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.
$0$ (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.
\[ x + 0 = x \]
\[ 1000 + 0 = 1000 \]
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 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.
\[ x * y = y * x \]
\[ 3 * 2 = 2 * 3 \]
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