Between any two distinct real numbers there is a rational number

In summary, the student is trying to find a rational number between x and y such that x < p/q < y. They are led down an argument for choosing q such that q(y-x)>1. They need to provide a proof that if they have q satisfying part 1 then such a p exists.
  • #1
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Homework Statement



Let x and y be real numbers with x<y and write an inequality involving a rational
number p/q capturing what we need to prove. Multiply everything in your inequality by q,
then explain why this means you want q to be large enough so that q(y-x)>1 . Explain
how you can rewrite this inequality and use the Archimedean property to find such a q.

The Attempt at a Solution


So, this is a question on a worksheet our teacher gave us to go along with the theorem in the book. Here is what I did so far:

x < p/q < y
Then multiply both sides by q as the question states:
qx < p < yq
0 < p-qx < yq-qx
0 < p-qx < q(y-x)

I am having trouble seeing why q(y-x)>1. It is obviously great than zero as the inequality states, but can someone help me see why it has to be >1??
 
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  • #2
What you are trying to do is exhibit a rational number [tex]p/q[/tex] between [tex]x[/tex] and [tex]y[/tex], that is, to choose [tex]p[/tex] and [tex]q[/tex] so that [tex]x < p/q < y[/tex]. The argument you're being led down is a strategy for choosing [tex]q[/tex], then [tex]p[/tex], so that they satisfy the conditions you want.

So, what you need to do is:
1. prove that you can choose [tex]q[/tex] so that [tex]q(y - x) > 1[/tex], i.e., prove that such a [tex]q[/tex] exists; then
2. figure out why that means that you can choose [tex]p[/tex] so that [tex]0 < p - qx < q(y - x)[/tex], i.e., give a proof that if you have [tex]q[/tex] satisfying part 1 then such a [tex]p[/tex] exists.
 
  • #3
So, we chose q>1/(y-x)?

So then,
0 < p-(x/(y-x)) < 1
0 < p < (y/(y-x))

I'm still lost on how we choose p then.
 
  • #4
You need to give one sentence (really, one phrase) of proof for why one can choose [tex]q > 1/(y - x)[/tex].

Now you need [tex]p[/tex] to satisfy [tex]0 < p - qx < q(y - x)[/tex]. Rewrite this as [tex]qx < p < qy[/tex]. You need to find an integer [tex]p[/tex] satisfying this inequality. Why does your choice of [tex]q[/tex] guarantee that there is at least one such [tex]p[/tex]? (This is the only condition you need [tex]p[/tex] to satisfy!)
 

1. What is a rational number?

A rational number is a number that can be expressed as a ratio of two integers, where the denominator is not equal to zero. For example, 1/2, 3/4, and 5/6 are all rational numbers.

2. How is a rational number different from an irrational number?

A rational number can be expressed as a fraction, while an irrational number cannot. Irrational numbers cannot be expressed as a ratio of two integers and have an infinite number of non-repeating decimal places.

3. Why is it important to know that there is a rational number between any two distinct real numbers?

Knowing that there is a rational number between any two distinct real numbers helps us to understand the concept of infinity and the density of the real number line. It also allows us to better approximate real numbers and make calculations more precise.

4. How do you find a rational number between two given real numbers?

To find a rational number between two given real numbers, you can use the following formula: (a+b)/2, where a and b are the given real numbers. This will give you the average of the two numbers, which will always be a rational number between them.

5. Can there be more than one rational number between two real numbers?

Yes, there can be an infinite number of rational numbers between two real numbers. This is because there is an infinite number of possible fractions that can be formed between any two numbers on the real number line.

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