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Do primes become more or less frequent as you go further out on the number line? That is, are there more or fewer primes between 1 and 1,000,000 than between 1,000,000 and 2,000,000?
A proof or pointer to a proof would be appreciated. |
Distribution of primes? |
I keep looking at this picture and its driving me crazy. How can the smaller circle travel the same distance when its circumference is less than the entire wheel?
![demonstration image][1]
[1]: http://mathworld.wolfram.com/images/gifs/AristotlesWheel.gif |
How does the wheel paradox work? |
Since constructive mathematics allows us to avoid things like Russell's Paradox, then why don't they replace traditional proofs? How do we know the "regular" kind of mathematics are free of paradox without a proof construction? |
Aren't constructive math proofs more "sound"? |
Prime numbers are numbers with no factors other than one and itself.
Factors of a number are always lower or equal to than a given number; so, the larger the number is, the larger the pool of "possible factors" that number might have.
So the larger the number, it seems like the less likely the number is to be a p... |
Is there possibly a largest prime number? |
What exactly does it mean for a function to be "well-behaved"? |
I am learning geometric algebra, and it is incredible how much it helps me understand other branches of mathematics. I wish I had been exposed to it earlier.
Additionally I feel the same way about enumerative combinatorics.
What are some less popular mathematical subjects that you think should be more popular?
... |
Often in my studies (economics) the assumption of a "well-behaved" function will be invoked. I don't exactly know what that entails (I think twice continuously differentiability is one of the requirements), nor do I know why this is necessary (though I imagine the why will depend on each case).
Can someone explain i... |
Okay, so hopefully this isn't too hard or off-topic. Let's say I have a very simple LP filter, with a position variable and a cutoff variable (between 0 and 1). So, in every step, position = position*(1-c)+input*c. Basically, it moves a percentage of the distance between the current position and then input value, store... |
Top Prime's Divisors'
Product (Plus one)'s factors are...?
Q.E.D Bitches
[XKCD][1]
[1]: http://xkcd.com/622/ |
Are real numbers *"real"*? It's not even computationally possible to compare two real numbers for equality!
Interestingly enough, it is shown in Abstract Algebra courses that the idea of complex numbers arises naturally from the idea of real numbers - you could not say, for instance, that the real numbers are *vali... |
Okay, so hopefully this isn't too hard or off-topic. Let's say I have a very simple LP filter, with a position variable and a cutoff variable (between 0 and 1). So, in every step, `position = position*(1-c)+input*c`. Basically, it moves a percentage of the distance between the current position and then input value, sto... |
<i>The sum of two Gaussian variables is another Gaussian.</i>
It seems natural, but I could not find a proof using Google.
What's a short way to prove this?
Thanks! |
The Weyl equidistribution theorem states that the sequence of fractional parts $\{n \xi\}$, $n = 0, 1, 2, \dots$ is uniformly distributed for $\xi$ irrational.
This can be proved using a bit of ergodic theory, specifically the fact that an irrational rotation is uniquely ergodic with respect to Lebesgue measure. It... |
How do you prove that $p(n \xi)$ for $\xi$ irrational and $p$ a polynomial is uniformly distributed modulo 1? |
The normal train of logic goes like this:
- Prime numbers have two divisors.
- 1 has only one divisor
- Therefore, 1 is not prime
There are some more complex subtleties to that, but for most purposes, that reasoning will do. This comes from the accepted definition of a prime number.
Why is this the accepted... |
Suppose there are two cards each with a positive real number and with one twice the other and each with value equal to the number on the card. You are given one of the cards and and opportunity to swap. If you choose to swap, you are just getting another random number, and so your expected gain should be 0. However, th... |
Given a point's coordinates (x,y), what is the procedure for determining if it lies within a polygon whose vertices are (x1,y1), (x2,y2), ..., (xn,yn)? |
How do you determine if a point sits inside a polygon? |
When I was that age, I discovered Raymond Smullyan's classic logic puzzle books in the library (such as *What is the name of this book?*), and really got into it. I remember my amazement when I first understood how a complicated logic puzzle could become trivial, just symbolic manipulation really, with the right notati... |
I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs? |
Are the "proofs by contradiction" weaker than other proofs? |
What I really don't like about all the above answers, is the underlying assumption that `1/3=0.3333...`, How do you know that?. It seems to me like assuming the something which is already known.
A proof I really like is:
0.9999... x 10 = 9.999999...
0.9999... x 9 + 0.99999.... = 9.99999....
... |
The following is a quote from *Surely you're joking, Mr. Feynman* . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challange? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Appare... |
I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs?
Edit: To clarify the que... |
Can someone give a simple explanation for why the series 1 + 1/2 + 1/3 + ... doesn't converge, but just grows very slowly?
