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Can someone explain to me how there can be different kinds of infinities?
I was reading "The man who loved only numbers" by Paul Hoffman and came across the concept of countable and uncountable infinities, but they're only words to me.
Any help would be appreciated.
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mathfactor is one I listen to. Does anyone else have a recommendation?
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I have read a few proofs that $\sqrt{2}$ is irrational.
I have never, however, been able to really grasp what they were talking about.
Is there a simplified proof that $\sqrt{2}$ is irrational?
|
I'm looking for a nice, quick online graphing tool. The ability to link to, or embed the output would be handy, too.
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I was reading up on the Fibonacci Sequence, $1,1,2,3,5,8,13,\ldots $ when I noticed some were able to calculate specific numbers. So far I've only figured out creating an array and counting to the value, which is incredibly simple, but I reckon I can't find any formula for calculating a Fibonacci number based on it's p... |
I'm told by smart people that
$$0.999999999\ldots=1$$
and I believe them, but is there a proof that explains why this is?
|
Given the semi-major axis and a flattening factor, is it possible to calculate the semi-minor axis?
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In mathematics, there seem to be a lot of different types of numbers. What exactly are:
Real numbers
Integers
Rational numbers
Decimals
Complex numbers
Natural numbers
Cardinals
Ordinals
And as workmad3 points out, some more advanced types of numbers (I'd never heard of)
Hyper-reals
Quaternions
Imaginary numbers
Ar... |
By matrix-defined, I mean
$$\left<a,b,c\right>\times\left<d,e,f\right> = \left|
\begin{array}{ccc}
i & j & k\\
a & b & c\\
d & e & f
\end{array}
\right|$$
...instead of the definition of the product of the magnitudes multiplied by the sign of their angle, in the direction orthogonal)
If I try cross productin... |
I'm looking for an online or software calculator that can show me the history of items I typed in, much like an expensive Ti calculator.
Can you recommend any?
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What length of rope should be used to tie a cow to an exterior fence post of a circular field so that the cow can only graze half of the grass within that field?
updated: To be clear: the cow should be tied to a post on the exterior of the field, not a post at the center of the field.
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I've learned that the dot product is just one of many possible inner product spaces. Can someone explain this concept? When is it useful to define it as something other than the dot product?
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This is simply not clicking for me. I'm currently learning math during the summer vacation and I'm on the chapter for relations and functions.
There are five properties for a relation:
Reflexive - $R \rightarrow R$
Symmetrical - $R \rightarrow S$ ; $S \rightarrow R$
Antisymmetrical - $R \rightarrow S$ && ($R \rightar... |
I recently discovered Benford's Law. I find it very fascinating. I'm wondering what are some of the real life uses of Benford's law. Specific examples would be great.
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Given two points $p_1$ , $p_2$ around the origin $(0,0)$ in $2D$ space, how would you calculate the angle from $p_1$ to $p_2$?
How would this change in $3D$ space?
|
I hear a lot about Elliptic Curve Cryptography these days, but I'm still not quite sure what they are or how they relate to crypto...
|
I've recently started reading about Quaternions, and I keep reading that for example they're used in computer graphics and mechanics calculations to calculate movement and rotation, but without real explanations of the benefits of using them.
I'm wondering what exactly can be done with Quaternions that can't be done as... |
Let's say I know $X$ is a Gaussian variable.
Moreover, I know $Y$ is a Gaussian variable and $Y=X+Z$.
Let $X$ and $Z$ be independent.
How can I prove that $Y$ is a Gaussian random variable if and only if $Z$ is a Gaussian R.V.?
It's easy to show the other way around ($X$ and $Z$ are orthogonal and normal, hence create ... |
Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
|
What purposes do the Dot and Cross products serve?
Do you have any clear examples of when you would use them?
|
I have the one or other interesting Math blog in my feedreader that I follow. It would be interesting to compile a list of Math blogs that are interesting to read, and do not require research-level math skills.
