Question

are irrational numbers always real numbers?

Answer 1

Yes, all irrational numbers are real numbers. Irrational numbers are defined as numbers that cannot be expressed as a fraction of two integers (that is, they cannot be represented in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \)). Examples of irrational numbers include \( \sqrt{2} \), \( \pi \), and \( e \).

Real numbers, on the other hand, include all the rational numbers (numbers that can be expressed as fractions) and all the irrational numbers. Together, they form the set of real numbers.

Therefore, while all irrational numbers fall within the category of real numbers, not all real numbers are irrational; some are rational.