I'd prefer an easily comprehensible explanation rather than a rigorous proof of the type I could get from an undergraduate text book. |
Why does the series 1/1 + 1/2 + 1/3 + ... not converge? |
Let's say I know 'X' is a Gaussian Variable.
Moreover, I know 'Y' is a Gaussian Variable and Y=X+Z.
Let's X and Z are Independent.
How can I prove Z is a Gaussian Variable?
It's easy to show the other way around (X, Z Orthogonal and Normal hence create a Gaussian Vector hence any Linear Combination of the two... |
Let's say I know 'X' is a Gaussian Variable.
Moreover, I know 'Y' is a Gaussian Variable and Y=X+Z.
Let's X and Z are Independent.
How can I prove Y is a Gaussian Random Variable if and only if Z is a Gaussian R.V.?
It's easy to show the other way around (X, Z Orthogonal and Normal hence create a Gaussian Vec... |
If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be? |
What is the single most influential book every mathematician should read? |
Is there a way of taking a number known to limited precision (e.g. 1.644934) and finding out an "interesting" real number (e.g. $pi^2/6$) that's close to it?
I'm thinking of something like Sloane's Online Encyclopedia of Integer Sequences, only for real numbers.
The intended use would be: write a program to calcu... |
Is there a real number lookup algorithm or service? |
Try [Wolfram Alpha][1]. It actually does sequences as well.
[1]: http://www.wolframalpha.com/input/?i=1.644934 |
The following is a quote from *Surely you're joking, Mr. Feynman* . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge? Theorems should be totally counter-intuitive, and be easily translatable to everyday language. (Appare... |
I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get them "hooked".
So the question: What is a good book, for laymen, which ... |
[Journey Through Genius][1]
![alt text][2]
A brilliant combination of interesting storytelling and large amounts of actual Mathematics. It took my love of Maths to a whole other level.
[1]: http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X/ref=sr_1_1?ie=UTF8&s=books&qid=127969... |
Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?
It seems as though formerly 0 was considered in the set of natural numbers but now it seems more common to see definitions saying that the natural numbers are precisely ... |
Is 0 a natural number? |
Below is a visual proof (!) that 32.5 = 31.5. How could that be?
![alt text][1]
[1]: http://farm1.static.flickr.com/48/152036443_ca28c8d2a1_o.png |
Is 32.5 = 31.5 ? |
John D Cook writes <a href="http://www.johndcook.com/blog/">The Endeavor</a>
One of the MathWorks blogs: <a href="http://blogs.mathworks.com/loren/">Loren on the Art of Matlab</a>
... a few more:
<a href="http://unimodular.net/blog/?p=185">eon</a>
<a href="http://cameroncounts.wordpress.com/">Peter Cameron'... |
Simple answer: sometimes yes, sometimes no, it's usually stated (or impled by notation). From the [Wikipedia article](http://en.wikipedia.org/wiki/Natural_number):
> In mathematics, there are two
> conventions for the set of natural
> numbers: it is either the set of
> positive integers {1, 2, 3, ...}
> accordi... |
Are there any simple methods for calculating the digits of Pi? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that can be computed by hand in order to compute the first few digits? |
The arithmetic hierarchy defines the Pi-1 formulae of arithmetic to be formulae that are provably equivalent to a formula in [prenex normal form][1] that only has universal quantifiers, and Sigma-1 if it is provably equivalent to a prenex normal form with only existential quantifiers.
A formula is Delta-1 if it is b... |
Why are Delta-1 sentences of arithmetic called recursive? |
Any homomorphism φ between the rings Z_18 and Z_15 is completely defined by φ(1). So from
0 = φ(0) = φ(18) = φ(18 * 1) = 18 * φ(1) = 15 * φ(1) + 3 * φ(1) = 3 * φ(1)
we get that φ(1) is either 5 or 10. But how can I prove or disprove that these 2 are valid homomorphisms? |
What are all the homomorphisms between the rings Z_18 and Z_15? |
For basic mathematics [mathpage][1] has a long course, including the list below. Don't be put off by the elementary school style lesson names, it does go into some depth with each one.
<pre>
Lesson 1 Reading and Writing Whole Numbers
Lesson 2 The Meaning of Decimals
Lesson 3 Multiplying and Dividing
Les... |
How come 32.5 = 31.5? |
The arithmetic hierarchy defines the Π_1 formulae of arithmetic to be formulae that are provably equivalent to a formula in [prenex normal form][1] that only has universal quantifiers, and Σ_1 if it is provably equivalent to a prenex normal form with only existential quantifiers.
A formula is Δ_1 if i... |
How can I show that (n-1)! is congruent to -1 (mod n) iff n is prime?
Thanks. |
In category theory, a subobject of X is defined as an object Y with a monomorphism, from Y to X. If A is a subobject of B, and B a subobject of A, are they isomorphic? It is not true in general that having monomorphisms going both ways between two objects is sufficient for isomorphy, so it would seem the answer is no.
