, 06,
xx, 08, xx, 10, xx, 12,
xx, 14, xx, 16, xx, 18,
xx, 20, xx, 22, xx, 24,
xx, 26, xx, 28, xx, 30,
xx, 32, xx, 34, xx, 36,
multiples of 3:
xx, xx, 03, xx, xx, 06,
xx, xx, 09, xx, xx, 12,
xx, xx, 15, xx, xx, 18,
xx, xx, 21, xx, xx, 24,
xx, xx, 27, xx, xx, 30,
xx, xx, 33, xx, xx, 36,
after we combine those multiples we get:
xx, 02, 03, 04, xx, 06,
xx, 08, 09, 10, xx, 12,
xx, 14, 15, 16, xx, 18,
xx, 20, 21, 22, xx, 24,
xx, 26, 27, 28, xx, 30,
xx, 32, 33, 34, xx, 36,
Therefore we can see that the prime numbers must be located within the remaining columns that are not already occupied by the composite numbers.
For prime numbers greater than Pn their prime factors form diagonal patterns which define the gaps between the prime numbers in those remaining collumns.
xx, xx, xx, xx, 05, xx,
xx, xx, xx, 10, xx, xx,
xx, xx, 15, xx, xx, xx,
xx, 20, xx, xx, xx, xx,
25, xx, xx, xx, xx, 30,
xx, xx, xx, xx, 35, xx,
or
xx, xx, xx, xx, xx, xx,
07, xx, xx, xx, xx, xx,
xx, 14, xx, xx, xx, xx,
xx, xx, 21, xx, xx, xx,
xx, xx, xx, 28, xx, xx,
xx, xx, xx, xx, 35, xx,
combining all waves we get:
xx, 02, 03, 04, 05, 06,
07, 08, 09, 10, xx, 12,
xx, 14, 15, 16, xx, 18,
xx, 20, 21, 22, xx, 24,
25, 26, 27, 28, xx, 30,
xx, 32, 33, 34, 35, 36,
note the numbers in bold are recognized as primes as well as those numbers marked as xx i.e. no prime divisors. The exception is 01 which always shows up as a prime number but you can just ignore it.
With the numbers arranged in a 2D array, we can treat it like a matrix and therefore we can ignore the value of the numbers themselves and only define the relative positions of all composite numbers within the array which of course also defines the relative positions of all of the prime numbers within the array.
Since we are only treating positions of the array as prime (1) or non-prime (0), then we can alleviate the issues with computational complexity of long numbers by only dealing with their primality and position.
Example:
By arranging numbers into rows of N numbers then you will notice that all primes will become aligned into columns that number fewer than columns of composite numbers.
e.g.
For the first 2 primes determine the positions of the first 11 primes within the first 36 numbers.
N = 2x3 = 6
0,1,1,0,1,0,
1,0,0,0,1,0,
1,0,0,0,1,0,
1,0,0,0,1,0,
0,0,0,0,1,0,
1,0,0,0,0,0,
For the first 3 primes we can define the positions of the first 29 primes within the first 100 numbers.
N = 2x3x5 = 30
0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,
1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,
1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,
1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1...
where
1 ≈ prime number position
0 ≈ composite number position
We can scale the Node ranges to the size of the prime numbers which we are focused on by adding more primes to the composite node.
N₂ = 2x3 = 6 N² = 36
N₃ = N₂x5 = 30 N² = 900
N₄ = N₃x7 = 210 N² = 44,100
N₅ = N₄x11 = 2310 N² = 5,336,100
N₆ = N₅x13 = 30,030 N² = 901,800,900
N₇ = N₆x17 = 510,510 N² = 260,620,460,100
N₈ = N₇x19 = 9,699,690 N² = 94,083,986,096,100
N₉ = N₈x23 = 223,092,870 N² = 49,770,428,644,836,900
N₁₀ = N₉x27 = 6,023,507,490 N² = 36,282,642,482,086,100,100
N₁₁ = N₁₀x31 = 186,728,732,190 N² = 34,867,619,425,284,742,196,100
N₁₂ = N₁₁x37 = 6,908,963,091,030 N² = 47,733,770,993,214,812,066,460,900
...
Nn
Below is an excel spreadsheet that uses the first 4 prime numbers to define a node length:
N = 2x3x5x7 = 210
This node was then used to find the first 8555 prime number positions from within 88,200 natural numbers by using the wave patterns of all prime factors to eliminate all composite numbers leaving only primes behind.
I was able to validate the correctness of those prime numbers to be 100% correct compared to downloaded samples.
