Rules and properties

There are many mathematical rules and properties that are necessary or helpful to know when trying to solve math problems. Learning and understanding these rules helps students form a foundation they can use to solve problems and tackle more advanced mathematical concepts.

Basic mathematical properties

Some of the most basic but important properties of math include order of operations, the commutative, associative, and distributive properties, the identity properties of multiplication and addition, and many more. They are properties that are used throughout most areas of mathematics in some form or other.

Order of operations

Order of operations is often taught using one of two acronyms: PEMDAS or BODMAS. Both indicate the order in which operations should be carried out.

PEMDAS: Parentheses, exponents, multiplication, division, addition, subtraction

BODMAS: Brackets, order, division, multiplication, addition, subtraction

Notice that multiplication and division are in different positions in PE(MD)AS and BO(DM)AS; this is because multiplication and division, and addition and subtraction, can be performed in either order, and usually when deciding the order of performing these operations (assuming parentheses and exponents are already taken into account), they are carried through from left to right.

The mnemonic device "Please excuse my dear aunt Sally" is commonly used as a way to remember the acronym PEMDAS. It is not necessary if you can remember the acronym immediately, but can be helpful to remember just in case.

Commutative property

The commutative property states that changing the order in which two numbers are added or multiplied does not change the result.

a + b = b + a

a × b = b × a

Associative property

The associative property states that changing the way that numbers are grouped in addition and multiplication does not change the result.

(a + b) + c = a + (b + c)

(a × b) × c = a × (b × c)

Distributive property

The distributive property states that multiplying a group of numbers that are being added is the same as multiplying by each number individually, then adding them together.

a × (b + c) = a × b + a × c