Question
What is the step-by-step process used to prove that the sum of 3/4 and the square root of 10 is irrational?
Answers
Answered by
GPT-4o mini
To prove that the sum of \( \frac{3}{4} \) and \( \sqrt{10} \) is irrational, we can follow these steps:
1. **Understand the components**: Identify that \( \frac{3}{4} \) is a rational number (because it can be expressed as the fraction of two integers) and \( \sqrt{10} \) is an irrational number (since 10 is not a perfect square, meaning there is no integer that, when squared, gives 10).
2. **Recall the properties of rational and irrational numbers**:
- A rational number is a number that can be expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
- An irrational number is a number that cannot be expressed as a fraction of integers.
3. **Formulate the sum**: We want to evaluate \( \frac{3}{4} + \sqrt{10} \).
4. **Assume the contrary**: Suppose for the sake of contradiction that \( \frac{3}{4} + \sqrt{10} \) is rational. According to our assumption:
\[
\frac{3}{4} + \sqrt{10} = r \quad (r \text{ is rational})
\]
5. **Isolate the square root**: Rearrange the equation to isolate \( \sqrt{10} \):
\[
\sqrt{10} = r - \frac{3}{4}
\]
6. **Express \( r - \frac{3}{4} \) as a single fraction**: Since \( r \) is rational, we can express it as \( \frac{a}{b} \) for some integers \( a \) and \( b \) (with \( b \neq 0 \)):
\[
\sqrt{10} = \frac{a}{b} - \frac{3}{4} = \frac{4a - 3b}{4b}
\]
7. **Analyze the right-hand side**: The right-hand side, \( \frac{4a - 3b}{4b} \), is a fraction of two integers, hence it is rational.
8. **Conclude about the square root**: If \( \sqrt{10} \) is equal to a rational number \( \frac{4a - 3b}{4b} \), then \( \sqrt{10} \) must also be rational.
9. **Use the established fact about \( \sqrt{10} \)**: However, we know that \( \sqrt{10} \) is irrational. This leads to a contradiction since we assumed that \( \frac{3}{4} + \sqrt{10} \) was rational.
10. **Conclude the proof**: Since our initial assumption that \( \frac{3}{4} + \sqrt{10} \) is rational leads to a contradiction, we conclude that:
\[
\frac{3}{4} + \sqrt{10} \text{ is irrational.}
\]
Thus, we have proved that the sum \( \frac{3}{4} + \sqrt{10} \) is indeed irrational.
1. **Understand the components**: Identify that \( \frac{3}{4} \) is a rational number (because it can be expressed as the fraction of two integers) and \( \sqrt{10} \) is an irrational number (since 10 is not a perfect square, meaning there is no integer that, when squared, gives 10).
2. **Recall the properties of rational and irrational numbers**:
- A rational number is a number that can be expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
- An irrational number is a number that cannot be expressed as a fraction of integers.
3. **Formulate the sum**: We want to evaluate \( \frac{3}{4} + \sqrt{10} \).
4. **Assume the contrary**: Suppose for the sake of contradiction that \( \frac{3}{4} + \sqrt{10} \) is rational. According to our assumption:
\[
\frac{3}{4} + \sqrt{10} = r \quad (r \text{ is rational})
\]
5. **Isolate the square root**: Rearrange the equation to isolate \( \sqrt{10} \):
\[
\sqrt{10} = r - \frac{3}{4}
\]
6. **Express \( r - \frac{3}{4} \) as a single fraction**: Since \( r \) is rational, we can express it as \( \frac{a}{b} \) for some integers \( a \) and \( b \) (with \( b \neq 0 \)):
\[
\sqrt{10} = \frac{a}{b} - \frac{3}{4} = \frac{4a - 3b}{4b}
\]
7. **Analyze the right-hand side**: The right-hand side, \( \frac{4a - 3b}{4b} \), is a fraction of two integers, hence it is rational.
8. **Conclude about the square root**: If \( \sqrt{10} \) is equal to a rational number \( \frac{4a - 3b}{4b} \), then \( \sqrt{10} \) must also be rational.
9. **Use the established fact about \( \sqrt{10} \)**: However, we know that \( \sqrt{10} \) is irrational. This leads to a contradiction since we assumed that \( \frac{3}{4} + \sqrt{10} \) was rational.
10. **Conclude the proof**: Since our initial assumption that \( \frac{3}{4} + \sqrt{10} \) is rational leads to a contradiction, we conclude that:
\[
\frac{3}{4} + \sqrt{10} \text{ is irrational.}
\]
Thus, we have proved that the sum \( \frac{3}{4} + \sqrt{10} \) is indeed irrational.
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