Commutative Property

Definition

The commutative property is a fundamental mathematical principle that states the order of the operands in an addition or multiplication operation does not affect the result. It allows the terms in an expression to be rearranged without changing the final value.

5 Must Know Facts For Your Next Test

1. The commutative property applies to both addition and multiplication of real numbers, including integers, fractions, and polynomials.
2. For addition, the commutative property states that $a + b = b + a$, where $a$ and $b$ are any real numbers.
3. For multiplication, the commutative property states that $a \cdot b = b \cdot a$, where $a$ and $b$ are any real numbers.
4. The commutative property simplifies the process of adding or multiplying numbers, as the order of the operands does not affect the final result.
5. The commutative property is a fundamental assumption in many algebraic proofs and problem-solving techniques.

Review Questions

• Explain how the commutative property applies to the addition and subtraction of integers.
  • The commutative property states that the order of the addends in an addition operation does not affect the final result. For example, $5 + 3 = 3 + 5$. This property also extends to the subtraction of integers, as subtraction can be rewritten as the addition of the opposite. Therefore, $a - b = a + (-b)$, and the commutative property allows the order of the terms to be rearranged without changing the final value.
• Describe how the commutative property is used in the multiplication and division of fractions.
  • The commutative property applies to the multiplication of fractions, stating that $\frac{a}{b} \cdot \frac{c}{d} = \frac{c}{d} \cdot \frac{a}{b}$. This property simplifies the process of multiplying fractions, as the order of the factors does not affect the final product. Additionally, the commutative property can be extended to division of fractions, as division can be rewritten as the multiplication of the dividend by the reciprocal of the divisor. Therefore, $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$, and the commutative property allows the order of the factors to be rearranged.
• Analyze how the commutative property is utilized in the addition and subtraction of polynomials.
  • The commutative property applies to the addition and subtraction of polynomials, just as it does with the addition and subtraction of real numbers. When adding or subtracting polynomial terms, the order of the terms does not affect the final result. For example, $(2x^2 + 3x - 4) + (5x^2 - 2x + 1) = (5x^2 + 2x^2) + (3x - 2x) + (-4 + 1)$, where the commutative property allows the rearrangement of the terms without changing the overall value of the expression. This property simplifies the process of adding and subtracting polynomials, as the order of the terms can be adjusted as needed.

Related terms

Associative Property: The associative property states that the grouping of terms in an addition or multiplication operation does not affect the result.

Distributive Property: The distributive property allows for the multiplication of a sum or difference to be distributed across the addends or subtrahends.

Identity Property: The identity property states that adding 0 to a number or multiplying a number by 1 does not change the original value.