Mathematics

Hi, Over the course of the last few weeks I've been attempting to learn about the foundations of mathematics and yet I've been unable to find out what the single most fundamental branch of mathematics is. I've read a few books that covered mathematical logic, mathematical philosophy, and elementry set theory, and yet the answer still eludes me. Could someone please point me in the right direction and tell me what the single most fundamental branch of mathematics is, and what branch of logic does the whole of mathematics stem from? Any help would be very much appreciated. Edit: Clarified question

So, I tried to determine if this is a prime number: [math]2^{257885161} - 1[/math] If this is a prime number, it will be the largest known prime, but if not then okay. Is there a quick way to tell? I tried dividing it by other numbers, yet after a long, long set of calculations I have found no factors yet(though I did this in Mathematica so it may be wrong).

Hello everyone I'd like to know which formula can express the number of unique possibilities in which a number, consistent of [math]m[/math] cyphers and [math]n[/math] different numbers. For example, a number with [math]n=3[/math] and [math]m=4[/math] can be written in 11 unique ways. [math]\left(=\frac{n!+m!}{2}-m\right)[/math] Be [math]n=3[/math] and [math]m=3[/math], then there are 6 possibilities. [math]\left( = 3!\right)[/math] Be [math]n=1[/math] and [math]m=3[/math], then there's only 1 possibility. Thanks! Function

in calculus, I learned that 1/infinity will approaches 0, but not equal to 0 like 1/0 is approaching infinity, but not equal to infinite tan 90 = sin(90)/cos(90) = 1/0 = infinite? 1/tan 90 = 1/infinite = 0? how does cot graph is valid (instead of using the "undefined" value or line up an asymptote there) just because the assumed value of 1/tan(90) = 0, while tan 90 = infinite..

Hello everyone In math class, we solved the next problem: How many numbers, composed out of 5 different digits, varying from 0 to 6, can be formed? So we concluded that the number of possibilities = [math]6\cdot 6\cdot 5\cdot 4\cdot 3[/math], excluding 0 as first digit. Conclusion: [math]N=6\cdot V^4_6[/math]. Then, suddenly, something came in my head; as we were working with variation formulas, I wanted to put the number of possibilities in a formula, solely using variation formulas; the first thing that came in my head was: [math]V^5_7-V^4_6[/math] My math teacher did some thinking and accepted my resolution (she said it was also a good solution), my …

Hello everyone Just messing around with some numbers and formulas, I got to the (pretty useless) idea to simplify [math]\lim_{x\to\infty}{\left[\frac{\log{x^a}+\log{x^b}}{\log{\left(x^a+x^b\right)}}\right]}[/math] The most simplified form I get is (after using the rule of de L'Hôpital once and just using some rules concering calculations with limits) [math]1+\lim_{x\to\infty}{\left[\frac{a\cdot x^b+b\cdot x^a}{a\cdot x^a+b\cdot x^b}\right]}[/math] Is there any way to simplify this even more? Thanks. Function

Hi Can some one help me in proving this statement. Show that in a Boolean algebra, for any a,b and c, if any a≤b, then a∨(b∧c)=b∧(a∨c). I got the answer thanks. Logic is X or X = X

Hello everyone I have this strange feeling that [math]\frac{(a+b)!}{a!+b!}\in\mathbb{N}[/math]. Can this be proven? If yes, how? Never mind. Counterproof: (15+6)!/(15!+6!) [math]\notin\mathbb{N}[/math]

Let's say you have a set A which consists of a subset B. Now, let's say that subset B contains infinite elements within it, but within the bounds that it would be finite in order to fit within the constraints of the finite of A. Would this be possible? I am trying to get me head around the idea that something can be infinitely small, like how there are infinitely many branches within a finite area of a fractal and yet it can't be infinitely larger than the set that contains it because it would seem a paradox would arise. I might need to clarify, so if it is confusing just ask me to clarify this.

