The property of numbers that states that [tex]\(1 + 3\)[/tex] is the same as [tex]\(3 + 1\)[/tex] is called the Commutative property.
Let’s walk through this step by step:
1. Definitions of the Properties:
- Associative Property: This property states that the way numbers are grouped in an addition or multiplication equation does not change their sum or product. For example, [tex]\((1 + 2) + 3 = 1 + (2 + 3)\)[/tex] and [tex]\((1 \cdot 2) \cdot 3 = 1 \cdot (2 \cdot 3)\)[/tex].
- Distributive Property: This property is used to multiply a single term and two or more terms inside a set of parentheses. For example, [tex]\(a(b + c) = ab + ac\)[/tex].
- Inverse Property: This property refers to adding a number and its opposite (additive inverse) which results in zero, or multiplying a number by its reciprocal (multiplicative inverse) which results in one. For example, [tex]\(a + (-a) = 0\)[/tex] or [tex]\(a \cdot \frac{1}{a} = 1\)[/tex] (assuming [tex]\(a \neq 0\)[/tex]).
- Commutative Property: This property states that the order of the numbers does not change the sum or product. For example, [tex]\(a + b = b + a\)[/tex] and [tex]\(a \cdot b = b \cdot a\)[/tex].
2. Analyzing the Expression [tex]\( 1 + 3 = 3 + 1 \)[/tex]:
- Here, the expression [tex]\(1 + 3\)[/tex] and [tex]\(3 + 1\)[/tex] both evaluate to the same result.
- This demonstrates that the order in which the numbers are added does not affect the sum.
- According to the definitions above, this matches the Commutative Property since it specifically addresses the change in order in addition (or multiplication) without affecting the outcome.
Therefore, the property that states [tex]\(1 + 3\)[/tex] is the same as [tex]\(3 + 1\)[/tex] is called the Commutative property.