While this is not important for finding small primes, the method uses the same process for finding prime number positions without needing access or operate on the large numbers themselves.
All prime number positions can be predifined using prime factor wave-like patterns so that accessing large prime numbers should be usable in real time.
While the spreadsheet had over 70 sheets involved, I show sample sheets for two prime factor patterns for 11 and 101.
The third sheet shows the prime number positions are shown in red within a field of natural numbers shown in grey, although the numbers themselves are just there to provide context for the result.
The actual numbers used in formulas are simply a 0 or 1 to show primality as shown in the fourth sheet.
Figure 3: Proof of concept in using an Excel spreadsheet which uses wave patterns of prime factors shown as the patterns of "1" intersecting harmonic columns shown as the white cells in order to derive the positions for the first 8555 prime numbers within the first 88,200 natural numbers.
While creating this kind of matrix for the largest prime numbers in question would be a large undertaking the memory and speed cost can be greatly reduced relative to current methods by dealing with relative prime number positions only without needing to store or perform operations on large prime numbers directly.
It may even be possible to model prime number positions using electronic signal waves or perhaps light wave frequencies at wavelengths that correspond with the prime factor periodicity in order to determining where they intersect with prime harmonic standing waves in order to identify the prime number positions.
Ultimately, with access to large primes in near real time, it should be possible to use methods of infinite compression, by taking large strings of binary data, and converting it to small number keys using universally accessable large mersenne primes to compress and decompress the data at either end.
]]>
I could go on and on, but I'll go onto explain some of the basic syntax of LaTeX.
Syntax
Functions & General Syntax
Basically put, if you want to write a math equation in LaTeX, you just write it. If you wanted f(x) = 3, then bung that between to math tags and you're done, producing [math]f(x)=3[/math]. Don't worry about extra spaces or carriage returns, because in general LaTeX will ignore them. It does get a little more complex than this, but don't worry about that for now. Remember that any letters you type in will be presumed to be some kind of variable and hence will be italicised.
We also have functions to display more complex things like matrices and fractions, and they have the syntax of having a \ before them, usually followed by some kind of argument. For example, \sin will produce the function sin and \frac{num}{denom} will produce a fraction with a specified numerator and denominator. More on these later.
Also remember that LaTeX is case sensitive, so \sigma is NOT the same as \Sigma.
Subscripts and Superscripts
This is perhaps one of the easiest things to do in LaTeX, and one of the most useful. Let's, for the sake of argument, say you wanted to write x^{2}. Then you'd write x^{2}, producing [math]x^2[/math]. Notice that you don't necessarily need the { and } in cases where you only have 1 thing in the index, for example x^2. But it does care if you want to write something like [math]x^{3x+2}[/math]. Subscripts are done similarly, but you use the _ operator instead of ^. If you want both subscript and superscript, then use the syntax x^{2}_{1} - which is equivalent to x_{1}^{2}.
Fractions and functions
As I've mentioned, fractions are generated by using the function \frac{num}{denom}. For example:
[math]\frac{1}{3}[/math]
[math]\frac{7}{x^2}[/math]
If you want smaller fractions, you can use \tfrac, to produce things like [math]\tfrac{1}{2}[/math] which will fit into a line nicely without having to seperate it.
LaTeX has some nice in-built functions like \sin, \cos, etc. I'm not going to write them all down here, but I'll point you to a website at the end of the document that contains them. Likewise, you can write symbols (such as infinity by using \infty) and Greek letters (e.g. \phi, \Sigma, \sigma, etc)
Bracketing
You can get all your usual brackets just by typing them straight in; for instance, (, |, [, etc. However, sometimes they won't be the right size, especially if you want to write something like (1/2)^{n}. You can get around this by using the \left and \right commands, and then placing your favourite brackets after them. For instance, to write (1/2)^{n}, we have:
[math]\left( \frac{1}{2} \right)^{n}[/math]
Integrals, Summations and Limits
Integrals can be produced by using \int, summations by \sum and limits by \lim. You can put limits on them all in the right places by using the normal subscript/superscript commands. For instance:
[math]\int_a^b x^2 \,dx[/math]
[math]\lim_{n\to\infty} \frac{1}{n} = 0[/math]
[math]\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}.[/math]
Summary
There's a lot more things you can do with LaTeX, and I'll try to add to this as time goes by. Have a look at:
http://www.maths.tcd.ie/~dwilkins/LaTeXPrimer/'>http://www.maths.tcd...ns/LaTeXPrimer/ - the LaTeX primer
http://omega.albany.edu:8008/Symbols.html'>http://omega.albany....08/Symbols.html - some symbols that you might find useful.
