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Fundamentals of Maths: Identity, Associative and Commutative Properties Of Numbers

Shaukat Mahmood Ahmad Shaukat Mahmood Ahmad Follow Jun 02, 2019 · 2 mins read
Fundamentals of Maths: Identity, Associative and Commutative Properties Of Numbers
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Identity Properties / Identity Numbers

Identity number for a mathematical operation is such a number that will have no impact on the result, identity numbers are also known as identity properties. In this article we will explore what identity numbers are in context of addition and multiplication.

Additive Identity

$0$ (zero) is the identity number for addition, adding $0$ to any number will have no effect on the result. You can see in following examples that adding $0$ to any other number has no effect at all.

\[ x + 0 = x \]

\[ 1000 + 0 = 1000 \]

Multiplicative Identity

For multiplication, the identity number is $1$ (one), A number or expression will retain it’s identify if we *multiply it with $1$, following examples show this behavior.

\[ x \times 1 = x \]

\[ 1000 \times 1 = 1000 \]

Associative, Commutative and Distributive Properties of Addition and Multiplication

In this section we will explore the laws for associative, commutative and distributive properties applied to addition and multiplication operations in math.

Associativity: Associative Property / Law

In mathematics, the associative property[1] is a property of some binary operations. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs Wikipedia. The word associative comes from the word associate or associate. Associative law states that the answer of an algebraic expression will remain same no matter what the order of it’s elements is. The Associative law or Associative property is applicable to both addition and multiplication expressions, let’s make it more clear with some practical examples.

Associative Property / Law of Addition

Following equations explain the associative property of addition operation.

\[ (x + y ) + z = x + (y + z) \]

\[ (3 + 2 ) + 5 = 3 + (2 + 5) \]

Associative Property / Law of Multiplication

Following equations explain the associative property of multiplication operation.

\[ (x * y ) * z = x * (y * z) \]

\[ (3 * 2 ) * 5 = 3 * (2 * 5) \]

Commutativity: Commutative Property

According to Commutative law the answer of multiplication operation in an algebraic expression will remain same, even if we change the order of the members of this expression Wikipedia. The word Commutative is derived from commute and in mathematics it states that re-arranging the elements of an algebraic expression will not affect the resultant value. Commutative law is also applicable to both addition and multiplication expressions, let’s make it more clear with some practical examples.

Commutativity Property of Addition / Commutative Law of Addition

Following equations explain the commutative property of addition operation.

\[ x + y = y + x \]

\[ 3 + 2 = 2 + 3 \]

Commutativity Property of Multiplication / Commutative Law of Multiplication

Following equations explain the commutative property of multiplication operation.

\[ x * y = y * x \]

\[ 3 * 2 = 2 * 3 \]

That’s it, hope you enjoyed it. You like this article, have any questions or suggestions please let us know in the comments section.

Thanks and Happy Learning!

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Shaukat Mahmood Ahmad
Written by Shaukat Mahmood Ahmad Follow
Hi, I am Shaukat Mahmood Ahmad, the author of SMA's blog and CTO / Co Founder at wizlinx.com