Suppose you were teaching and some students asked you if there is a smallest positive number. Write a sequence of questions that you would ask the students to help convince them that there is no smallest positive number.
1. Let's start by considering the concept of a positive number. Can anyone define what a positive number is?
2. Are positive numbers ever represented by negative signs? Why or why not?
3. Imagine we have a set of positive numbers, such as {1, 2, 3, 4, 5}. Can you identify the property that all these numbers share?
4. Now, think about the number 1. Is it the smallest positive number?
5. What about numbers less than 1? For example, 0.5 or 0.1? Are these considered positive numbers?
6. Can someone provide a number that is smaller than any other positive number?
7. If someone suggests a number like 0.0001, can you always find a smaller positive number?
8. Let's imagine a scenario where we already have a number, say 0.00001, that is considered the "smallest positive number." Could we divide this number by 2 and obtain a smaller positive number?
9. If we continued dividing our "smallest positive number" by 2, what would happen?
10. From what we've discussed, does it seem possible to find a smallest positive number? Why or why not?
11. Can you summarize why we concluded that there is no smallest positive number?
12. What implications does this have for mathematical concepts that involve positive numbers? For example, comparing numbers or finding limits?
13. Can you think of any real-life examples where the absence of a smallest positive number might be relevant or useful?
14. Do you have any remaining questions or concerns about the concept of a smallest positive number?