... |
If A is a subobject of B, and B a subobject of A, are they isomorphic? |
I've read about about [higher-order logics](http://en.wikipedia.org/wiki/Higher_order_logic) (i.e. those that build on first-order predicate logic) but am not too clear on their applications. While they are capable of expressing a greater range of proofs (though never *all*, by Godel's Incompleteness theorem), they are... |
When I tried to approximate $\int_0^1 (1-x^7)^(1/5) - (1-x^5)^(1/7) dx$, I kept getting answers that were really close to 0, so I think it might be true. But why? When I [ask Mathematica][1], I get a bunch of symbols I don't understand!
[1]: http://integrals.wolfram.com/index.jsp?expr=%281-x%5E7%29%5E%281%2F5%2... |
why is $\int_0^1 (1-x^7)^(1/5) - (1-x^5)^(1/7) dx=0$? |
Why is $\int_0^1 (1-x^7)^(1/5) - (1-x^5)^(1/7) dx=0$? |
Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain **to a non-mathematician** that complex numbers are necessary and m... |
Are complex numbers quantities? |
What is a unital homomorphism? Why are they important? |
Which is the single best book for [Number Theory][1] that everyone who loves Mathematics should read?
[1]: http://en.wikipedia.org/wiki/Number_theory |
**Background:** Many (if not all) of the transformation matrices used in 3D computer graphics are 4x4, including the three values for `x`, `y` and `z`, plus an additional term which usually has a value of 1.
Given the extra computing effort required to multiply 4x4 matrices instead of 3x3 matrices, there must be a s... |
Why are 3D transformation matrices [4]x[4] instead of [3]x[3]? |
> even though 3x3 matrices should (?) be sufficient to describe points and transformations in 3D space.
No, they aren't enough! Suppose you represent points in space using 3D vectors. You can transform these using 3x3 matrices. But if you examine the definition of matrix multiplication you should see immediately tha... |
Similar to the Monty Hall problem, but trickier: at the latest Gathering 4 Gardner, Gary Foshee asked
> I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?
We are assuming that births are equally distributed during the week, that every child is a boy or girl with probabi... |
Wikipedia has a closed-form function called "Binet's formula".
http://en.wikipedia.org/wiki/Fibonacci_number#Relation_to_the_golden_ratio
![$F\left(n\right) = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}$][1]
This is based on the Golden Ratio.
[1]: http://chart.apis.google.com/chart?cht=tx&chl=F%5Cleft(n%5... |
It's not clear what the question is asking. Is it asking for logical relations among the properties? e.g. a relation that is transitive and symmetric MUST be reflexive (unless the relation holds between no two objects).
Or is it asking for mathematical relations that have these properties? ">" is transitive, asymmet... |
[Godel's proof][1] is one I enjoyed. It's was a little hard to understand but there is nothing in this book that makes it inaccessible to someone without a strong math background.
Keeping with Godel in the title, [Godel, Escher, Bach: An Eternal Golden Braid][2] while not just about math was a good read (a bit long ... |
One's that were suggested to me by my Calculus teacher in High School. Even my wife liked them and she hates math now:
- [The Education of T.C. Mits: What
modern mathematics means to you][1]
- [Infinity: Beyond the Beyond the
Beyond][1]
Written and illustrated(Pictures are great ;p) by a couple: L... |
I'm very interested in Computer Science (computational complexity, etc.). I've already finished a University course in the subject (using Sipser's "Introduction to the Theory of Computation").
I know the basics, i.e. Turing Machines, Computability (Halting problem and related reductions), Complexity classes (time an... |
I am vaguely familiar with the broad strokes of the development of group theory, first when ideas of geometric symmetries were studied in concrete settings without the abstract notion of a group available, and later as it was formalized by Cayley, Lagrange, etc (and later, infinite groups being well-developed). In any ... |
There's some of the history here in Bourbaki's _Commutative Algebra,_ in the appendix. Basically, a fair bit of ring theory was developed for algebraic number theory. This in turn was because people were trying to prove Fermat's last theorem.
