Proving the Existence of Irrational Numbers Between Two Reals

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In summary, the conversation discusses the proofs of two theorems: the existence of an irrational number between any two real numbers, and the existence of infinitely many rational and irrational numbers between any two real numbers. The first proof involves choosing specific values for $a$, $b$, and $m$, but it is pointed out that this may not be the most convincing approach. The second proof combines the previous theorem with the existence of a rational between any two real numbers. It is noted that this proof is dependent on the theorem being used.
  • #1
OhMyMarkov
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Hello everyone!

I want to prove that between two reals, there exists an irrational. This is what I got:

$\forall x \in R$, $\exists m \in Z$ s.t. $m-1 < x < m$. In particular, $x\notin Q$.
$\exists a,b \in R$ s.t. $ax$ and $ax+b$ are irrational. Also, $a(m-1)<ax<am$, $a(m-1)+b<ax+b<am+b$.
End of proof.I also want to prove that between two reals, there are infinitely many rationals and irrationals. This is my proof:

(Using (1) the above theorem, and (2) the theorem that says that there exists a rational between every two reals)

Combining (1) and (2), between two reals, there exists a real. $\forall x,z \in R$ where $x<z$, $\exists y \in R$ s.t. $x<y<z$. In other words, $\forall x_0, x_1 \in R$ where $x_0 < x_1$, $\exists x_2 \in R$ s.t. $x_0<x_1<x_2$. Apply this again, $x_1<x_2<x_3$. Apply this N-times, $x_N<x_{n+1}<x_{N+2}$. If we let N grow indefinitely, we can say that there are infinitely many reals between two reals.Are these proofs correct? I am very new to establishing proofs in analysis. Any comments or criticism is highly appreciated. :)
 
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  • #2
OhMyMarkov said:
I want to prove that between two reals, there exists an irrational. This is what I got:

$\forall x \in R$, $\exists m \in Z$ s.t. $m-1 < x < m$. In particular, $x\notin Q$.
$\exists a,b \in R$ s.t. $ax$ and $ax+b$ are irrational. Also, $a(m-1)<ax<am$, $a(m-1)+b<ax+b<am+b$.
End of proof.

Hi OhMyMarkov, :)

You have finally obtained,

\[a(m-1)+b<ax+b<am+b\]

From this, I don't understand how you came to the final conclusion that between any two reals there is an irrational number. Can you please elaborate?

Kind Regards,
Sudharaka.
 
  • #3
Of course, $x$ is irrational, and so is $ax+b$ for choice of $ax+b$. Also, $a(m-1)+b$ and $am+b$ are real.
 
  • #4
OhMyMarkov said:
Of course, $x$ is irrational, and so is $ax+b$ for choice of $ax+b$. Also, $a(m-1)+b$ and $am+b$ are real.

There is something incorrect in your proof. You are choosing specific values for \(a\), \(b\) and \(m\). Hence the numbers, $a(m-1)+b$ and $am+b$ are dependent upon your choice of \(x\). This is not what you need to prove. You should take any two real numbers and show that in between those two reals there is an irrational.
 
  • #6
OhMyMarkov said:
Hello everyone!
I want to prove that between two reals, there exists an irrational. This is what I got:
$\forall x \in R$, $\exists m \in Z$ s.t. $m-1 < x < m$. In particular, $x\notin Q$.
$\exists a,b \in R$ s.t. $ax$ and $ax+b$ are irrational. Also, $a(m-1)<ax<am$, $a(m-1)+b<ax+b<am+b$.
End of proof.
This problem depends on the theorem Between any two numbers there is a rational number.
With that theorem having been done this problem is trival.

If $a<b$ then $\sqrt2 a<\sqrt2 b$. So $\exists r\in\mathbb{Q}\setminus\{0\}$ such that $\sqrt2 a< r <\sqrt2 b$.

Now the irrational number $\dfrac{r}{\sqrt2}$ is between $a~\&~b~.$
 

FAQ: Proving the Existence of Irrational Numbers Between Two Reals

What are numbers between two given numbers?

Numbers between two given numbers are any numbers that fall between the two given numbers on a number line. They can be whole numbers, decimals, or fractions.

How do you find numbers between two given numbers?

To find numbers between two given numbers, you can count up or down on a number line or use a mathematical equation. For example, to find numbers between 1 and 10, you can count 2, 3, 4, 5, 6, 7, 8, 9 or use an equation such as (1+10)/2 = 5.5.

What is the difference between numbers between and consecutive numbers?

Numbers between are any numbers that fall between two given numbers, while consecutive numbers are numbers that come one after the other with no gaps in between. For example, 3 and 5 are numbers between 1 and 10, while 3 and 4 are consecutive numbers.

Can there be an infinite amount of numbers between two given numbers?

No, there is a finite number of numbers between any two given numbers. This is because numbers between two given numbers are considered to be part of a set of numbers, and sets have a limited number of elements.

What is the importance of understanding numbers between two given numbers?

Understanding numbers between two given numbers is important in various mathematical concepts such as fractions, decimals, and measurement. It also helps with number sense and being able to estimate and approximate numbers.

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