Hello In Belgium, you have to pass an exam in order to commence Medicine at university. Here's an example question: Be [math]8x^4+10x^3-7px^2-5qx+9r[/math] dividable by [math]4x^3+7x^2-21x-18[/math], then [math]p+q+r=?[/math] 12 13 14 15 In order to do this, I divided the first polynome by the second, resulting in quotient [math]2x-1[/math] and rest [math](49-7p)x^2+(15-5q)x-(9r-18)[/math] and as the first polynome is dividable by the second one, [math](49-7p)x^2+(15-5q)x-(9r-18)=0[/math] Can someone help me on solving this problem? Thanks. Function

Hello everyone I was wondering if there was a proof for this theorem: Be [math]\Delta x = \sqrt{x}-\sqrt{x-1}[/math] then [math]\lim_{x\to\infty}{\Delta x}=0[/math] Thanks! Oh wait... Would this be a plausible proof: [math]\lim_{x\to\infty}{\left(\sqrt{x}-\sqrt{x-1}\right)}[/math] [math]=\lim_{x\to\infty}{\left[\frac{\left(\sqrt{x}-\sqrt{x-1}\right)\left(\sqrt{x}+\sqrt{x-1}\right)}{\sqrt{x}+\sqrt{x-1}}\right]}[/math] [math]=\lim_{x\to\infty}{\left[\frac{x-(x-1)}{\sqrt{x}+\sqrt{x-1}}\right]}[/math] [math]=\lim_{x\to\infty}{\left[\frac{1}{\sqrt{x}+\sqrt{x-1}}\right]}\left(=\frac{1}{\infty}\right)=0[/math]

Hello everyone I have 2 questions: First of all, I'd like to know if these (notations) are correct: [math]\left(u_n=\sum_{i=0}^{+\infty}{10^i}\right)\equiv (1,0,1,0,1,0,\cdots )\bmod11[/math] [math]\left(u_n=\sum_{i=0}^{+\infty}{10^i}\right)^2\equiv (1,0,1,0,1,0,\cdots )\bmod11[/math] Secondly and finally, I'd like to know if it's possible to prove that the row [math]\left(u_n=\sum_{i=0}^{+\infty}{10^i}\right)=(1,11,111,1111,11111,111111,\cdots)[/math] has no prime numbers in it (11 excluded)? Thank you! -Function P.S. Could the title please be changed to "2 characteristics of the row (1, 11, 111, 1 111, ...)"? Thanks.

Hi everybody, Can anyone name some good solid geometry books please? My field of study is Computer sciences, so I khow basic mathematics and geometry and I have a course on basic manifold geometry, so I'm looking for some advanced books, and because I'm working on computer graphics, I prefere the books to be new and to have some new stuff. My purpose is to find the mathematical discription of shapes in 3D space. In the links below I'm going to show you what my fields of interests are: http://en.wikipedia.org/wiki/B%C3%A9zier_curve http://en.wikipedia.org/wiki/B%C3%A9zier_surface http://en.wikipedia.org/wiki/Graphical_projection http://en.wikipedia.org/wiki…

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So, light can be explained by math by equations, heat can, water droplets can with surface tension, motion can be explained. What can't be, or is life just one big math equation?

Hello everyone, I have found that a certain strange truth value, different from true and false, appears in all of Russell's paradox, the liar paradox, Cantor's diagonal argument, the Grelling-Nelson paradox, the crocodile dilemma, a problem related to the halting problem I came up with, and Berry's paradox. I am certainly not an expert in these problems, but I have not seen my arguments anywhere else. I think that my arguments might be new and that they say something important about these problems. I would really like to see what you think. My article is on my web page as "The Infinitely Recursive Truth Value" at http://www4.ncsu.edu/~tjarmst3/ . I am hoping some pe…

The line of numbers goes like this, from negatives to positives through zero: .....-3 -2 -1 0 +1 +2 +3 ...... The symbols are mirrored around zero. They are "opposite" The negative numbers are less than zero. And minus 3 is evidently a lesser value than +3 Because the arrow of small to big goes from left to right small...................0......................BIG > from-small-to-big does not change direction. but the symbols of the line are mirrored. Minus 3 is the "opposite" of +3, but that is only a symbolic feature. .....-3 -2 -1 0 +1 +2 +3 ...... Though mathematical operations keep the small/big arrow …

I don't know how to ask this question, so I'll just give something; my question: is there a formula which can put this 'phenomenon' in a general form? [math]1^2 = 1[/math] [math]11^2 = 121 \rightarrow 1+2+1=4=2^2[/math] [math]111^2=12321\rightarrow 1+2+3+2+1=9=3^2[/math] [math]1,111^2=1234321\rightarrow 1+2+3+4+3+2+1=16=4^2[/math] [math]\cdots[/math] How can this be put in a general, mathematically more plausible expression? I think that this is a 'crucial' part of it: (I found it about a few minutes ago; sorry if it's wrong, haven't seen summations in school yet..) (To be honest, I'm pretty happy with the result that I found this ^^ ) [math]\left[\sum…