I'll start with my entries:
Division By Zero
Tanya Khovanova’s Math Blog
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I'm talking in the range of 10-12 years old, but this question isn't limited to only that range.
Do you have any advice on cool things to show kids that might spark their interest in spending more time with math? The difficulty for some to learn math can be pretty overwhelming. Do you have any teaching techniques that ... |
Not sure if this is the place for it, but there are similar posts for podcasts and blogs, so I'll post this one. I'd be interested in seeing a list of online resources for mathematics learning.
As someone doing a non-maths degree in college I'd be interested in finding some resources for learning more maths online, mos... |
I am looking for an accurate algorithm to calculate the error function
$$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\ dt$$
I have tried using the following formula,
(Handbook of Mathematical Functions, formula 7.1.26), but the results are not accurate enough for the application.
|
I am looking for a reference (book or article) that poses a problem that seems to be a classic, in that I've heard it posed many times, but that I've never seen written anywhere: that of the possibility of a man in a circular pen with a lion, each with some maximum speed, avoiding capture by that lion.
References to p... |
This is in relation to the Euler Problem $13$ from http://www.ProjectEuler.net.
Work out the first ten digits of the sum of the following one-hundred $50$-digit numbers.
$37107287533902102798797998220837590246510135740250$
Now, this was my thinking:
I can freely discard the last fourty digits and leave the last ten.
$... |
It is $10$ o'clock, and I have a box.
Inside the box is a ball marked $1$.
At $10$:$30$, I will remove the ball marked $1$, and add two balls, labeled $2$ and $3$.
At $10$:$45$, I will remove the balls labeled $2$ and $3$, and add $4$ balls, marked $4$, $5$, $6$, and $7$.
$7.5$ minutes before $11$, I will remove ... |
I keep seeing this symbol $\nabla$ around and I know enough to understand that it represents the term "gradient." But what is a gradient? When would I want to use one mathematically?
|
If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already:
Let $a = b$.
Then $a^2 = ab$
$a^2 - b^2 = ab - b^2$
Factor to $(a-b)(a+b) = b(a-b)$
Then divide out $(a-b)$ to get $a+b = b$
Since $a = b$, then $b+b = b$
The... |
I keep hearing about this weird type of math called calculus. I only have experience with geometry and algebra. Can you try to explain what it is to me?
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Why is $1$ not considered a prime number?
Or, why is the definition of prime numbers given for integers greater than $1$?
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I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful.
Is there any good examples of their uses outside academia?
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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 can ... |
Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.
|
I was playing around with the squares and saw an interesting pattern in their differences.
$0^2 = 0$
+ 1
$1^2 = 1$
+ 3
$2^2 = 4$
+ 5
$3^2 = 9$
+ 7
$4^2 = 16$
+ 9
$5^2 = 25$
+ 11
$6^2 = 36$
etc.
(Also, in a very related question, which major Math Research Journal should I... |
I have a square that's $10\mathrm{m} \times 10\mathrm{m}$. I want to cut it in half so that I have a square with half the area. But if I cut it from top to bottom or left to right, I don't get a square, I get a rectangle!
I know the area of the small square is supposed to be $50\mathrm{m}^{2}$, so I can use my calcul... |
It seems like functions that are continuous always seem to be differentiable, to me. I can't imagine one that is not. Are there any examples of functions that are continuous, yet not differentiable?
The other way around seems a bit simpler -- a differentiable function is obviously always going to be continuous. But ... |
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 meani... |
In the past, I've come across statements along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations:
addition/subtraction
multiplication/division
raising to powers and roots
trigonometric functions
exponential functions
logarithmic functions
... |
I've heard of some other paradoxes involving sets (i.e., "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand.
Why is "the set of all sets" a paradox? It seems like it would be fine, to me. There is nothing paradoxical about a set cont... |
I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! How could somebody guess something like this for the formula?
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Consider the task of generating random points uniformly distributed within a circle of a given radius $r$ that is centered at the origin. Assume that we are given a random number generator $R$ that generates a floating point number uniformly distributed in the range $[0, 1)$.
