Square root of 3 is irrational

In summary, the conversation discusses proving that the square root of 3 is irrational using a similar method as the proof for the square root of 2. However, when trying to show that p^2 is even, a contradiction arises. The conversation then suggests using divisibility by three instead.
  • #1
Thecla
132
10
I am trying to prove sqrt(3) is irrational. I figured I would do it the same way that sqrt(2) is irrational is proved:
Assume sqrt(2)=p/q
You square both sides.
and you get p^2 is even, therefore p is even.
Also q^2 is shown to be even along with q.
This leads to a contradiction.
However doing this with sqrt(3), you get 3q^2=p^2. Now you can't show p^2 is even.Also now I am stuck. How do I continue from here?
 
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  • #2
So maybe when going from 2 to 3, you don't need to use evenness but 'divisible by three' ;)
 
  • #3
Thanks jacobrhcp. If p^2 =3q^2, then p must be a multiple of 3. The same goes for q^2 and q, both are multiples of 3.
 

FAQ 1: What does it mean for a number to be irrational?

When a number is irrational, it means that it cannot be expressed as a fraction of two integers. In other words, there is no whole number or fraction that can represent the number.

FAQ 2: How do we know that the square root of 3 is irrational?

The proof for the irrationality of the square root of 3 is known as the "proof by contradiction". It involves assuming that the square root of 3 can be expressed as a fraction, and then showing that this assumption leads to a contradiction.

FAQ 3: Can you give an example of why the square root of 3 is irrational?

Sure! Let's assume that the square root of 3 can be expressed as a fraction, say a/b. Then, we can square both sides to get 3 = (a/b)^2. This can be rearranged to 3b^2 = a^2. Since both a and b are integers, this means that a^2 is divisible by 3, which also means that a must be divisible by 3. Similarly, b^2 must also be divisible by 3, meaning that b must also be divisible by 3. But this contradicts our assumption that a and b have no common factors, since both a and b are divisible by 3.

FAQ 4: Are there other numbers besides the square root of 3 that are also irrational?

Yes, there are infinitely many irrational numbers, and in fact, most real numbers are irrational. Some other famous examples of irrational numbers include pi and the golden ratio.

FAQ 5: Why is it important to know that the square root of 3 is irrational?

Understanding the concept of irrational numbers, such as the square root of 3, helps us to better understand the properties of numbers and their relationships. It also has practical applications in fields such as mathematics, physics, and engineering.

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