## Top new questions this week:

Start with any positive fraction $\frac{a}{b}$. First add the denominator to the numerator: $$\frac{a}{b} \rightarrow \frac{a+b}{b}$$ Then add the (new) numerator to the denominator: $$\frac{a+b}{b} ... sequences-and-series golden-ratio  asked by Joseph O'Rourke 32 votes  answered by Brady Gilg 73 votes ### There is a number divisible by all integers from 1 to 200, except for two consecutive numbers. What are the two? To reiterate the question, basically there is some number, n that exists that is divisible all the integers 1, \dots, 200, except for two consecutive numbers in that range. The goal is to find ... elementary-number-theory prime-numbers divisibility gcd-and-lcm  asked by Slade 20 votes  answered by slbtab 40 votes ### How does Peano Postulates construct Natural numbers only? I am beginning real analysis and got stuck on the first page (Peano Postulates). It reads as follows, at least in my textbook. Axiom 1.2.1 (Peano Postulates). There exists a set \Bbb N with an ... natural-numbers peano-axioms  asked by Solomon Tessema 14 votes  answered by Ethan Bolker 30 votes ### Does removing finitely many points from an open set yield an open set? Removing finitely many point from an open set in \mathbb{R}^n gives an open set. Is this true in general for any space? My intuition is that this is the case, however, how does one (dis)prove ... general-topology separation-axioms  asked by Stephen 14 votes  answered by Maximilian Janisch 19 votes ### How are the logarithmic integrals \int_{-\pi}^{\pi} \ln^n(2\operatorname{cos}(x/2))dx related to \zeta(n)? Suppose we have defined the "cochord" of an angle \theta \in (-\pi,\pi) as$$\operatorname{coc}(\theta) := 2\cos\left(\frac \theta 2\right),$$and set$$c_n := \frac 1 \pi \int_{-\pi}^{\pi} \ln^n ...

integration fourier-series riemann-zeta

### Is there any intuition why the following matrix is positive semidefinite?

I have the following 8 by 8 square matrix, which is positive semidefinite: \begin{bmatrix}3&1&1&-1&1&-1&-1&-3\\1&3&-1&1&-1&1&-3&-1& \\ ...

linear-algebra matrices symmetric-matrices positive-semidefinite

### Suppose that $x^5$ and $20x+\frac {19}x$ are rational numbers. Then $x$ is also rational

Let $x\neq0$ be a real number such that $x^5$ and $20x+\frac {19}x$ are rational. How can we prove that $x$ is also rational? (This was a question from the RMO 2019 in India.) My attempt: Let ...

contest-math roots rational-numbers

## Greatest hits from previous weeks:

### A loss and gain problem

This is a very simple but confusing puzzle. A customer buys goods worth $200$ rupees from a shop. The shopkeeper selling these goods makes zero profit from this purchase. The lady gives him a $1000$ ...

puzzle

### why does e raised to the power of negative infinity equal 0?

Why is it that e raised to the power of negative infinity would equal 0 instead of negative infinity? I am working on problems with regards to limits of integration, specifically improper integrals ...

calculus limits exponentiation infinity

### What does "∈" mean?

I have started seeing the "∈" symbol in math. What exactly does it mean? I have tried googling it but google takes the symbol out of the search.

notation

### Derivative of square root

What would be the derivative of square roots? For example if I have $2 \sqrt{x}$ or $\sqrt{x}$. I'm unsure how to find the derivative of these and include them especially in something like implicit.

calculus derivatives

### Finding two numbers given their sum and their product

Which two numbers when added together yield $16$, and when multiplied together yield $55$. I know the $x$ and $y$ are $5$ and $11$ but I wanted to see if I could algebraically solve it, and found ...

algebra-precalculus systems-of-equations

### How to find perpendicular vector to another vector?

How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please? I have a solution to this when I have ...

linear-algebra geometry vector-spaces vectors

### How many times are the hands of a clock at $90$ degrees.

How many times are the hands of a clock at right angle in a day? Initially, I worked this out to be $2$ times every hour. The answer came to $48$. However, in the cases of $3$ o'clock and $9$ ...

geometry puzzle

## Can you answer these questions?

### Equation-to-Text Converter

I was just thinking today... I was reading a book called "Forecasting: Principles and Practice", and found myself reading mostly the theoretical paragraphs, and skipping most of the mathematical ...

notation

### Collatz Conjecture Numbers up to n

I was playing around with the Collatz sequences of numbers up to a number. My question was, for which numbers $n$ does no integer below $n$ reach $n$ in its iteration? So I wrote a program to find ...

number-theory collatz