Why's this? Let $p$ be a prime. Then the equation $x^p + y^p = z^p$ can... |
One's that were suggested to me by my Calculus teacher in High School. Even my wife liked them and she hates math now:
- [The Education of T.C. Mits: What
modern mathematics means to you][1]
- [Infinity: Beyond the Beyond the
Beyond][1]
Written and illustrated(Pictures are great ;p) by a couple: L... |
[Paul Nahin][1] has a number of accessible mathematics books written for non-mathematicians, the most famous being
* [An Imaginary Tale: The Story of $\sqrt{-1}$][2]
* [Dr. Euler's Fabulous Formula (Cures Many Mathematical Ills!)][3]
[Professor Ian Stewart][4] also has many books which give laymen explanations o... |
I'm looking to find out if there's any easy way to calculate the number of ways to tile a $3 \times 2n$ rectangle with dominoes. I was able to do it with the two codependent recurrences
f(0) = g(0) = 1
f(n) = f(n-1) + 2g(n-1)
g(n) = f(n) + g(n-1)
where $f(n)$ is the actual answer and $g(n)$ is a hel... |
I'm looking to find out if there's any easy way to calculate the number of ways to tile a $3 \times 2n$ rectangle with dominoes. I was able to do it with the two codependent recurrences
f(0) = g(0) = 1
f(n) = f(n-1) + 2g(n-1)
g(n) = f(n) + g(n-1)
where $f(n)$ is the actual answer and $g(n)$ is a hel... |
[Paul Nahin][1] has a number of accessible mathematics books written for non-mathematicians, the most famous being
* [An Imaginary Tale: The Story of $\sqrt{-1}$][2]
* [Dr. Euler's Fabulous Formula (Cures Many Mathematical Ills!)][3]
[Professor Ian Stewart][4] also has many books which each give laymen overviews... |
I know that the Fibonacci sequence can be described via the Binet's formula.
However, I was wondering if there was a similar formula for $n!$.
**Is this possible? If not, why not?** |
Is there a closed-form equation for $n!$? If not, why not? |
what is the best book or script on category theory? |
Are x × 0 = 0 and x × 1 = 1 and -(-x) = x axioms? |
Are x × 0 = 0 and x × 1 = x and -(-x) = x axioms? |
When it comes to textbooks, the Kenneth Rosen text [Discrete Mathematics and its Applications][1] is highly recommended. I was first introduced to it at my university, but I've seen it cited in several places.
[1]: http://www.amazon.com/gp/product/0073229725/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=486539851&pf_rd_s=lpo-to... |
The question is more profound than is initially seems, and is really about algebraic structures. The first question you have to ask yourself is *where you're working*:
In general, addition and multiplication are defined on a *structure*, which in this case is a *set* (basically a collection of "things") with two *o... |
The fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after `n` generations if they each produce a new pair every generation. Is there any *other* reason you would care about the fibonacci sequence? |
what is the best book or lecture notes on category theory? |
I know that the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges. I also know that the sum of the inverse of prime numbers 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ... diverges too, even more slowly since it's O(log log n).
But I think I read that if we consider the numbers whose decimal representation do not have a certain... |
I know that the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges. I also know that the sum of the inverse of prime numbers 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ... diverges too, even if really slowly since it's O(log log n).
But I think I read that if we consider the numbers whose decimal representation do not have a ce... |
One of the first things ever taught in a differential calculus class:
- The derivative of sin(x) is cos(x)
- The derivative of cos(x) is -sin(x)
This leads to a rather neat (and convenient?) chain of derivatives:
<pre>
sin(x)
cos(x)
-sin(x)
-cos(x)
sin(x)
...
</pre>
An analysis of the shape of their... |
There's some of the history here in Bourbaki's _Commutative Algebra,_ in the appendix. Basically, a fair bit of ring theory was developed for algebraic number theory. This in turn was because people were trying to prove Fermat's last theorem.
Why's this? Let $p$ be a prime. Then the equation $x^p + y^p = z^p$ can... |
As a Physics Major, I would like to propose an answer that comes from my understanding of seeing sine and cosine in the real world.
In doing this, I will examine uniform circular motion.
Because of the point-on-a-unit-circle definition of sine and cosine, we can say that:
r(t) = < cos(t), sin(t) >
Is a ... |
As a Physics Major, I would like to propose an answer that comes from my understanding of seeing sine and cosine in the real world.
In doing this, I will examine uniform circular motion.
Because of the point-on-a-unit-circle definition of sine and cosine, we can say that:
r(t) = < cos(t), sin(t) >
Is a ... |
0^x = 0, x^0 = 1
both are true when x > 0
what happens when x=0? undefined, because there is no way to chose one definition over the other.
Some people define 0^0 = 1 in their books, like Knuth, because 0^x is less 'useful' than x^0. |
We will call the set of all positive even numbers `E` and the set of all positive integers `N`.
At first glance, it seems obvious that `E` is smaller than `N`, because for `E` is basically `N` with half of its terms taken out. The size of `E` is the size of `N` divided by two.
You could see this as, for every it... |
What is larger -- the set of all positive even numbers, or the set of all positive integers? |
One of the first things ever taught in a differential calculus class:
- The derivative of sin(x) is cos(x)
- The derivative of cos(x) is -sin(x)
This leads to a rather neat (and convenient?) chain of derivatives:
<pre>
sin(x)
cos(x)
-sin(x)
-cos(x)
sin(x)
...
</pre>
An analysis of the shape of their... |
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