Question
Real numbers that cannot be expressed as one integer divided by another
Answers
GPT-4o mini
Real numbers that cannot be expressed as the ratio of one integer divided by another are known as **irrational numbers**. An irrational number cannot be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).
Some common examples of irrational numbers include:
- **The square root of non-perfect squares**: For instance, \( \sqrt{2}, \sqrt{3}, \) and \( \sqrt{5} \) are irrational.
- **Mathematical constants**: For example, \( \pi \) (the ratio of the circumference of a circle to its diameter) and \( e \) (the base of natural logarithms) are both irrational.
In contrast, rational numbers can be expressed as the ratio of two integers (e.g., \( \frac{1}{2}, \frac{-3}{4}, \) etc.). The set of real numbers includes both rational and irrational numbers.
Some common examples of irrational numbers include:
- **The square root of non-perfect squares**: For instance, \( \sqrt{2}, \sqrt{3}, \) and \( \sqrt{5} \) are irrational.
- **Mathematical constants**: For example, \( \pi \) (the ratio of the circumference of a circle to its diameter) and \( e \) (the base of natural logarithms) are both irrational.
In contrast, rational numbers can be expressed as the ratio of two integers (e.g., \( \frac{1}{2}, \frac{-3}{4}, \) etc.). The set of real numbers includes both rational and irrational numbers.