# Commutative Swap

The `Commutative Property`

of numbers says that we can re-order two `addition`

or `multiplication`

terms so that one occurs before the other in the expression without changing the value of the expression.

The formulation of this property is the same for addition and multiplication:

- Addition
`a + b = b + a`

- Multiplication
`a * b = b * a`

The commutative property is used for re-arranging the order of parts of an expression and is, as such, very important for working with mathematical expressions.

### Transformations¶

Given a common parent node, this rule switches the order of the children of that node. It can only be applied to addition or multiplication nodes.

#### Addition¶

`a + b = b + a`

```
+ +
/ \ / \
/ \ -> / \
/ \ / \
a b b a
```

#### Multiplication¶

`a * b = b * a`

```
* *
/ \ / \
/ \ -> / \
/ \ / \
a b b a
```

### Examples¶

Info

All the examples shown below are drawn from the mathy test suite that verifies the expected input/output combinations for rule transformations.

Input | Output | Valid |
---|---|---|

2x = 6x - 8 | 6x - 8 = 2x | ✔ |

2x + 1y^3 + 7j + -2q + 93m + 6x | 2x + 1y^3 + 7j + -2q + 6x + 93m | ✔ |

2x + 1y^3 + 7j + -2q + 6x + 93m | 2x + 1y^3 + 7j + 6x + -2q + 93m | ✔ |

2x + 1y^3 + 7j + 6x + -2q + 93m | 2x + 1y^3 + 6x + 7j + -2q + 93m | ✔ |

2x + 1y^3 + 6x + 7j + -2q + 93m | 2x + 6x + 1y^3 + 7j + -2q + 93m | ✔ |

12x * 10y | (x * 12) * 10y | ✔ |

2530z + 1m + 3.5x + 2z + 8.9c | 2530z + 3.5x + 1m + 2z + 8.9c | ✔ |

(5 + 12) * a | a * (5 + 12) | ✔ |

2b^4 * 3x | (b^4 * 2) * 3x | ✔ |

4 + 17 | 17 + 4 | ✔ |

4x | x * 4 | ✔ |

2^4 * 8 | 8 * 2^4 | ✔ |

(7 + x) + 2 | x + 7 + 2 | ✔ |

12x + 10y | 10y + 12x | ✔ |

8y^4 | --- | --- |

4x | --- | --- |

4 / 3 | --- | --- |

7 / x | --- | --- |