Question
What does the Closure Property tell you about the rational and irrational numbers under subtraction?"
The multiple-choice answers provided are:
The difference of two rational numbers will always be a rational number.
The difference of two irrational numbers will always be an irrational number.
The difference of a rational number and an irrational number will always be a rational number.
The difference of a rational number and an irrational number cannot be determined.
The multiple-choice answers provided are:
The difference of two rational numbers will always be a rational number.
The difference of two irrational numbers will always be an irrational number.
The difference of a rational number and an irrational number will always be a rational number.
The difference of a rational number and an irrational number cannot be determined.
Answers
Answered by
GPT-4o mini
The Closure Property in mathematics refers to the idea that when you perform an operation (like addition, subtraction, multiplication, or division) on numbers from a certain set, the result will also belong to that set.
Now, let's examine the statements you provided concerning the closure property for rational and irrational numbers under subtraction:
1. **The difference of two rational numbers will always be a rational number.**
- This statement is true. The set of rational numbers is closed under subtraction.
2. **The difference of two irrational numbers will always be an irrational number.**
- This statement is false. For example, \( \sqrt{2} - \sqrt{2} = 0 \), which is a rational number. Thus, two irrational numbers can yield a rational result.
3. **The difference of a rational number and an irrational number will always be a rational number.**
- This statement is false. For example, \( 1 - \sqrt{2} \) is irrational.
4. **The difference of a rational number and an irrational number cannot be determined.**
- This statement is misleading. While the result can be determined, it will always be irrational when you subtract a rational number from an irrational number.
Given this analysis, the first statement is correct:
**The difference of two rational numbers will always be a rational number.**
Now, let's examine the statements you provided concerning the closure property for rational and irrational numbers under subtraction:
1. **The difference of two rational numbers will always be a rational number.**
- This statement is true. The set of rational numbers is closed under subtraction.
2. **The difference of two irrational numbers will always be an irrational number.**
- This statement is false. For example, \( \sqrt{2} - \sqrt{2} = 0 \), which is a rational number. Thus, two irrational numbers can yield a rational result.
3. **The difference of a rational number and an irrational number will always be a rational number.**
- This statement is false. For example, \( 1 - \sqrt{2} \) is irrational.
4. **The difference of a rational number and an irrational number cannot be determined.**
- This statement is misleading. While the result can be determined, it will always be irrational when you subtract a rational number from an irrational number.
Given this analysis, the first statement is correct:
**The difference of two rational numbers will always be a rational number.**