Yes, common differences can be irrational.
The provided reference explicitly states that the common difference in a sequence or series can be an irrational number. It also highlights that it can be rational, but it cannot be zero.
To elaborate:
- Definition of Common Difference: The common difference refers to the constant value added or subtracted between consecutive terms in an arithmetic sequence.
- Types of Numbers:
- Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers, and q ≠ 0. Examples include 1/2, -3, 0.75.
- Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers. Examples include √2, π, e.
Therefore, the common difference, being a numerical value, can fall into either the rational or irrational categories.
