Prove: Irrational Numbers Have Rational Numbers Between

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In summary, irrational numbers are numbers that cannot be expressed as a ratio of two integers, while rational numbers can. The density of rational numbers theorem proves that there are always rational numbers between any two irrational numbers, but there is no specific method to find them. This proof is important for understanding the relationship between irrational and rational numbers and their interconnected nature.
  • #1
sedaw
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need to prove that : between any two irrational numbers there is at least one rational number .

TNX .
 
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  • #2
It's not true, there is none between [itex]\sqrt{2}[/itex] and [itex]\sqrt{2}[/itex].

Also, what would you like to use for a proof? Analysis? Topology?
 
  • #3
CompuChip said:
It's not true, there is none between [itex]\sqrt{2}[/itex] and [itex]\sqrt{2}[/itex].

Also, what would you like to use for a proof? Analysis? Topology?


Analysis, for x and y that x<y
 

1. What are irrational numbers?

Irrational numbers are numbers that cannot be written as a ratio of two integers. They are numbers that cannot be expressed as terminating or repeating decimals.

2. What are rational numbers?

Rational numbers are numbers that can be expressed as a ratio of two integers. They can be written as terminating or repeating decimals.

3. How can you prove that irrational numbers have rational numbers between them?

This can be proven using the theorem called the "density of rational numbers". This theorem states that between any two irrational numbers, there exists an infinite number of rational numbers.

4. Is there a specific method to find a rational number between two irrational numbers?

There is no specific method to find a rational number between two irrational numbers. However, by using the density of rational numbers theorem, we can always find a rational number between any two irrational numbers.

5. Why is it important to prove that irrational numbers have rational numbers between them?

This proof is important because it helps us understand the relationship between irrational and rational numbers. It also shows that these two types of numbers are not completely separate, but rather, they are interconnected and can be used to describe the same quantities in different ways.

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