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In order to determine whether or not a number is rational, we much see whether or not the number can be expressed as a fraction with an integer numerator and denominator.
Clearly, `3/4`, 2, and -1 are rational because each can easily be expressed as a fraction.
There are plenty of ways to show that the square root of 2 is irrational. We'll summarize one here:
Suppose `sqrt(2)` is rational and can be expressed as a fully-reduced fraction `n/m`. Now, the following must hold true:
`sqrt(2)^2 = n^2/m^2`
`2 = n^2/m^2`
`2m^2 = n^2`
Because `n^2` is equivalent to an even number, we know that n, itself, must be even. Therefore, we can express n as another number, `2k.`
Now, we can restate the fraction:
`sqrt(2) = (2k)/m`
Again, let's take the square of both sides and solve for `m^2` :
`sqrt(2)^2 = (2k)^2/m^2`
`2 = (4k^2)/m^2`
`2m^2 = 4k^2`
`m^2 = 2k^2`
Now, we come up on the fact that `m` must be even, just like `n`, and it can be expressed as 2l.
However, now we come up on a contradiction! Remember how we defined `n/m` as the most-reduced form of the fraction that could represent `sqrt2`. However, the fact that both n and m are even allow `n/m` to be further-reduced. contradicting our assumption. Therefore, `sqrt(2)` must be irrational.
There are other proofs at the link below.
In a similar way, you can prove that `sqrt(3)` is also irrational.
So, our final answer is `sqrt2` and `sqrt3`.
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