Question

which list shows an accurate hierarchy of number sets

Answer 1

The hierarchy of number sets can be represented as follows, from the most specific (smallest set) to the most general (largest set):

1. **Natural Numbers (\(\mathbb{N}\))**: The set of positive integers (1, 2, 3, ...). Depending on the definition, it may or may not include 0.

2. **Whole Numbers**: Often includes all natural numbers plus zero (0, 1, 2, 3, ...).

3. **Integers (\(\mathbb{Z}\))**: The set of all whole numbers, both positive and negative, including zero (..., -3, -2, -1, 0, 1, 2, 3, ...).

4. **Rational Numbers (\(\mathbb{Q}\))**: The set of numbers that can be expressed as a fraction of two integers, where the denominator is not zero. This includes integers, finite decimals, and repeating decimals.

5. **Irrational Numbers**: Numbers that cannot be expressed as a fraction of two integers, such as \(\sqrt{2}\) or \(\pi\).

6. **Real Numbers (\(\mathbb{R}\))**: The set of all rational and irrational numbers.

7. **Complex Numbers (\(\mathbb{C}\))**: Numbers that have a real part and an imaginary part (a + bi, where \(i\) is the imaginary unit and \(a\) and \(b\) are real numbers).

In summary, the hierarchy can be illustrated as:
\[ \mathbb{N} \subset \text{Whole Numbers} \subset \mathbb{Z} \subset \mathbb{Q} \cup \text{Irrational Numbers} \subset \mathbb{R} \subset \mathbb{C} \]

This hierarchy reflects how each set builds on the previous one, encompassing more numbers.