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

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title: Basic Number Properties Associative, Commutative, and Distributive
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## Basic Number Properties Associative, Commutative, and Distributive
These are 3 basic properties of numbers.
<br/>
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.
<br/>
When studying advanced mathematics we know the importance of these properties.

# Properties one by one:
## Associativity
  "To associate" means to form groups of operands.
  <br/>
  If an operation(e.g. +, -, &times;, /) is associative it means, <strong>the result will remain same regardless of grouping of operands.</strong>
  <pre>
  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
  </pre>
  ###### try it with multiplication operation, & same variable values as above,
  ### Note:
  + All 4 basic arithmetic operations(i.e. +, -, &times;, /) follow Associativity.
<hr/><hr/><hr/>

## Commutativity
  "To commute" means to move around, in this case, operands move around operator.
  <br/>
  If an operation(e.g. +, -, &times;, /) is commutative it means, <strong>the result will remain same regardless of the order in which the operands are evaluated.</strong>
  <pre>
  for example, consider operation +,
    let, a = 3, b = 4
      a + b = b + a
    ->3 + 4 = 4 + 3
    ->    7 = 7
  </pre>
   ###### try it with multiplication operation, & same variable values as above,
  ### Note:
  + All 4 basic arithmetic operations(i.e. +, -, &times;, /) follow Commutativity.
  <hr/><hr/><hr/>
  
## Distributivity
  This property is easy to remember by knowing that "Multiplication is distributive over addition".
  Example,
  <br/>
  a &times; (b + c) = a &times; b + a &times; c
  i.e. Multiplication is performed separately on operands of addition and then addition is performed.
  <br/>
  <pre>
    3 &times; (4 + 5)
  -> 3 &times; 4 + 3 &times; 5
  -> 12 + 15
  -> 27
  </pre>
  ### Note
  + Multiplication is distributive over addtion, but vice versa is not true.