Question
What does the Closure property tell you about the rational and irrational numbers under subtraction?
A. The difference of two rational numbers will always be a rational number.
B. The difference of a rational number and an irrational number cannot be determined.
C. The difference of a rational number and an irrational number will always be a rational number.
D. The difference of two irrational numbers will always be an irrational number.
A. The difference of two rational numbers will always be a rational number.
B. The difference of a rational number and an irrational number cannot be determined.
C. The difference of a rational number and an irrational number will always be a rational number.
D. The difference of two irrational numbers will always be an irrational number.
Answers
Answered by
GPT-4o mini
The Closure Property states that when you perform a specific operation (like addition, subtraction, multiplication, or division) on members of a particular set, the result will also be a member of that set.
Now let's analyze the options with regard to the sets of rational and irrational numbers under subtraction:
A. **The difference of two rational numbers will always be a rational number.**
This statement is true because the set of rational numbers is closed under subtraction.
B. **The difference of a rational number and an irrational number cannot be determined.**
This statement is misleading. The difference will be a specific number, but it will not be rational.
C. **The difference of a rational number and an irrational number will always be a rational number.**
This statement is false. If you subtract an irrational number from a rational number, the result is always irrational.
D. **The difference of two irrational numbers will always be an irrational number.**
This statement is also false. The difference of two irrational numbers can be rational (for example, \( \sqrt{2} - \sqrt{2} = 0 \)).
Thus, the correct conclusion regarding the Closure property and rational and irrational numbers under subtraction is:
**A. The difference of two rational numbers will always be a rational number.**
Now let's analyze the options with regard to the sets of rational and irrational numbers under subtraction:
A. **The difference of two rational numbers will always be a rational number.**
This statement is true because the set of rational numbers is closed under subtraction.
B. **The difference of a rational number and an irrational number cannot be determined.**
This statement is misleading. The difference will be a specific number, but it will not be rational.
C. **The difference of a rational number and an irrational number will always be a rational number.**
This statement is false. If you subtract an irrational number from a rational number, the result is always irrational.
D. **The difference of two irrational numbers will always be an irrational number.**
This statement is also false. The difference of two irrational numbers can be rational (for example, \( \sqrt{2} - \sqrt{2} = 0 \)).
Thus, the correct conclusion regarding the Closure property and rational and irrational numbers under subtraction is:
**A. The difference of two rational numbers will always be a rational number.**