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A rational number is a number that can be written as , where and are integers.
An irrational number is a real number that is not rational.
How do we know that irrational numbers exists? Back during Pythagoras' time, they did not believe that there were irrational numbers (they were being irrational!) and you could be drowned for believing something that was true!
Let's show that is irrational:
Proof: We will prove this by contradiction. Suppose that is not irrational. Then it is rational, and hence is of the form . We may make the assumption that the fraction is in the lowest terms, or that and have no common factors.
Squaring the equation, we get that . Since is even, hence is even. Let , where is an integer. Substituting this in, we get , or that . We repeat the above argument. Since is even, thus is even.
But this says that and are both even, and hence are both divisible by 2. This contradicts the original assumption that and have no common factors!
Thus we know that is irrational, and thanks to the advance of mathematics, we do not need to sacrifice our lives.
How do you show that is irrational? What about ?