If you have any questions about the system, send me a PM and I'll try to help
Cheers.
So I'll start off:
1) http://linuxfreak87.googlepages.com/
1) Covers a lot of stuff, Maths and some physics.
2) http://mathworld.wolfram.com/
2) Amazing maths resource, lot of advanced stuff.
3) Basic and advanced maths here. Good tutorials.
4) Again more good tutorias and weekly challanges.
5) http://mathforum.org/dr.math/
5) LOTS of question solutions here, examples too. This one has helped me a lot in the past and still does
6) http://www.ics.uci.edu/~eppstein/junkyard/
6) Lots of fun geometry, useful stuff and interesting stuff here.
7) http://en.wikipedia.org/wiki/Category:Mathematics
7) As always Wikipedia is a great resource for one and all.
8) http://www.research.att.com/~njas/sequences/Seis.html
8) If your interest is number sequences this is the place to go. Useful for research.
9) Equations, equations and yes you guessed - MORE equations. Very useful resorce for reference.
10) http://home.att.net/~numericana/
10) Lots of interesting stuff and some other useful links too.
11) http://www.mcs.surrey.ac.uk/Personal/R.Knott/
11) Lots of interesting stuff, the mysteries of the Fibonacci Numbers etc.
12) http://integrals.wolfram.com/
12) Very useful too, online integral solver!
13) Maths in music, what next?
If you have more to add pease share them
Cheers,
Ryan Jones
]]>a^{3} = 345 / n^{2}
Is there any way to calculate da/dt?
]]>
This issue is offtopic in a thread about electrons so I have started a new thread for discussion.
https://www.scienceforums.net/topic/115443-electrons-how-do-they-work/?tab=comments#comment-1062243
Quote1 hour ago, studiot said:Indeed, which is why I based my figure smack in the middle of the range that can can currently be found.
This range is very large, larger than the range of atomic sizes for instance.
Furthermore zero is a finite number.
Finite is often used to mean a non-zero number
https://en.wikipedia.org/wiki/Finite_number
"In mathematical parlance, a value other than infinite or infinitesimal values and distinct from the value 0"
I prefer the definitions of
Dedekind : not infinite
Russell : Able to be counted using a terminating sequence of natural numbers.
Consider the equation
x2 - 2x + 1= 0
Is the difference between the two roots of this quadratic or infinite?
]]>
This statement is where we could use my own new type of maths I am trying to bring into secondary education through my company "Hydra." I have spent a lot of time planning new approached to maths, at secondary and tertiary levels, so, please bear with me and observe where the strengths are of my new system? This is just an example of how it can be applied, of course.
[9H] * [6K] * [2X] * [1Q] = [108A]. This is because the powers can be applied to the symbols and then taken as '[X1]'. This leads to a simple set of symbols to multiply.
Then, we take the [108A] divided into the symbols on the left that we recognize, being [H], [X] and [K], coming to [HXK] = [108].
This leaves the [DG] being equal to 'the left over bits.' This means we can say [HKX] = [108] and [DG] = [X], so, that means the sum on the left equals [HK2X] = [108].
Or, did I make a mistake somewhere... I am in a heck of a hurry!
]]>This would be [infinity] * [2n] = [3] * [2] = [6] * [x]
Then, [5] * [2] = [10] * [x]
Then, [7] * [2] = [14] * [x]
So, the answer is double prime times by [x].
Or,
[N] * [2] *[X].
]]>I have had the urge for years so if anyone could help that would be great.
]]>
I am interested to know in the context of intrinsic curvature but feel I need to get this concept well understood first.
For example must a mathematical "surface" in a 4-D space be 2-dimensional (like a skin) or is it 3-dimensional (like a volume)?
If it is 3-dimensional,what defines it as a surface?
]]>
This is my final work on the Prime product problem.
I know it is just x^2 * y^2 = PNP^2
However the terms would just cancel out. Instead I have decided to let x^2 equal a pattern of x and PNP. So I just substituted the equation which is more complex and will not equal the right side of the equation for x^2.
In calculus where you have a complex derivative where you let du/dx equal a portion of the derivative so you can understand and simplify the manipulation of the integral. I am instead taking a more complex pattern and leaving it so x^2 does not cancel x squared. By doing this I hope it solves the pattern.
So if you could solve this polynomial equation you would solve the factorization problem.
If you couldn’t solve the polynomial? Well you could just write an algorithm that plugged in Prime numbers from smallest to largest. And because the polynomial is set up to find PNP you would get a feel for the range x was in. I mean, that this time when you try a number how far away the computed value is from PNP is significant.
