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Skip to 0 minutes and 11 secondsHello. Welcome back to a step in practice. We are dealing here with rational numbers. In exercise 1, we are asked to prove that square root of 5 is not a rational number. Well, we proceed as Francis showed us, that square root of 2 is not a rational number. So we assume the opposite. We assume, by contradiction, that the opposite is true. So assume that square root of 5 belongs to the set of rational numbers, and we are going to find a contradiction.

Skip to 0 minutes and 57 secondsWell, stating that the square root of 5 is a rational number means that it can be written as a quotient of two natural numbers a and b, and we may assume that there are no common factors in a and b.

Skip to 1 minute and 14 secondsWithout common factors.

Skip to 1 minute and 22 secondsThis implies that b times square root of 5 equals a. So by taking the square, we get that 5 times b to the square is the square of a. In particular, 5 divides a to the square.

Skip to 1 minute and 49 secondsBut if 5 divides a to the square, necessarily 5 divides a. Otherwise, if 5 does not appear into the decomposition of a, it cannot appear in the decomposition of a to the square. So as I say, really 5 divides a. Well, but then 5 to the square divides a to the square. Which is 5 times b square. And so 5 divides b to the square. And again we get that 5 divides b.

Skip to 2 minutes and 31 secondsBut now, 5 divides, well, a, and also divides b. This means that 5 appears into the composition of a and in the composition of b. So they have a common factor.

Skip to 2 minutes and 51 seconds5 is a common factor, to a and b. But this is a contradiction, because we assumed that they had no common factors. So we get the contradiction.

Skip to 3 minutes and 11 secondsSo it was not correct to assume that square root of 5 is a rational number. Thus square root of 5 does not belong to the set of rational numbers.

Skip to 3 minutes and 25 secondsIn exercise 2, we are asked to prove that 1 plus square root of 2 is not a rational number. Again, assume it is a rational number. Assume that 1 to square root of 2 is a rational number. Well, we get that square root of 2 equals q minus 1. And q is a rational number, 1 is a rational number, so the difference is again a rational number.

Skip to 4 minutes and 4 secondsBut it is false. Because square root of 2, as Francis showed us, is not a rational number. So contradiction-- we get a contradiction.

Skip to 4 minutes and 17 secondsSo it was not correct to assume that 1 plus square root of 2 is a rational number.

Skip to 4 minutes and 25 secondsThus 1 plus square root of 2 is not a rational number. Finally, in exercise 3, we want to show that 3 times square root of 2 is not a rational number. Again, assume it is a rational number. If 3 times square root of 2 is a rational number-- call it q-- then we get 3 times square root of 2 equals q, rational number, and thus square root of 2 is the quotient of q with 3, which is again a rational number. But this is a contradiction, because square root of 2 is not a rational number.

Skip to 5 minutes and 13 secondsThus 3 times square root of 2 is not a rational number. And this ends exercise 3 and this step in practice. See you in the next step.

Rational numbers in practice

The following exercises are solved in this step.

We invite you to try to solve them before watching the video.

In any case, you will find below a PDF file with the solutions.

Exercise 1.

Prove that \(\sqrt{5}\) is irrational.

Exercise 2.

Prove that \(1+\sqrt 2\) is irrational.

Exercise 3.

Prove that \(3\sqrt 2\) is irrational.

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Precalculus: the Mathematics of Numbers, Functions and Equations

University of Padova

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