Question

What type of number is
\[3\pi+1\]?
Choose all answers that apply:
Choose all answers that apply:
(Choice A) Whole number
A
Whole number
(Choice B) Integer
B
Integer
(Choice C) Rational
C
Rational
(Choice D) Irrational
D
Irrational

Answer 1

To determine the type of number represented by \(3\pi + 1\), we start by analyzing the components of the expression.

1. **Identify the nature of \(\pi\)**:
\(\pi\) is known to be an irrational number, meaning it cannot be expressed as a fraction of two integers.

2. **Multiply \(\pi\) by 3**:
When you multiply an irrational number like \(\pi\) by a rational number (in this case, 3), the result is still irrational. Therefore, \(3\pi\) is irrational.

3. **Add 1 to \(3\pi\)**:
Adding a rational number (1 is rational) to an irrational number (\(3\pi\)) will always yield an irrational number. Thus, \(3\pi + 1\) remains irrational.

Now, we can evaluate the answer choices based on this information:

- **Whole number**: A whole number is a non-negative integer (0, 1, 2, ...). Since \(3\pi + 1\) is irrational, it cannot be a whole number.
- **Choice A** is incorrect.

- **Integer**: An integer includes whole numbers, negative whole numbers, and zero. Since \(3\pi + 1\) is irrational, it cannot be an integer.
- **Choice B** is incorrect.

- **Rational**: A rational number can be expressed as a fraction of two integers. Since \(3\pi + 1\) is irrational, it cannot be rational.
- **Choice C** is incorrect.

- **Irrational**: As we've established, \(3\pi + 1\) is an irrational number.
- **Choice D** is correct.

Therefore, the answer is:
**(Choice D) Irrational.**