SOLUTION: If x < y, then x<(x+y)/2 < y. Use this fact to give a convincing argument for the following: Between any two rational numbers there are infinitely many rational numbers.

Question 442256: If x < y, then x<(x+y)/2 < y. Use this fact to give a convincing argument for the following: Between any two rational numbers there are infinitely many rational numbers.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Keep in mind that if x and y are rational, then

is also a rational number. The proof is quite simple; simply let

and

, where a,b,c,d are integers and b and d are nonzero. Then

. This is a ratio of two integers, so the average of x and y is a rational number.

Hence, if we let

, we may let

, and in general,

. All of these numbers are rational, and we can go on indefinitely, so there must be infinitely many rational numbers, regardless of how far x and y are.

RELATED QUESTIONS
"Tell whether this equation represents a direct variation. If so, identify the constant... (answered by rothauserc)

if x and y are positive real numbers, use the fact that t+(1/t)>=2 for positive numbers... (answered by venugopalramana)

Construct a regular proof to derive the conclusion of the following argument: 1. X >Y... (answered by Edwin McCravy)

How would I answer these problems I am really stuck. Any help would be greatly... (answered by stanbon)

Use the quadratic formula to determine the x-intercepts (if any) of the following... (answered by Earlsdon)

Use the quadratic formula to determine the x-intercepts (if any) of the following... (answered by scott8148)

Hello and thank you, My question is..... can you give me step by step instructions on... (answered by bucky)

I am given a problem and I must use a t-chart and pick any number for x, and solving it... (answered by rapaljer)

Click on the link to solve this problem: I'm doing the review section in my calculus... (answered by ichudov)