So if this works it is faster than using division to factor.
But of course I await any disagreeing opinions. I now this problem gets that. But it was my final attempt before moving on to a different problem to pursue.
(((((x^2*PNP^4 + 2*PNP^2 * x^5) + x^8)/ PNP^4 ) - ((1 - x^2/(2*PNP)))) * ((PNP^2/x^2 ))) == PNP^2 Above is the pattern of x^2 * y^2 = PNP^2 It is not to be simplified yet x put and tested in that place. It is faster than division since the equation approaches PNP as the proper x is used. Smallest to largest Prime numbers are to be used. PNP = 85 x = 5 (((((x^2*PNP^4 + 2*PNP^2 * x^5) + x^8)/ PNP^4 ) - ((1 - x^2/(2*PNP)))) * ((PNP^2/x^2 ))) 85 5 4179323/578 N[4179323/578, 14] Sqrt[7230.66262975778546712802768166089965397924`14.] 85.033303062728 ((((x^2*PNP^4 + 2*PNP^2 * x^5) + x^8)/ PNP^4 ) - ((1 - x^2/(2*PNP)))) 4179323/167042 N[4179323/167042, 14] 25.019593874594 ((PNP^2/x^2)) 289 Questions to ask: Is it unique to factors of PNP or does it just give a decimal to all real numbers? Does it work for other values of PNP and x? Is it faster than factoring (recursion)?. Is it just x = x and as so not a useful pattern? Can the error be programmed? Verify then post. That is what I need to do. But I actually believe there is a pattern here. The question is does it work for all PNP and x values. I have many patterns. Some answer some of the questions. But I haven't found a polynomial I can solve after these questions have been answered. For example if this worked, I would need to solve the given equation. And this proves to be challenging. PNP = 85 x = 3 (((((x^2*PNP^4 + 2*PNP^2 * x^5) + x^8)/ PNP^4 ) - ((1 - x^2/(2*PNP)))) * ((PNP^2/x^2 ))) 85 3 847772947/130050 N[847772947/130050, 14] 6518.8231218762 Sqrt[6518.8231218762] 80.739229138481]]>
Now let's say you want there to be a chance of a red marble appearing instead of a green one. In order to make this happen, you must define a separate 'mean time to happen' for the red marbles. But you do not want to change the overall chance of any marble to appear; the combined probability of a red or green marble appearing must be the same as the previous probability of a green one appearing.
Aim at, say, a fifteen percent probability of a red marble appearing instead of a green one, with the current 'mean time to happen' being one thousand seconds. What 'mean times to happen' would you have to assign to both green and red?
]]>Is there any way to calculate from those numbers that the semi-major axis is 26534.9039306654?
]]>
[math]\phi^{(\displaystyle\frac{\pi + \phi}{2})}= \pi[/math]
which manages to roughly approximate [math]\pi[/math]. I then found if you did
[math]\phi^{(\displaystyle\frac{\pi + \phi}{x})}= \pi[/math] with [math]x = 2.000811416[/math],
the equation exactly reached [math]\pi[/math]. But [math]x = 2.000811416[/math] seems too random to me, is there any connection between [math]\pi and \phi[/math] that would produce [math]x = 2.000811416[/math]?
On the slight chance you understood what i said, do you know where [math]x = 2.000811416[/math] can be derived from?
Cheers,
Rob
]]>everything dean has shown was known at the time godel did his proof but no one meantioned any of it
http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
look
godel used the 2nd ed of PM he says
“A. Whitehead and B. Russell, Principia Mathematica, 2nd edition, Cambridge 1925. In particular, we also reckon among the axioms of PM the axiom of infinity (in the form: there exist denumerably many individuals), and the axioms of reducibility and of choice (for all types)”
note he says he is going to use AR
but
Russell following wittgenstien took it out of the 2nd ed due to it being invalid
godel would have know that
russell and wittgenstien new godel used it but said nothing
ramsey points out AR is invalid before godel did his proof
godel would have know ramseys arguments
ramsey would have known godel used AR but said nothing
Ramsey says
Such an axiom has no place in mathematics, and anything which cannot be
proved without using it cannot be regarded as proved at all.
This axiom there is no reason to suppose true; and if it were true, this
would be a happy accident and not a logical necessity, for it is not a
tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY
every one knew AR was invalid
they all knew godel used it
but nooooooooooooo one said -or has said anything for 76 years untill dean
the theorem is a fraud the way godel presents it in his proof it is crap
]]>