Question

Prove that if x and y are rational then their product is also rational. First write a proof on your own and then put the following statements into order to obtain the proof. Put N next to the statements that should not be used in the proof. 1. Then, xy=acbd . 2. Suppose x and y are arbitrary rational numbers. 3. Since a, c are integers, e=ac is also an integer.

Answer 1

Answer:

Step-by-step explanation:

The aim of this question is to prove that the product of two rationals is rational.

To proof:

If $\mathbf{x \ and \ y}$ are arbitrary rational numbers.

Thus, going by the definition of rational, there exist integers

a, b, ≠ 0, c, and d ≠ 0 ; $x = \dfrac{a}{b} \ and \ y = \dfrac{c}{d}$

Hence, $xy = (\dfrac{a}{b})(\dfrac{c}{d}) = \dfrac{ac}{bd}$

Since a and c are integers, then e = ac appears to be an integer as well.

Also, provided that b and d are non-zero integers;

f =bd appears to be a non-zero integer.

Therefore, $xy = \dfrac{e}{f}$, and going by the definition of rational, xy is rational.

Hence, from the complete question:

The order of the statement is:

7,6,3,5,2,4

The statements that should not be used in the proof are:

1 N

8 N

9 N

10 N

11 N