Question
According to the Closure Property, which of these sums results in an irrational number?(1 point)
Responses
1+2–√
1 plus square root of 2
0.668357+0.488712197
0 point 6 6 8 3 5 7 plus 0 point 4 8 8 7 1 2 1 9 7
97+53
9 sevenths plus 5 thirds
4–√+5
Responses
1+2–√
1 plus square root of 2
0.668357+0.488712197
0 point 6 6 8 3 5 7 plus 0 point 4 8 8 7 1 2 1 9 7
97+53
9 sevenths plus 5 thirds
4–√+5
Answers
Answered by
GPT-4o mini
To determine which of these sums results in an irrational number, let's analyze each option:
1. **1 + 2 - √2**
- This simplifies to \(3 - \sqrt{2}\). Since \( \sqrt{2} \) is an irrational number, \(3 - \sqrt{2}\) is also irrational.
2. **0.668357 + 0.488712197**
- Both addends are rational numbers (as they can be expressed as fractions), and the sum of two rational numbers is a rational number. Thus, this sum is rational.
3. **97 + 53**
- The sum of these two integers is \(150\), which is a rational number.
4. **4 - √5 + 5**
- This simplifies to \(9 - \sqrt{5}\). Since \( \sqrt{5} \) is an irrational number, \(9 - \sqrt{5}\) is also irrational.
From the above analysis, both **1 + 2 - √2** and **4 - √5 + 5** result in irrational numbers. However, if choosing just one option based on which sum is explicitly asked, either of these would be an appropriate choice.
For a direct answer to the prompt: **1 + 2 - √2** or **4 - √5 + 5** both yield irrational sums, but **1 + 2 - √2** provides a more straightforward example. If required to select only one from a list, you can go with **1 + 2 - √2**.
1. **1 + 2 - √2**
- This simplifies to \(3 - \sqrt{2}\). Since \( \sqrt{2} \) is an irrational number, \(3 - \sqrt{2}\) is also irrational.
2. **0.668357 + 0.488712197**
- Both addends are rational numbers (as they can be expressed as fractions), and the sum of two rational numbers is a rational number. Thus, this sum is rational.
3. **97 + 53**
- The sum of these two integers is \(150\), which is a rational number.
4. **4 - √5 + 5**
- This simplifies to \(9 - \sqrt{5}\). Since \( \sqrt{5} \) is an irrational number, \(9 - \sqrt{5}\) is also irrational.
From the above analysis, both **1 + 2 - √2** and **4 - √5 + 5** result in irrational numbers. However, if choosing just one option based on which sum is explicitly asked, either of these would be an appropriate choice.
For a direct answer to the prompt: **1 + 2 - √2** or **4 - √5 + 5** both yield irrational sums, but **1 + 2 - √2** provides a more straightforward example. If required to select only one from a list, you can go with **1 + 2 - √2**.