Irrationality of Difference of two numbers

In summary, using the fact that \sqrt{3} is an irrational number, it can be shown that the square root of 8 minus the square root of 3 is an irrational number.
  • #1
smiles988
6
0
Proof of Irrationality

How can I prove that the square root of 8 minus the square root of 3 is an irrational number using the fact that the square root of 3 is an irrational number? I know I need to use a proof by contradiction, but I am stuck after that.
 
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  • #2

Homework Statement


[tex]\sqrt{8}[/tex]-[tex]\sqrt{3}[/tex] is an irrational number.
Use the fact that [tex]\sqrt{3}[/tex] is an irrational number to prove the following theorem.


Homework Equations


A rational number can be written in the form [tex]\frac{p}{q}[/tex] where p and q are integers in lowest terms.


The Attempt at a Solution


I know that I need to use a proof by contradiction to solve this problem. Therefore, we would assume that [tex]\sqrt{8}[/tex]-[tex]\sqrt{3}[/tex] is an rational number and that [tex]\sqrt{3}[/tex] is an irrational number and try and reason to a contradiction. I am stuck and don't know how to get to the contradiction.
 
  • #3
well it's actually asking to prove that
(sqrt(2)+sqrt(3))+sqrt(2)
is irrational.
assume it equals p/q where (p,q)=1
then multiply by sqrt(8)-sqrt(3)
youll get that sqrt(8)-sqrt(3)=5q/p
and you also have sqrt(8)+sqrt(3)=p/q
so you get that 4sqrt(2)=5q/p+p/q
which yields: sqrt(2)=(5q/p+p/q)/4 which is a contradiction.
you could have easily have done it with sqrt(3) instead but it doesn't matter.
 
Last edited:
  • #4
well, i had mistaken sqrt(8)-sqrt(3) with sqrt(8)+sqrt(3) but it doenst matter the same idea will work as well as in this case.
 
  • #5
Suppose that it is, then this implies that the square root of 8 is what?
 
  • #6
Just to add something to review my number theory.Another idea:

I think a nice general result is that for x a pos. integer, sqr(x) is rational iff x is a perfect square (an integer, of course). Think x=a^2/b^2 , so a^2x=b^2 . Then think of what the factorization of x needs to satisfy in order for a^2x to be a perfect square.

x=p_1^e_1...p_ne^n .



Then , re your problem, think of what happens when you square your expression.
 
  • #7
I believe this is the third time you have posted this same question in a separate thread!
 

1. What is the irrationality of difference of two numbers?

The irrationality of difference of two numbers refers to the fact that the result of subtracting two irrational numbers may also be irrational. This means that the difference cannot be expressed as a simple fraction or decimal, and the digits after the decimal point will continue infinitely without repeating.

2. Can the difference of two rational numbers be irrational?

Yes, it is possible for the difference of two rational numbers to be irrational. This occurs when the two rational numbers have a large difference in their decimal values, resulting in an infinite and non-repeating decimal. For example, the difference between 3.257 and 3.256 can be expressed as 0.001, an irrational number.

3. How do we know if the difference of two numbers is irrational?

To determine if the difference of two numbers is irrational, we can use the rational root theorem. This theorem states that if a polynomial has rational coefficients, any rational root of the polynomial must have a numerator that is a factor of the constant term and a denominator that is a factor of the leading coefficient. If the difference of two numbers does not satisfy this condition, then it is irrational.

4. What is an example of an irrational difference of two numbers?

A classic example of an irrational difference of two numbers is the difference between the square root of 2 and the square root of 3. Both of these numbers are irrational, and their difference is also irrational, as it cannot be expressed as a simple fraction or decimal.

5. Why is the concept of irrationality of difference of two numbers important?

The concept of irrationality of difference of two numbers is important in mathematics because it helps us understand the complexity and infinite nature of numbers. It also has practical applications in fields such as physics and engineering, where precise measurements and calculations are essential. Understanding irrational numbers and their properties can also lead to new mathematical discoveries and advancements in various fields of science.

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