Properties of Real Numbers
The six properties that govern real numbers: commutative, associative, identity, inverse, closure and distributive, with worked examples of each.
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The six properties that govern real numbers: commutative, associative, identity, inverse, closure and distributive, with worked examples of each.
Real numbers obey a small set of rules that hold without exception. These properties are what license every rearrangement you make while solving a problem — they are the reason a step is allowed rather than merely convenient.
Reordering the terms leaves the result unchanged.
This holds for addition and multiplication only. Subtraction and division are not commutative.
Where you place the brackets makes no difference to the answer.
Commutativity moves the numbers; associativity moves the brackets. Together they let you tackle a long sum in whatever order is easiest.
Each operation has one value that leaves any number exactly as it was.
Every number has a partner that cancels it, bringing you back to the identity.
Zero is the one exception: it has no multiplicative inverse, because is undefined.
Adding or multiplying two real numbers always produces another real number. The operation never takes you outside the set.
This is why you can chain operations freely without ever checking whether the answer still counts as a number you may use.
Multiplication spreads over addition: .
This is the only property here that connects two different operations, which is what makes it so useful when expanding expressions.