freeCodeCamp/guide/english/mathematics/basic-number-properties-associative-commutative-and-distributive/index.md

title
Basic Number Properties Associative, Commutative, and Distributive

Basic Number Properties Associative, Commutative, and Distributive

These are 3 basic properties of numbers.
These properties play an important role in advanced mathematics. Textbooks generally don't discuss them in detail because all number systems we use upto high school follow these properties by default.
When studying advanced mathematics we know the importance of these properties.

Properties one by one:

Associativity

"To associate" means to form groups of operands.
If an operation(e.g. +, -, ×, /) is associative it means, the result will remain same regardless of grouping of operands.

  for example, consider operation +,
    let, a = 3, b = 4, c = 5
      (a + b) + c = a + (b + c)
    ->(3 + 4) + 5 = 3 + (4 + 5)
    ->          12=12
  

try it with multiplication operation, & same variable values as above,

Note:

• All 4 basic arithmetic operations(i.e. +, -, ×, /) follow Associativity.

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Commutativity

"To commute" means to move around, in this case, operands move around operator.
If an operation(e.g. +, -, ×, /) is commutative it means, the result will remain same regardless of the order in which the operands are evaluated.

  for example, consider operation +,
    let, a = 3, b = 4
      a + b = b + a
    ->3 + 4 = 4 + 3
    ->    7 = 7
  

try it with multiplication operation, & same variable values as above,

Note:

• All 4 basic arithmetic operations(i.e. +, -, ×, /) follow Commutativity.

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Distributivity

This property is easy to remember by knowing that "Multiplication is distributive over addition". Example,
a × (b + c) = a × b + a × c i.e. Multiplication is performed separately on operands of addition and then addition is performed.

    3 × (4 + 5)
  -> 3 × 4 + 3 × 5
  -> 12 + 15
  -> 27
  

Note

• Multiplication is distributive over addtion, but vice versa is not true.