Question

Which list shows an accurate hierarchy of number sets within the real number system from largest to smallest?

Answer 1

In the real number system, the hierarchy of number sets from largest to smallest is as follows:

1. **Real Numbers (ℝ)** - This includes all rational and irrational numbers.
- Example: All integers, fractions, and irrational numbers like √2, π.

2. **Rational Numbers (ℚ)** - This includes all numbers that can be expressed as a fraction of two integers (where the denominator is not zero).
- Example: 1/2, 3.75, -1 (which can be expressed as -1/1).

3. **Irrational Numbers** - This is a subset of real numbers that cannot be expressed as a simple fraction. They cannot be written as the ratio of two integers.
- Example: √2, π, e.

4. **Integers (ℤ)** - This includes whole numbers and their negatives.
- Example: -3, 0, 2.

5. **Whole Numbers** - This is a subset of integers that includes all non-negative integers.
- Example: 0, 1, 2, 3, ...

6. **Natural Numbers (ℕ)** - This includes the positive integers, often starting from 1.
- Example: 1, 2, 3, ...

So, a complete hierarchy from largest to smallest would be:

**Real Numbers (ℝ) > Rational Numbers (ℚ) > Irrational Numbers > Integers (ℤ) > Whole Numbers > Natural Numbers (ℕ).**