Consider the following procedure:
Generate... |
I have a collection of 3D points in the standard $x$, $y$, $z$ vector space. Now I pick one of the points $p$ as a new origin and two other points $a$ and $b$ such that $a - p$ and $b - p$ form two vectors of a new vector space. The third vector of the space I will call $x$ and calculate that as the cross product of th... |
Basic definitions: a tiling of d-dimensional Euclidean space is a decomposition of that space into polyhedra such that there is no overlap between their interiors, and every point in the space is contained in some one of the polyhedra.
A vertex-uniform tiling is a tiling such that each vertex figure is the same: each v... |
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.
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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?
|
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?
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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 prime.
... |
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 it t... |
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?
|
Okay, so hopefully this isn't too hard or off-topic. Let's say I have a very simple lowpass filter (something that smooths out a signal), and the filter object has a position variable and a cutoff variable (between 0 and 1). In every step, a value is put into the following bit of pseudocode as "input": position = posit... |
The sum of two Gaussian variables is another Gaussian.
It seems natural, but I could not find a proof using Google.
What's a short way to prove this?
Thanks!
Edit: Provided the two variables are independent.
|
Suppose there are two face down cards each with a positive real number and with one twice the other. Each card has value equal to its number. You are given one of the cards (with value $x$) and after you have seen it, the dealer offers you an opportunity to swap without anyone having looked at the other card.
If you ch... |
Given the coordinates of a point $(x, y)$, what is a procedure for determining if it lies within a polygon whose vertices are $(x_1, y_1), (x_2, y_2), \ldots , (x_n,y_n)$?
|
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?
Is there any reason that one... |
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. (Apparentl... |
Can someone give a simple explanation as to why the harmonic series
$$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$
doesn't converge, on the other hand it grows very slowly?
I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks... |
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?
|
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. $\displaystyle\frac{\pi^2}{6}$) 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 ... |
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 are good books, for laymen, which tea... |
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 ... |
Below is a visual proof (!) that $32.5 = 31.5$. How could that be?
(As noted in a comment and answer, this is known as the "Missing Square" puzzle.)
|
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 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 ... |
Any homomorphism $φ$ between the rings $\mathbb{Z}_{18}$ and $\mathbb{Z}_{15}$ is completely defined by $φ(1)$. So from
$$0 = φ(0) = φ(18) = φ(18 \cdot 1) = 18 \cdot φ(1) = 15 \cdot φ(1) + 3 \cdot φ(1) = 3 \cdot φ(1)$$
we get that $φ(1)$ is either $5$ or $10$. But how can I prove or disprove that these two are valid ho... |
How can I show that $(n-1)!\equiv-1 \pmod{n}$ if and only if $n$ 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 th... |
I've read about about higher-order logics (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 often said to be less "well-behaved".
Mathematicians... |
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, I get a bunch of symbols I don't understand!
|
What is a unital homomorphism? Why are they important?
|
So you have $2$ dice and you want to get at least a $1$ or a $5$ (on the dice not added). How do you go about calculating the answer for this question.
This question comes from the game farkle.
|
Books on Number Theory for anyone who loves Mathematics?
(Beginner to Advanced & just for someone who has a basic grasp of math)
|
Background: Many (if not all) of the transformation matrices used in $3D$ computer graphics are $4\times 4$, 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 $4\times 4$ matrices instead of $3\times 3$ matric... |
I'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago.
|
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 and s... |
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 ... |
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 helper function that r... |
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?
|
What are the best books and lecture notes on category theory?
|
Context: Rings.
Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms?
Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...
|
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?
|
I know that the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges. I also know that the sum of the inverse of prime numbers $\frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \cdots$ diverges too, even if really slowly since it's $O(\log \log n)$.
But I think I read that if we consider the numbers ... |
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:
sin(x)
cos(x)
-sin(x)
-cos(x)
sin(x)
...
