Question

If x and y are irrational number show that x+y and x*y are irrational.

Answer 1

Answer on Question #56670 – Math – Real Analysis

If $x$ and $y$ are irrational numbers, show that $x + y$ and $x * y$ are irrational.

Solution

This statement is not necessarily true.

Let's consider some of irrational numbers. For instance

$$\sqrt{2}; \quad 2\sqrt{2}; \quad 2 - \sqrt{3}; \quad 2 + \sqrt{3}$$

Now, find the sum and the multiplication of these numbers.

$$\sqrt{2} + 2\sqrt{2} = 3\sqrt{2}$$

In this example a sum of two irrational is irrational.

But:

$$(2 - \sqrt{3}) + (2 + \sqrt{3}) = 4$$

In this case sum of two irrationals is rational.

The same situation for the multiplication of two irrational numbers. For example

$$\sqrt{2} * 2\sqrt{2} = 4$$

Four is rational

But for multiplication of $\sqrt{2}$ and $2 + \sqrt{3}$ we have

$$\sqrt{2} * (2 + \sqrt{3}) = 2\sqrt{2} + \sqrt{6}$$

It is irrational.

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