# Symbols:Greek/Pi

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## Pi

The $16$th letter of the Greek alphabet.

Minuscules: $\pi$ and $\varpi$
Majuscule: $\Pi$

The $\LaTeX$ code for $\pi$ is \pi .
The $\LaTeX$ code for $\varpi$ is \varpi .

The $\LaTeX$ code for $\Pi$ is \Pi .

### Constant

$\pi$

The real number $\pi$ (pronounced pie) is an irrational number (see proof here) whose value is approximately $3.14159\ 26535\ 89793\ 23846\ 2643 \ldots$

### Prime-Counting Function

$\map \pi x$

The prime-counting function is the function $\pi: \R \to \Z$ which counts the number of primes less than or equal to some real number.

That is:

$\displaystyle \forall x \in \R: \map \pi x = \sum_{\substack {p \mathop \in \mathbb P \\ p \mathop \le x} } 1$

where $\mathbb P$ denotes the set of prime numbers.

The $\LaTeX$ code for $\map \pi x$ is \map \pi x .

### Projection

$\pi_i$

The notation $\pi_i$ is often used for the $i$th projection.

The $\LaTeX$ code for $\pi_i$ is \pi_i .

### Rectangle Function

$\map \Pi x$

The rectangle function is the real function $\Pi: \R \to \R$ defined as:

$\forall x \in \R: \map \Pi x := \begin {cases} 1 : & \size x \le \dfrac 1 2 \\ 0 : & \size x > \dfrac 1 2 \end {cases}$

The $\LaTeX$ code for $\map \Pi x$ is \map \Pi x .

### Probability Generating Function

$\map {\Pi_X} s$

Let $X$ be a discrete random variable whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$.

Let $p_X$ be the probability mass function for $X$.

The probability generating function for $X$, denoted $\map {\Pi_X} s$, is the formal power series defined by:

$\displaystyle \map {\Pi_X} s := \sum_{n \mathop = 0}^\infty \map {p_X} n s^n \in \R \left[\left[{s}\right]\right]$

The $\LaTeX$ code for $\map {\Pi_X} s$ is \map {\Pi_X} s .

### Product Notation

$\displaystyle \prod_{j \mathop = 1}^n a_j$

Let $\struct {S, \times}$ be an algebraic structure where the operation $\times$ is an operation derived from, or arising from, the multiplication operation on the natural numbers.

Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$.

The composite is called the product of $\tuple {a_1, a_2, \ldots, a_n}$, and is written:

$\displaystyle \prod_{j \mathop = 1}^n a_j = \paren {a_1 \times a_2 \times \cdots \times a_n}$

The $\LaTeX$ code for $\displaystyle \prod_{j \mathop = 1}^n a_j$ is \displaystyle \prod_{j \mathop = 1}^n a_j .

The $\LaTeX$ code for $\displaystyle \prod_{1 \mathop \le j \mathop \le n} a_j$ is \displaystyle \prod_{1 \mathop \le j \mathop \le n} a_j .

The $\LaTeX$ code for $\displaystyle \prod_{\map \Phi j} a_j$ is \displaystyle \prod_{\map \Phi j} a_j .

## Linguistic Note

While the conventional contemporary prounciation of $\pi$ in Western English is pie, it is worth noting that the "correct" Greek pronunciation of the name of the letter $\pi$ is in fact the same as the letter p is pronounced in English.

It is just as well that $\pi$ is pronounced pie, otherwise the opportunity for confusion between $\pi$ and $p$ in spoken language would be too great.

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