The set of numbers that are real numbers but cannot be expressed as one integer divided by another is the set of irrational numbers.
Irrational numbers are real numbers that cannot be represented as the quotient of two integers, where an integer divided by another integer results in a rational number. In other words, if a number cannot be expressed in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \), then it is classified as irrational.
Examples of irrational numbers include:
- The square root of non-perfect squares (e.g., \( \sqrt{2}, \sqrt{3}, \sqrt{5} \), etc.)
- The number \( \pi \)
- The number \( e \) (Euler's number)
In summary, irrational numbers are the real numbers that cannot be represented as a fraction of two integers.