View Single Post
CRGreathouse
CRGreathouse is offline
#8
Oct8-05, 11:15 AM
Sci Advisor
HW Helper
P: 3,680
Quote Quote by QuantumTheory
I always asked my algebra teacher math questions, namely, caluclus questions
She told me that there are infinite number of numbers between 1 and 2
Is this true? Secondly, would this be a correct limit for it?
To answer your first question, I need to make explicit one of your assumptions. This can be true or false depending on what number system you are using. There are an infinite number of numbers between 1 and 2 in [tex]\mathbb{R}[/tex], the real numbers, but only a finite number (0 or 2, depending on whether you include the endpoints) in [tex]\mathbb{Z}[/tex], the integers.

To phrase your question "mathematically", you're looking for the cardinality (size) of the interval [tex](1,2)[/tex]. This cardinality is finite (0 or 2) in the integers, infinite ([tex]\aleph_0[/tex], as large as the cardinality of the integers) in the rational numbers [tex]\mathbb{Q}[/tex], and infinite ([tex]\mathfrak{c}[/tex], as large as the cardinality of the real numbers) in the real numbers. Yes, there are as many real numbers between 1 and 2 as there are in total! Isn't that crazy?

For the second question, there is no limit because there is no order. If you pick a particular sequence between 1 and 2 you can find a limit, but there needs to be an order before this even makes sense. So, to answer your question, there is no limit.