An analysis of the shape of their graphs confirms some p... |
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 item in E, two items in ... |
I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are mathematically. Can anyone explain what a monad is using as little category theory as possible?
|
Say we have a function $\displaystyle f(z)=\sum_{n=0}^\infty a_n z^n$ with radius of convergence $R>0$. Why is the radius of convergence only $R$? Can we conclude that there must be a pole, branch cut or discontinuity for some $z_0$ with $|z_0|=R$? What does that mean for functions like
$$f(z)=\begin{cases}
0 & \text... |
We define Lie algebras abstractly as algebras whose multiplication satisfies anti-commutativity and Jacobi's Identity. A particular instance of this is an associative algebra equipped with the commutator bracket: $[a,b]=ab-ba$. However, the notation suggests that this bracket is the one we think about. Additionally, t... |
It appears to me that only Triangles, Squares, and Pentagons are able to "tessellate" (is that the proper word in this context?) to become regular 3D convex polytopes.
What property of those regular polygons themselves allow them to faces of regular convex polyhedron? Is it something in their angles? Their number of ... |
Two questions:
Find a bijective function from $(0,1)$ to $[0,1]$. I haven't found the solution to this since I saw it a few days ago. It strikes me as odd--mapping a open set into a closed set.
$S$ is countable. It's trivial to find a bijective function $f:\mathbb{N}\to\mathbb{N}\setminus S$ when $|\mathbb{N}| = |\mat... |
I just got my statistics test back and I am totally confused about one of the questions!
A study was done that took a simple random sample of 40 people and measured whether
the subjects were right-handed or
left-handed, as well as their ages.
The study showed that the proportion
of left-handed people and the... |
I've sort of gotten a grasp on the Chain rule with one variable. If you hike up a mountain at 2 feet an hour, and the temperature decreases at 2 degrees per feet, the temperature would be decreasing for you at $2\times 2 = 4$ degrees per hour.
But I'm having a bit more trouble understanding the Chain Rule as applied t... |
The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes, but also that there are three distinct, real roots if $\Delta > 0$, and that there is one real root and two complex roots (complex ... |
This answer suggests that there are explicit polynomial equations for which the existence
(or nonexistence) of integer solutions is unprovable. How can this be?
|
I know there must be something unmathematical in the following but I don't know where it is:
\begin{align}
\sqrt{-1} &= i \\\\\
\frac1{\sqrt{-1}} &= \frac1i \\\\
\frac{\sqrt1}{\sqrt{-1}} &= \frac1i \\\\
\sqrt{\frac1{-1}} &= \frac1i \\\\
\sqrt{\frac{-1}1} &= \frac1i \\\\
\sqrt{-1} &= \frac1i \\\\
i &= \frac1i \\\\ ... |
An equivalence relation is defined by three properties: reflexivity, symmetry and transitivity.
Doesn't symmetry and transitivity implies reflexivity? Consider the following argument.
For any $a$ and $b$,
$a R b$ implies $b R a$ by symmetry. Using transitivity, we have $a R a$.
Source: Exercise 8.46, P195 of Mathematic... |
$\frac17 = 0.(142857)$...
with the digits in the parentheses repeating.
I understand that the reason it's a repeating fraction is because $7$ and $10$ are coprime. But this...cyclical nature is something that is not observed by any other reciprocal of any natural number that I know of (besides multiples of $7$). (if I... |
Let's say, I have 4 yellow and 5 blue balls. How do I calculate in how many different orders I can place them? And what if I also have 3 red balls?
|
I just came back from an intense linear algebra lecture which showed that linear transformations could be represented by transformation matrices; with more generalization, it was later shown that affine transformations (linear + translation) could be represented by matrix multiplication as well.
This got me to thinking... |
I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to use.
I'm looking for nontrivial ideas in abstract mathematics that can be demonstrated with some contraption, constructi... |
Are there any interesting and natural examples of semigroups that are not monoids (that is, they don't have an identity element)?
To be a bit more precise, I guess I should ask if there are any interesting examples of semigroups $(X, \ast)$ for which there is not a monoid $(X, \ast, e)$ where $e$ is in $X$. I don't con... |
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