
This short guide is intended to help developers understand mathematical notation
using familiar programming concepts.

## Summation: The Trusty For-Loop in TypeScript

You've likely used a `for` loop to sum up numbers; it's practically a rite of
passage. Here's how it typically appears in TypeScript:

```ts
let sum = 0; // Initialize sum to zero
// Loop from 1 to n (inclusive)
for (let i = 1; i <= n; i++) {
  sum += i; // Increment sum by i
}
```

In mathematics, the Sigma symbol $ \Sigma $ accomplishes the same task, albeit
more elegantly:

$$
\text{sum} = \sum_{i=1}^{n} i
$$

**KaTeX Code**

```katex
\text{sum} = \sum_{i=1}^{n} i
```

### KaTeX Legend

- $ \sum $: Meet our sum variable.
- $ \Sigma $: This Greek Sigma is basically the `for` loop of math.
- $ i=1 $: The starting point of our loop.
- $ n $: The ending point.
- $ i $: The value added during each iteration.

## Product: Multiplying Like a Pro

Now let's multiply numbers! Here's a for loop in TypeScript that multiplies
numbers from 1 to $ n $.

```ts
let product = 1; // Initialize product to 1
// Loop from 1 to n (inclusive)
for (let i = 1; i <= n; i++) {
  product *= i; // Multiply product by i
}
```

Mathematics offers an equally elegant symbol for this: the Pi symbol $$ \Pi $$.

$$
\text{product} = \prod_{i=1}^{n} i
$$

**KaTeX Code:**

```katex
\text{product} = \prod_{i=1}^{n} i
```

### KaTeX Legend

- $ \text{product} $: Say hello to our product variable.
- $ \Pi $: Greek Pi, the multiplicative counterpart to Sigma.
- $ i=1 $: The starting point.
- $ n $: The ending point.
- $ i $: What's being multiplied at each step.

## For Complex Iterations: General Iterative Process

Sometimes your loops involve more than basic addition or multiplication. For
instance, you might add when a condition is met and subtract otherwise.

```ts
let x = 0; // Initialize x to 0
// Loop from 1 to n (inclusive)
for (let i = 1; i <= n; i++) {
  if (i % 2 === 0) {
    x += i; // Add i to x if i is even
  } else {
    x -= i; // Subtract i from x if i is odd
  }
}
```

Mathematics isn't left behind in this regard. It employs "piecewise" or "case"
notation to describe such complex operations:

$$
x = \begin{cases}
  x + i & \text{if $ i $ is even} \\
  x - i & \text{if $ i $ is odd}
\end{cases}, \quad \text{for $ i = 1, 2, \ldots, n $}
$$

**KaTeX Code:**

```katex
x = \begin{cases}
  x + i & \text{if $ i $ is even} \\
  x - i & \text{if $ i $ is odd}
\end{cases}, \quad \text{for $ i = 1, 2, \ldots, n $}
```

### KaTeX Legend

- $ x $: Our dynamically changing variable.
- $ \begin{cases} \end{cases} $: The mathematical equivalent of an `if-else`
  block.
- $ i $: Our loop counter, as always.
- $ n $: The loop's endpoint.


A Developer's Guide to Mathematical Notation