## Top new questions this week:

### Why do we study real numbers?

I apologize if this is a somewhat naive question, but is there any particular reason mathematicians disproportionately study the field $\mathbb{R}$ and its subsets (as opposed to any other algebraic ...

(real-analysis) (soft-question)

### Are there an infinite number of prime numbers where removing any number of digits leaves a prime?

Suppose for the purpose of this question that number $1$ is a prime number. Consider the prime number $311$. If we remove one $1$ from the number we arrive at the number $31$ which is also prime. If ...

(elementary-number-theory) (prime-numbers)

I recently watched a video about different infinities. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, ... (cardinals) (infinity) (ordinals)  asked by KKZiomek 27 votes  answered by jaska 25 votes ### You have to estimate$\binom{63}{19}$in$2$minutes to save your life. This is from the lecture notes in this course of discrete mathematics I am following. The professor is writing about how fast binomial coefficients grow. "So, suppose you had 2 minutes to save your ... (combinatorics) (discrete-mathematics) (binomial-coefficients)  asked by capablanca79 27 votes  answered by Carry on Smiling 32 votes ### Are there theoretical applications of trigonometry? I am a high school student currently taking pre-calculus. We have just finished a unit on analytic trig. I am curious to know if there are any purely theoretical uses for trigonometry. More ... (trigonometry)  asked by Conan G. 26 votes  answered by David C. Ullrich 38 votes ### Can you use both sides of an equation to prove equality? For example:$\color{red}{\text{Show that}}$$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$ In high school my maths teacher told me To prove equality of an equation; you start on ...

(soft-question) (proof-writing)

### How do we know the ratio between circumference and diameter is the same for all circles?

The number $\pi$ is defined as the ratio between the circumeference and diameter of a circle. How do we know the value $\pi$ is correct for every circle? How do we truly know the value is the same ...

(geometry) (pi)

## Greatest hits from previous weeks:

### A family has three children. What is the probability that at least one of them is a boy?

According to me there are $4$ possible outcomes: $$GGG \ \ BBB \ \ BGG \ \ BBG$$ Out of these four outcomes, $3$ are favorable. So the probability should be $\frac{3}{4}$. But should you take ...

(probability)

### How many squares actually ARE in this picture? Is this a trick question with no right answer?

This is one of those popular pictures on sites like Facebook. I always see a huge variation of answers such as $8, 9, 16, 17, 24, 28, 30, 40, 41, 52,$ etc., yet I've never seen a definitive answer on ...

(puzzle)

An integer sided triangle has an area $A$. Heronian triangle areas have no radical, or radical 1. Otherwise, $4 A$ will always be of the form $a\sqrt{r}$, where $r$ is the squarefree radical of the ...

(geometry) (number-theory) (recreational-mathematics) (triangle)

### Measure on $\omega$ defined in the generic extension by an atomless measure algebra is atomless

Work in Cantor space with standard probability measure $m$. Suppose we are given a sequence of measurable sets $\bar{A}=\langle A_n : n\in \omega\rangle$ and a non-principal ultrafilter $U$ and the ...

(measure-theory) (set-theory)
This is a (bit funny) ultra-soft question regarded to a type of thinking that is puzzling me. Suppose $a(p_{1}, p_{2}, p_{3}, p_{4}, ... , p_{n}, Q)$ denote: from the items $p_{i}$, find a common ...