Question
set of numbers that are real and cannot be expressed by one interger divided by another
Answers
GPT-4o mini
The set of numbers that are real and cannot be expressed as one integer divided by another consists of irrational numbers.
Irrational numbers are real numbers that cannot be expressed as the ratio of two integers (i.e., \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b \neq 0 \)). Some common examples of irrational numbers include:
1. **The square root of numbers that are not perfect squares**:
- \( \sqrt{2} \)
- \( \sqrt{3} \)
- \( \sqrt{5} \)
2. **Mathematical constants**:
- \( \pi \) (the ratio of the circumference of a circle to its diameter)
- \( e \) (the base of natural logarithms)
3. **Non-repeating, non-terminating decimals**:
- 0.101001000100001... (a decimal that does not repeat or terminate)
In summary, the set of real numbers that cannot be expressed as a ratio of two integers is composed of irrational numbers.
Irrational numbers are real numbers that cannot be expressed as the ratio of two integers (i.e., \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b \neq 0 \)). Some common examples of irrational numbers include:
1. **The square root of numbers that are not perfect squares**:
- \( \sqrt{2} \)
- \( \sqrt{3} \)
- \( \sqrt{5} \)
2. **Mathematical constants**:
- \( \pi \) (the ratio of the circumference of a circle to its diameter)
- \( e \) (the base of natural logarithms)
3. **Non-repeating, non-terminating decimals**:
- 0.101001000100001... (a decimal that does not repeat or terminate)
In summary, the set of real numbers that cannot be expressed as a ratio of two integers is composed of irrational numbers.