Can you prove that #sqrt(3)# is irrational?
#sqrt(3)# is irrational by contradiction. E.g. show that it is not rational
The idea behind the proof is pretty simple, although depending on how rigorous you want to be, the setup can make it a little lengthy. The idea is that you suppose it's of the form
The same technique can be used to show that
Suppose, to the contrary, that
Substituting those in, we get
However, as the left hand side of this is an odd number and the right is even, we have reached a contradiction. Thus our initial premise was false, meaning
Well, we'll prove it by proving that it's not a rational number. I'm really sorry this is long but hopefully I explain every step thoroughly.
So, let's start with the definition of a rational number:
Rational Number: a real number that can be expressed through a fraction of REAL, INTEGER, NONZERO numbers. The fraction has to be in it's lowest form (
Remember an integer is a whole number (1, 2, 3, etc) and not a fraction of any sort (so not 3.592 or
We know that 5 is rational because it can be expressed as
So, let's prove that
Let's start with the statement in the previous paragraph.
And now, we divide both sides by 3.
Ok so now we need to use some logic. We know both
Now, this also means that
We figured that out through simple substitution of "an integer" and
Ok, so back to the problem. We just figured out that
Mathematical Rule: A square of an integer is divisible by the same prime numbers that it's square root is divisible by.
This rule means that
Here's a pretty simple way to look at it. Let's say that
So, now we know that
Let's keep going here. Since we know that
If we square both sides in the equation we made, we get that:
Now, let's substitute the equation we just made above into the original equation. The original is:
Our new one is:
Let's divide both sides by 3 to get:
And divide both sides by 3 again:
And suddenly we find ourselves repeating ourselves. Remember when we proved that
We can prove that b is divisible by 3 since c is an integer just like we proved that
So, I'm going to skip over that step but re-read the part where we proved a was divisible by 3 if you want to jog your memory.
Ok, we're getting close to finishing. Let's examine the definition of a rational number again:
Rational Number: a real number that can be expressed through a fraction of REAL, INTEGER, NONZERO numbers. The fraction has to be in it's lowest form (1/2 instead of 2/4).
Based on what we have proven,
Since it's not rational, it has to be irrational.
Thanks for reading, sorry for it being so long!!!!