In order to prove that there is no rational number whose square is 2, we will use indirect reasoning and we will show that this will lead to a contradiction. Keep in mind that what we are trying to prove here is that the square root of 2 cannot be a rational number. So, in order to do an indirect proof, we will assume that the square of 2 can be a rational number.
Indirect reasoning : Suppose that there is a rational number a/b
such that (a/b)^{2} = 2
a^{2}/b^{2} = 2
After multiplying both sides by b^{2}, we get a^{2} = 2b^{2}
Since a^{2} = 2b^{2} they must have the same prime factorization. (Does this make sense? Well, if 10 = 10, then 10 has the same prime factorization as 10)
Now, we will not prove it, but any number raised to the second power has an even number of prime factors. (4^{2} = 16 = 2 × 2 × 2 × 2. Then, 16 has 4 prime factors and 4 is an even number)
b^{2 }will also have an even number of prime factors
If b^{2} has an even number of prime factors, 2b^{2} will have an odd number of prime factors (the 2 next to b^{2} adds an extra prime factor)
This is a contraction because we said before in blue that they a^{2} and 2b^{2} have the same prime factorization. However, now we are saying that they don't in the sentence shown in red.
Therefore, it was wrong for us to assume that the square of 2 can be a rational number.
Conclusion: There is no rational number whose square is 2.
Mar 13, 19 11:50 AM
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Mar 13, 19 11:50 AM
Learn how to derive the equation of an ellipse when the center of the ellipse is at the origin.