Commutative Property

Although the official use of commutative property or commutative law began at the end of 18th century, the use of commutative property was known even in the ancient era. This usage has now been organized properly with rules and is known as commutative law or commutative property.

Commutative Property

Commutative, word originated from the French word “commute or commuter” means to switch or move around combined with the suffix “-ative” means tend to. Therefore, literal meaning of word is tending to switch or move around. If you take real life instance, commutation has a role to play. Like, putting buttons of a shirt; whether you put first button or last button first doesn’t matter. It is a commutative action.

In mathematical computation, commutative property explains that order of terms doesn’t matters, result will be same. That is, suppose two numbers \(A\) and \(B\) on addition gives a sum \(C\), then:

\(A ~+~ B\) = \(B ~+~ A\) = \(C\)

For example, \(4~ +~ 3\) = \(7\) = \(3 ~+~ 4\); here, whether \(3\) come before or after the plus sign, the sum of \(4\) and \(3\) will be always \(7\) irrespective of their order.

Commutative law is applicable to multiplication also i.e. the multiplication properties of commutative law. For example, \(2~ \times ~3\) = \(6\) = \(3 ~\times~ 2\)

Associative Property

According to associative law, regardless of how the numbers are grouped, you can add or multiply them. In other words, the placement of parenthesis does not matter when it comes to adding or multiplying.

Distributive Property of Multiplication

The distributive property of Multiplication states that multiplying a sum by a number is same as multiplying every addend by the value and adding the products then.

According to the Distributive Property, if a, b, c are real numbers:

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

Non-Commutative Property

Some operations are non-commutative. By non-commutative we mean the switching of order will give different results. The mathematical operations, subtraction and division are the two non-commutative operations. Unlike addition, in subtraction switching of orders of terms results in different answers.

Example, \(4 ~-~ 3\) = \( 1\) but \(3 ~-~ 4\) = \(-1\) which are two different integers.

Also, division does not follow the commutative property. That is,

\(6~ ÷ ~2\) = \(3\)

\(2~ ÷~ 6\) = \(\frac{1}{3}\)

\(\Rightarrow~ 6 ~÷ ~2~ ≠~ 2 ~÷ ~6\)

Now we know why commutative property works for addition and multiplication only but not for subtraction and division.

Solve: Which of the following follows commutative law?

  1. \(3~\times~ 12\)
  2. \(4 ~+~ 20\)
  3. \(36 ~÷~ 6\)
  4. \(36 ~- ~6\)
  5. \(-3~\times~4\)<

Solution: i, ii and v follows the commutative law

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