eolas/zk/Distributivity.md

---
tags:
  - theorems
  - prealgebra
---

# The Distributive Property of Multiplication

**Let $a$, $b$ represent members of $\mathbb{W}$ or $\mathbb{Z}$ then:**

$$ a \cdot (b + c) = a \cdot b + a \cdot c $$

### Demonstration

When faced with $4(2\cdot3)$ we may proceed with the official order of
operations in algebra, namely:

```
4 x (2 + 3) = 4 x (5)
= 20
```

In other words we find the sum of the values in parentheses and then multiply
this by the value outside of the brackets.

When we use distributive property we _distribute_ each value in the parentheses
against the value outside of the parentheses:

```
4 x (2 + 3) = (4 x 2) + (4 x 3)
8 + 12 = 20
```