Is There Always a Real Number Between Two Arbitrary Real Numbers?

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  • #1
courtrigrad
1,236
2
Hello all

I need help with the following proofs

1. If x and y are arbitrary real numbers, prove that there is at least one real z satisfying x < z < y. (Do I just use the Archimedian Property?)

Thanks
 
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  • #2
courtrigrad said:
(Do I just use the Archimedian Property?)

Yep. :smile:
 
  • #3
No. and it isn't true as stated.

x and y must be distinct, and the requirement of being R is unnecessary, since it is true for Q (and C).


what is (x+y)/2?
 
  • #4
Good catch. I was thinking of rational numbers between arbitrary reals.
 

FAQ: Is There Always a Real Number Between Two Arbitrary Real Numbers?

What is number theory?

Number theory is a branch of mathematics that deals with the properties and relationships of integers, or whole numbers.

What is the importance of proof in number theory?

Proof is essential in number theory because it provides rigorous and logical reasoning for mathematical statements and helps to establish their validity and truth.

What are some common techniques for proving statements in number theory?

Common techniques for proof in number theory include direct proof, proof by contradiction, proof by induction, and proof by cases.

What are prime numbers and why are they important in number theory?

Prime numbers are positive integers that are only divisible by 1 and themselves. They are important in number theory because they are the building blocks of all other integers and play a crucial role in many mathematical concepts and applications.

What is the Goldbach Conjecture and why is it significant in number theory?

The Goldbach Conjecture is a famous unsolved problem in number theory that states that every even integer greater than 2 can be expressed as the sum of two prime numbers. It is significant because it has been tested for millions of cases, but has yet to be proven or disproven, making it a fascinating challenge for mathematicians.

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