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In mathematics, commutativity applies to operations such as addition and multiplication of whole numbers, integers, and rational numbers but does not hold for subtraction and division. This section explores the commutative property across different number sets with examples.
Commutativity is a fundamental property in mathematics, primarily involving operations like addition and multiplication. When we say an operation is commutative, it means that changing the order of the numbers involved does not change the result of the operation. In this section, we will review how commutativity applies to whole numbers, integers, and rational numbers, using examples and exercises to reinforce understanding.
In summary, commutativity applies to addition and multiplication across these number types but does not apply to subtraction and division.
Commutativity: The principle that the order of numbers does not affect the result in certain operations.
Addition: A commutative operation where a + b = b + a.
Multiplication: A commutative operation where a × b = b × a.
Subtraction: A non-commutative operation where a - b ≠ b - a.
Division: A non-commutative operation where a ÷ b ≠ b ÷ a.
Commutative is neat, it can't be beat, add or multiply, the order's a treat!
In a land where numbers lived, two friends named Addy and Multiply loved to swap places and play games without sadness, for their results stayed the same!
A for Addition, M for Multiply, both can swap without fear, but S for Subtraction must steer clear.
Example 1: 3 + 5 = 5 + 3 = 8 (Addition is commutative.)
Example 2: 4 × 6 = 6 × 4 = 24 (Multiplication is commutative.)
Example 3: 10 - 5 ≠ 5 - 10 (Subtraction is not commutative.)
Example 4: 15 ÷ 3 ≠ 3 ÷ 15 (Division is not commutative.)
Term: Commutativity
Definition: A property of operations that states altering the order of the elements does not change the outcome (e.g., a + b = b + a).
A property of operations that states altering the order of the elements does not change the outcome (e.g., a + b = b + a).
Term: Rational Numbers
Definition: Numbers that can be expressed as the quotient of two integers, with the denominator not being zero.
Numbers that can be expressed as the quotient of two integers, with the denominator not being zero.
Term: Addition
Definition: An arithmetic operation that combines two numbers to yield a sum.
An arithmetic operation that combines two numbers to yield a sum.
Term: Multiplication
Definition: An arithmetic operation that combines two numbers to yield a product.
An arithmetic operation that combines two numbers to yield a product.
Term: Subtraction
Definition: An arithmetic operation that represents the removal of one number from another.
An arithmetic operation that represents the removal of one number from another.
Term: Division
Definition: An arithmetic operation that determines how many times one number is contained within another.
An arithmetic operation that determines how many times one number is contained within another.