Skip to 0 minutes and 11 seconds Hello. 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 seconds Well, 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 seconds Without common factors.

Skip to 1 minute and 22 seconds This 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 seconds But 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 seconds But 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 seconds 5 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 seconds So 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 seconds In 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 seconds But 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 seconds So it was not correct to assume that 1 plus square root of 2 is a rational number.

Skip to 4 minutes and 25 seconds Thus 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 seconds Thus 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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