I want to clarify on my idea of language as a computational system. I think that there are a few basic premises behind language acquisition, and that in order to make it computational, a new and unrecognized field that can be described as math is emerging. Here are the basic premises behind mathematical linguistics. One universal premise is pattern recognition. Pattern recognition is the ability to determine the statistical impracticality of randomness. This introduces the other premise, statistics plays a major part in this. In mathematical linguistics, there are four basic computational objects. 1. A set (the role this object plays is unknown, if it plays an…

Just recently I was messing around with fractals, I came across these by self replicating the shape and then rotating it clockwise (or counter clockwise) 90 degrees each time. It created these beautiful 'frost' alike patterns. I was wondering if anybody has meet these fractals before, and if so the name of them. 1st Fractal, rotated anti clockwise 90 degrees each iteration 2nd Fractal, rotated clockwise 90 degrees each iteration Thanks!

Hello everyone I'd like to know if following notations are correct, as first steps in order to get the canonical equations for a parabola, ellips and hyperbola: Parabola Given the focus [math]F\left(0,\frac{p}{2}\right)[/math] and directive [math]d\leftrightarrow y=\frac{-p}{2}[/math] of a parabola [math]\mathcal{P}[/math]. [math]\mathcal{P}:=\forall P(x,y):\left|PF\right|=d(P,d)[/math]. Ellips Given the focusses [math]F_1\left(c,0\right)[/math] and [math]F_2\left(-c,0\right)[/math] of an ellips [math]\mathcal{E}[/math] with main axis [math]2a[/math]. [math]\mathcal{E}:=\forall P(x,y):\left|PF_1\right|+\left|PF_2\right|=2a[/math] Hyperbola …

Hello, I was doing my math team practice packet, and in the writers choose section, I had three questions on the problems. Lets see, for the first question was this: How many integers (10<x<100) are increased by nine when their digits are reversed. So my answer was eight numbers; the numbers 12, 23, 34, 45, 56, 67, 78, and 89, and I know that is right. I had to do it all out by hand, until I noticed a pattern which was you take the last digit of the number, make that the first digit of the next number, and have the number in the tens place be one less than the one in the ones. It took me about five minutes to reach that conclusion. Now my question would…

Hello everyone Pretty stupid: I was wondering how you can write "the derivative of a function f(x) in a" (to use in proofs), using [math]\frac{d}{dx}[/math], not just [math]f'(a)[/math]: [math]\frac{df}{dx}_{x=a}[/math]? [math]\frac{d}{dx}f(a)[/math]? [math]\frac{df}{dx}(a)[/math]? [math]\frac{df}{dx}_a[/math]? Something else? Can someone help me on this one? Thanks. Function.

Does (a+b)1/2 equal a1/2 + b1/2 - (2ab)1/2? If you have ( (a+b)2)1/2, then because of the binomial theorem, then (a+b)2=a2+b2+2ab. Inserting this into the equation, then you get (a2+b2+2ab)1/2, which gives you a+b, since a2+b2+2ab=(a+b)2. If you set x=a2 and y=b2, then you have (x+y+2x1/2y1/2)1/2, which gives you x1/2+y1/2. This is also equal to x1/2 + y1/2 + (2x1/2y1/2)1/2 - (2x1/2y1/2)1/2, since (2x1/2y1/2)1/2 and -(2x1/2y1/2)1/2 cancel out. What if you removed 2x1/2y1/2 from the square root? Then, would you get x1/2 + y1/2 - (2x1/2y1/2)1/2?

I have worked on a number system, that works, in order to label points of a perimeter and be connected to produce vertical symmetry, mostly. I find more relevance in using an odd number of points, dispersed evenly around a geometric perimeter. I gravitate to the results in triangle perimeters, since it is applicable to create other shapes with its multiplication. i.e. Four triangles equals a square, five a pentagon, and six triangles making up a hexagon, and so fourth. In compounding triangles into grids that use the symmetric numbering system, there is formation of wave patterns and very interesting results. I am not, but an enthusiast for math. I have no crident…