Question
How does the Closure Property prove that the sum of a rational and irrational number is irrational?
Irrational numbers can be added or subtracted.
Irrational numbers can be added or subtracted.
The sum or difference of a rational and irrational number is a rational number.
The sum or difference of a rational and irrational number is a rational number.
The sum or difference of two rational numbers is a rational number.
The sum or difference of two rational numbers is a rational number.
The sum of an irrational and rational number can’t be predicted.
Irrational numbers can be added or subtracted.
Irrational numbers can be added or subtracted.
The sum or difference of a rational and irrational number is a rational number.
The sum or difference of a rational and irrational number is a rational number.
The sum or difference of two rational numbers is a rational number.
The sum or difference of two rational numbers is a rational number.
The sum of an irrational and rational number can’t be predicted.
Answers
Answered by
GPT-4o mini
To prove that the sum of a rational and an irrational number is irrational, we can use the properties of rational and irrational numbers along with the definition of each type.
### Definitions:
1. **Rational Numbers**: A number is rational if it can be expressed as the quotient of two integers \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).
2. **Irrational Numbers**: A number is irrational if it cannot be expressed as a fraction \( \frac{a}{b} \) of two integers.
### Proof:
Let \( r \) be a rational number and \( i \) be an irrational number. We want to show that the sum \( r + i \) is irrational.
Assume for contradiction that \( r + i \) is rational. Then, we can express it as:
\[
r + i = q
\]
for some rational number \( q \).
Rearranging gives:
\[
i = q - r
\]
Here, \( q \) is rational (by our assumption) and \( r \) is rational by definition. The difference of two rational numbers \( q - r \) is also a rational number (this is based on the closure property of rational numbers under subtraction).
Thus, if both \( q \) and \( r \) are rational, then \( i \) (as expressed by the equation \( i = q - r \)) is also rational.
This leads to a contradiction because we started by assuming \( i \) is irrational. Therefore, our assumption that \( r + i \) is rational must be false.
### Conclusion:
Since our assumption that the sum \( r + i \) could be rational leads to a logical contradiction, we conclude that the sum of a rational number and an irrational number must be irrational. Thus, we have shown using properties of rational and irrational numbers, particularly the closure property of rational numbers under addition and subtraction, that the sum of a rational and an irrational number is indeed irrational.
### Definitions:
1. **Rational Numbers**: A number is rational if it can be expressed as the quotient of two integers \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).
2. **Irrational Numbers**: A number is irrational if it cannot be expressed as a fraction \( \frac{a}{b} \) of two integers.
### Proof:
Let \( r \) be a rational number and \( i \) be an irrational number. We want to show that the sum \( r + i \) is irrational.
Assume for contradiction that \( r + i \) is rational. Then, we can express it as:
\[
r + i = q
\]
for some rational number \( q \).
Rearranging gives:
\[
i = q - r
\]
Here, \( q \) is rational (by our assumption) and \( r \) is rational by definition. The difference of two rational numbers \( q - r \) is also a rational number (this is based on the closure property of rational numbers under subtraction).
Thus, if both \( q \) and \( r \) are rational, then \( i \) (as expressed by the equation \( i = q - r \)) is also rational.
This leads to a contradiction because we started by assuming \( i \) is irrational. Therefore, our assumption that \( r + i \) is rational must be false.
### Conclusion:
Since our assumption that the sum \( r + i \) could be rational leads to a logical contradiction, we conclude that the sum of a rational number and an irrational number must be irrational. Thus, we have shown using properties of rational and irrational numbers, particularly the closure property of rational numbers under addition and subtraction, that the sum of a rational and an irrational number is indeed irrational.