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

In summary, two people are discussing the importance of setting goals in order to achieve success. The first person explains that setting specific and measurable goals can help to keep one focused and motivated. The second person agrees and adds that it is important to have a clear plan and to regularly review and adjust goals as needed.
  • #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
 
Mathematics news on Phys.org
  • #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.
 

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.

Similar threads

MHB Inequality involving positive real numbers
    • General Math
Replies
1
Views
896
MHB What is the Sum of x, y, and z in a Non-Negative Real Number System?
    • General Math
Replies
4
Views
927
B Principal square root of a complex number, why is it unique?
    • General Math
Replies
13
Views
3K
I What is the role of geometry and dimensional operations?
    • General Math
Replies
3
Views
263
I Partial Differential Equation solved using Products
    • General Math
Replies
2
Views
247
I Can the same argument be used for both radians and degrees in the sine function?
    • General Math
Replies
3
Views
752
I Alternating Harmonic Numbers are cool, spread the word!
    • General Math
Replies
4
Views
409
MHB Solving problem with a middle number between 2 numbers
    • General Math
Replies
1
Views
1K
A Explicit example of two complex numbers for double cover U(1) of SO(2) for a specified angle
    • General Math
Replies
2
Views
399
Proof of Infinite Set of Real Numbers Between Two Unequal Real Numbers
    • General Math
Replies
18
Views
1K

Hot Threads

Recent Insights

Back
Top