https://file.aiquickdraw.com/mathbot/file/1725900464292zel1g4iq.jpg

To solve this problem, we need to determine the type of number represented by \(3\pi + 1\).

### Step-by-step Analysis

1. **Understanding \( \pi \):**  
   \( \pi \) is an irrational number, meaning it cannot be expressed as a fraction of two integers. It has a non-terminating, non-repeating decimal representation (approximately 3.14159...).

2. **Calculating \( 3\pi \):**  
   Multiplying \( \pi \) by 3, we still get an irrational number. An irrational number multiplied by a rational number (other than zero) remains irrational. Therefore, \( 3\pi \) is irrational.

3. **Adding 1 to \( 3\pi \):**  
   When an irrational number (like \( 3\pi \)) is added to a rational number (like 1), the result remains irrational. Thus, \( 3\pi + 1 \) is still an irrational number.

### Conclusion
Based on this analysis, the correct answer is:
- **D. Irrational**

### Incorrect Choices:
- **A. Whole number**: Whole numbers are non-negative integers (0, 1, 2, ...). \(3\pi + 1\) is not a whole number.
- **B. Integer**: Integers include both positive and negative whole numbers and zero. \(3\pi + 1\) is not an integer.
- **C. Rational**: Rational numbers can be expressed as the quotient of two integers. Since \(3\pi + 1\) is irrational, it cannot be rational.

Would you like more details on this explanation or have any questions? 

### Related Questions:
1. What is the difference between rational and irrational numbers?
2. How can we prove that a number like \( \pi \) is irrational?
3. What happens when you add two irrational numbers? Can the result be rational?
4. Are all non-repeating decimals irrational numbers?
5. How can you determine if a square root is rational or irrational?

### Tip:
Remember that irrational numbers cannot be written as a simple fraction. When performing operations like addition or multiplication, the result is often still irrational.

Explore whether the expression 3π + 1 is a rational or irrational number. This problem delves into the properties of irrational numbers and real numbers, and provides a clear explanation of why 3π + 1 is irrational. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation