Prove that the square root of 3 is not rational

In summary, the conversation discusses proving that the square root of 3 is not a rational number. The definition of an irrational number is given, and it is explained that if p is the square root of 3 and q is not an element of z, then the square root of 3 cannot be rational. The conversation also mentions a previous discussion about proving that the square root of 2 is irrational. Overall, the conversation is focused on understanding and proving the irrationality of the square root of 3.
  • #1
chocolatelover
239
0

Homework Statement


Show that the square root of 3 is not rational


Homework Equations





The Attempt at a Solution



A number is irrational if χ is not ε. Q=p/q: p, q ε z and q is not=0, z=integers

If p/q: p, q is not ε or q=0, then square 3 is rational. If p=square root of 3 and q is not ε, then the square root of 3 cannot be rational.

Could someone please tell me if this is correct and if not show me what I need to do?

Thank you very much
 
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  • #2
What is your [tex]\epsilon[/tex]?
Do you know how to show that [tex]\sqrt{2}[/tex] is irrational?
 
  • #3
x ε is "x is an element of x" Yes, I do.

Thank you very much

Now I know how to do it

Regards
 

1. What does it mean for a number to be rational?

A rational number is any number that can be written as a fraction of two integers. This includes whole numbers, fractions, and terminating or repeating decimals.

2. Why is proving that the square root of 3 is not rational important?

Proving that the square root of 3 is not rational is important because it helps us understand the properties of real numbers and the concept of irrational numbers. It also has practical applications in various fields of science and mathematics.

3. How do you prove that the square root of 3 is not rational?

To prove that the square root of 3 is not rational, we use the method of proof by contradiction. We assume that the square root of 3 is rational and then show that this leads to a contradiction, thus proving that our initial assumption was false.

4. Can you give an example of proof by contradiction for the square root of 3?

Sure, let's assume that the square root of 3 is rational and can be expressed as a fraction a/b, where a and b are integers with no common factors. By squaring both sides, we get 3 = a^2/b^2. This means that 3b^2 = a^2. Since a^2 is divisible by 3, a must also be divisible by 3. This means that a^2 is divisible by 9. But then 3b^2 = a^2 implies that b^2 is also divisible by 3, and therefore b must also be divisible by 3. This contradicts our initial assumption that a and b have no common factors, thus proving that the square root of 3 is not rational.

5. What are the implications of proving that the square root of 3 is not rational?

Proving that the square root of 3 is not rational has implications in various fields such as number theory, geometry, and physics. It also highlights the fact that not all numbers can be expressed as a ratio of two integers, leading to a deeper understanding of the infinite and complex nature of numbers.

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