# Programming paradigms

One of the biggest hurdles to understanding functional programs when coming from
an imperative background is the shift in thinking. Imperative programs describe
**how** to do something, whereas declarative programs describe **what** to do.
Let's sum the numbers from 1 to 10 to show this.

## Imperative

```rust
let mut sum = 0;
for i in 1..11 {
    sum += i;
}
println!("{}", sum);
```

With imperative programs, we have to play compiler to see what is happening.
Here, we start with a `sum` of `0`. Next, we iterate through the range from 1
to 10. Each time through the loop, we add the corresponding value in the range.
Then we print it out.

| `i` | `sum` |
| :-: | :---: |
|  1  |   1   |
|  2  |   3   |
|  3  |   6   |
|  4  |  10   |
|  5  |  15   |
|  6  |  21   |
|  7  |  28   |
|  8  |  36   |
|  9  |  45   |
| 10  |  55   |

This is how most of us start out programming. We learn that a program is a set
of steps.

## Declarative

```rust
println!("{}", (1..11).fold(0, |a, b| a + b));
```

Whoa! This is really different! What's going on here? Remember that with
declarative programs we are describing **what** to do, rather than **how** to do
it. `fold` is a function that
[composes](https://en.wikipedia.org/wiki/Function_composition) functions. The
name is a convention from Haskell.

Here, we are composing functions of addition (this closure: `|a, b| a + b`) with
a range from 1 to 10. The `0` is the starting point, so `a` is `0` at first. `b`
is the first element of the range, `1`. `0 + 1 = 1` is the result. So now we
`fold` again, with `a = 1`, `b = 2` and so `1 + 2 = 3` is the next result. This
process continues until we get to the last element in the range, `10`.

| `a` | `b` | result |
| :-: | :-: | :----: |
|  0  |  1  |   1    |
|  1  |  2  |   3    |
|  3  |  3  |   6    |
|  6  |  4  |   10   |
| 10  |  5  |   15   |
| 15  |  6  |   21   |
| 21  |  7  |   28   |
| 28  |  8  |   36   |
| 36  |  9  |   45   |
| 45  | 10  |   55   |