# What areas of math are related to both $\pi$ and $\varphi$? [closed]

What areas of math are related to both $\pi$ and $\varphi$?

There are at least two answers.

• In this question,is $\varphi$ the golden ratio? – Dragonemperor42 Mar 25 '16 at 17:26
• "There are at least two answers." - How is this question not too broad? – Bob Mar 25 '16 at 17:47
• @Bob I'm shaking my head at you. – question_asker Mar 25 '16 at 17:50
• @question_asker I bite my thumb at you. – Bob Mar 25 '16 at 18:20
• I'm voting to close this question as off-topic because it's not related to puzzling. – Deusovi Mar 25 '16 at 18:41

Number theory,

because

in number theory, $\pi(n)$ denotes the prime-counting function (the number of primes $\le n$), and $\varphi(n)$ denotes the Euler totient function (the number of integers between $1$ and $n-1$ that are relatively prime to $n$).

• This was the intended explanation for Number Theory. – wythagoras Mar 25 '16 at 20:09

I hope I'm missing the point, but I'll just answer this like it was a normal question:

Geometry and Number Theory

Because

Pi and the Golden Ratio show up in both Geometry and Number Theory. Though I feel like if this is a valid answer, there are dozens of other valid answers

• This were the two things I had in mind, but I don't think that Pi and the Golden Ratio are very common in Number Theory. However, there are other reasons why it is related to Number Theory, for which I refer to the answer by Big Black Box. – wythagoras Mar 25 '16 at 20:08
• Ok, but I'll make a couple comments: 1)This isn't really a puzzle. 2)The greek letters pi and phi are used in basically every branch of mathematics and physics with various meanings. I was really hoping the real answer would be a joke or a pun of some sort. – Solocutor Mar 25 '16 at 20:38

The area of a circle with radius $\phi$ is $\pi \phi^2$, which is very obviously related to both numbers.

In algebra (specfically polynomials) , the $n$-th cyclotomic polynomial is defined as $$\Phi_n(x) = \prod_{\stackrel{1\leq k \leq n}{\gcd(k,n)=1}} \left(x-e^{2i\pi\frac{k}{n}}\right)$$

Also,

In probability theory, the probability density function of the normal distribution. $$\phi(x) = (2\pi)^{-1/2}\cdot e^{-x^2/2}$$

Please see that $\varphi$ is the same as $ϕ$ for both the above mentioned functions.

• Why did I get a down vote? – Dragonemperor42 Mar 25 '16 at 17:49

Are they both

Transcendental

numbers?

• Assuming you take $\varphi$ to be the golden ratio, it is not transcedental number, since $\varphi^2-\varphi-1=0$. They are both irrational though. Also, your answer is not really an area of math. – wythagoras Mar 25 '16 at 17:25
• Hm. Then I'm out, but I'll leave it up. Unless of course the answer is a man. – Raystafarian Mar 25 '16 at 17:26