Number Properties

Adding or multiplying two numbers in either order gives the same result: 5 + 8 and 8 + 5 are both 13, and 4 × 7 and 7 × 4 are both 28. This is the commutative property. In letters, a + c = c + a and ac = ca, which is why x × 7 and 7x mean the same thing.

Grouping does not matter either: (2 + 3) + 4 and 2 + (3 + 4) are both 9, and (2 × 5) × 7 and 2 × (5 × 7) are both 70. This is the associative property, and it is what let you pair friendly numbers in Section 1.

Neither property holds for subtraction or division, because those operations have a direction. 10 - 4 ≠ 4 - 10: one is 6, the other -6. (10 - 4) - 2 is 4, but 10 - (4 - 2) is 8. And 12 divided by 4 is 3, while 4 divided by 12 is a third. Before you swap or regroup, check that the operation is + or ·.