Determine which categories that the following numbers belong a number may be in more than one category Neutral, intriguer, and irrational

To categorize the numbers you've listed into "neutral," "intriguer," and "irrational," let's first define each category:

1. **Neutral**: In mathematical contexts, the term "neutral" typically refers to the number zero, as it is neither positive nor negative.

2. **Intriguer**: While this term isn't standard in mathematics, it can possibly refer to interesting or notable numbers, but for this context let's assume it relates to numbers that can be expressed as fractions (rational numbers) as well.

3. **Irrational**: These are numbers that cannot be expressed as a simple fraction; they have non-repeating, non-terminating decimal expansions. Common examples include the square root of non-perfect squares, π, and e.

Now, let's analyze each of the given numbers:

- **Zero (0)**:
- **Neutral**: Yes (it is the neutral element in addition).
- **Intriguer**: Yes (it can be expressed as a fraction, for example, 0 = 0/1).
- **Irrational**: No.

- **One (1)**:
- **Neutral**: No.
- **Intriguer**: Yes (it can be expressed as a fraction, e.g., 1 = 1/1).
- **Irrational**: No.

- **Negative Four (-4)**:
- **Neutral**: No.
- **Intriguer**: Yes (it can be expressed as a fraction, e.g., -4 = -4/1).
- **Irrational**: No.

- **Seven over Negative Eleven (7 / -11)**:
- **Neutral**: No.
- **Intriguer**: Yes (it's a rational number).
- **Irrational**: No.

- **Square Root of Five (√5)**:
- **Neutral**: No.
- **Intriguer**: No (it cannot be expressed as a fraction and is approximately 2.236).
- **Irrational**: Yes (because it cannot be written as a fraction).

### Summary:
- **Neutral**: 0
- **Intriguer**: 0, 1, -4, 7/-11
- **Irrational**: √5

Thus:
- **0**: Neutral, Intriguer
- **1**: Intriguer
- **-4**: Intriguer
- **7/-11**: Intriguer
- **√5**: Irrational

No number is both neutral and irrational.