Key Concepts:
Natural Numbers (\(\mathbb{N}\)):
The set of positive counting numbers: \[ \mathbb{N} = \{1, 2, 3, 4, 5, \dots\} \] Some definitions include 0 as a natural number: \(\mathbb{N}_0 = \{0, 1, 2, 3, \dots\}\).Whole Numbers (\(\mathbb{W}\)):
The set of natural numbers including zero: \[ \mathbb{W} = \{0, 1, 2, 3, \dots\} \]Integers (\(\mathbb{Z}\)):
The set of whole numbers and their negative counterparts: \[ \mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\} \]Rational Numbers (\(\mathbb{Q}\)):
Numbers that can be expressed as a fraction \(\frac{a}{b}\), where \(a, b \in \mathbb{Z}\) and \(b \neq 0\). \[ \mathbb{Q} = \left\{\frac{a}{b} \mid a, b \in \mathbb{Z}, b \neq 0 \right\} \] Examples: \(\frac{1}{2}, -3, 0.75, 5\).Irrational Numbers:
Numbers that cannot be written as fractions, having non-repeating, non-terminating decimals. \[ \sqrt{2}, \pi, e \]Real Numbers (\(\mathbb{R}\)):
The set of all rational and irrational numbers. \[ \mathbb{R} = \mathbb{Q} \cup \text{Irrational Numbers} \]Complex Numbers (\(\mathbb{C}\)):
The set of numbers in the form: \[ a + bi, \quad \text{where } a, b \in \mathbb{R} \text{ and } i = \sqrt{-1}. \]Examples:
Example 1: Classify the number \(\frac{5}{2}\).
Solution:
\(\frac{5}{2}\) is a fraction of two integers, so it belongs to **rational numbers (\(\mathbb{Q}\))**. Since it is not a whole number, it is not an integer.Example 2: Determine whether \(\sqrt{16}\) and \(\sqrt{17}\) are rational.
Solution:
\[ \sqrt{16} = 4, \quad \text{which is an integer and therefore rational.} \] \[ \sqrt{17} \approx 4.123, \quad \text{which is a non-repeating, non-terminating decimal, so it is irrational.} \] Thus, \(\sqrt{16}\) is **rational**, while \(\sqrt{17}\) is **irrational**.