Mathematics

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1500 questions
101
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1 answer

Arithmetic-geometric mean of 3 numbers

The arithmetic-geometric mean $^{[1]}$ $\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad a_{n+1}=\frac{a_n+b_n}2,\quad b_{n+1}=\sqrt{a_n b_n}.$$…
Vladimir Reshetnikov
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101
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13 answers

Can I think of Algebra like this?

This year in Algebra we first got introduced to the concept of equations with variables. Our teacher is doing a great job of teaching us how to do them, except for one thing: He isn't telling us what we are actually doing when simplifying/solving…
Nico A
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101
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10 answers

Proof that a Combination is an integer

From its definition a combination $\binom{n}{k}$, is the number of distinct subsets of size $k$ from a set of $n$ elements. This is clearly an integer, however I was curious as to why the expression $$\frac{n!}{k!(n-k)!}$$ always evaluates to an…
Akusete
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101
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11 answers

What is a simple example of an unprovable statement?

Most of the systems mathematicians are interested in are consistent, which means, by Gödel's incompleteness theorems, that there must be unprovable statements. I've seen a simple natural language statement here and elsewhere that's supposed to…
Michael Harris
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101
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14 answers

How to find a general sum formula for the series: 5+55+555+5555+.....?

I have a question about finding the sum formula of n-th terms. Here's the series: $5+55+555+5555$+...... What is the general formula to find the sum of n-th terms? My attempts: I think I need to separate 5 from this series such that:…
akusaja
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101
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10 answers

Given a die, what is the probability that the second roll of a die will be less than the first roll?

If you are given a die and asked to roll it twice. What is the probability that the value of the second roll will be less than the value of the first roll?
Salman Paracha
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100
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7 answers

Polynomials such that roots=coefficients

Here is my question : Are there monic polynomials with degree $\geq 5$ such that they have the same real all non zero roots and coefficients ? Mathematically, prove or disprove the existence of $n \geq 5$ such that $$\exists (z_1,\ldots, z_n) \in…
Gabriel Romon
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100
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24 answers

100 blue-eyed islanders puzzle: 3 questions

I read the Blue Eyes puzzle here , and the solution which I find quite interesting. My questions: What is the quantified piece of information that the Guru provides that each person did not already have? Each person knows, from the beginning, no…
A Googler
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100
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15 answers

math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?

I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
newcomer
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100
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2 answers

Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question . I also asked a related question on MO , although hopefully this question should be easier. There…
Dan Rust
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100
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8 answers

Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$

Inadvertently, I find this interesting inequality. But this problem have nice solution? prove that $$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$ This problem have nice solution? Thank you. ago,I find this…
math110
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100
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7 answers

Density of sum of two independent uniform random variables on $[0,1]$

I am trying to understand an example from my textbook. Let's say $Z = X + Y$ , where $X$ and $Y$ are independent uniform random variables with range $[0,1]$ . Then the PDF is $$f(z) = \begin{cases} z & \text{for $0 < z < 1$} \\ 2-z & \text{for $1…
Zhulu
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100
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4 answers

Finding the transform matrix from 4 projected points (with JavaScript)

I'm working on a project using Chrome - JS and WebKit 3D CSS3 transform matrix. The final goal is to create a tool for artistic projects using projectors and animation - somewhat far away from using maths... I'm using a projector to project several…
mika
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100
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1 answer

$4494410$ and friends

The number $4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number. $3883544142410_{10}=3883544E24A_{16}$ is another. These numbers are in…
Ross Millikan
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100
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31 answers

Non-textbook Math book recommendation to read to my kids

I'm looking for a book to read to my kids. NOT a kids book, but not too mature for a kid. My youngest kid that reads with me is 6 and the eldest is 10. I'm looking for a book that is good literature, and is hard to put down. Also maybe a book that…
Jeremy Campbell
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