Is the sum of a rational number and an irrational number always irrational?

[chwala]

Homework Statement

show that if $x$ is rational and $y$ is irrational, then $x+y$ is irrational. Assume that $x$ is irrational and that $y$ is also irrational. Is $xy$ irrational?

Homework Equations

The Attempt at a Solution

lol
${1/2} +{√2}$ [/B]is irrational.
${{1/√2}}×{√2}=2$ which is rational and therefore not irrational

 

[member 587159]

Homework Statement

show that if $x$ is rational and $y$ is irrational, then $x+y$ is irrational. Assume that $x$ is irrational and that $y$ is also irrational. Is $xy$ irrational?

Homework Equations

The Attempt at a Solution

lol
${1/2} +{√2}$ [/B]is irrational.
${{1/√2}}×{√2}=2$ which is not rational

${{1/√2}}×{√2}=1$ in fact and this is rational. Do you know what a rational number is?

For the x*y part, with x and y irrational. They want you to proof or disproof that this product is irrational. Hint: you can disproof something by giving a counterexample.

 

[chwala]

${{1/√2}}×{√2}=1$ in fact and this is rational. Do you know what a rational number is?

For the x*y part, with x and y irrational. They want you to proof or disproof that this product is irrational. Hint: you can disproof something by giving a counterexample.

sorry typo error i have corrected i definitely know what a rational number is.

 

[BvU]

Chwala has a problem with brackets. I don't think it has to do with a broken keyboard .

In this thread I at first spied two exercises:

1. Show that if x is rational and y is irrational, then x + y is irrational
2. Assume that x is irrational and that y is also irrational. Is xy irrational?
​

Or am I playing dumb again and should I read:

Show that if x is rational and y is irrational, then x + y is irrational. (Hint: assume that x is irrational and that y is also irrational. Is xy irrational ? ).​

Anyway it's clear that xy is not necessarly irrational if x and y are. from the counter-example.

From post #3 I gather it is also clear that a rational number can be written as a ratio of two integers (hence the name rational...), right ?

In order to forward our lol (?) attempt on part 1, I propose we rewrite it as: show that x(rational) + y(irrational) can NOT be written as M(integer) / N(integer) . Would that be a good strategy ?

 

[haruspex]

show that if x is rational and y is irrational, then x+y is irrational.

1/2+√2 is irrational

Not sure I understand the attempt. Is that supposed to be a proof?

 

[Ray Vickson]

Homework Statement

show that if $x$ is rational and $y$ is irrational, then $x+y$ is irrational. Assume that $x$ is irrational and that $y$ is also irrational. Is $xy$ irrational?

Homework Equations

The Attempt at a Solution

lol
${1/2} +{√2}$ [/B]is irrational.
${{1/√2}}×{√2}=2$ which is rational and therefore not irrational

What happens if $x \neq 1/2$ and/or $y \neq \sqrt{2}$? Just showing the first result for two numerical examples of $x$ and $y$ does NOT constitute a proof.

 

[Incand]

To show that something is irrational it's usually easier to assume it's rational and arrive at a contradiction.
In general when doing proofs always go back to the definition.
What does it mean that $x$ is rational exactly? What's the definition?

A hint at solution steps:
Assume $x+y$ is rational. What does this mean?
What does this mean for $y$?

 

[chwala]

Ray how do we show this? in regards to your post number 6...

 

[BvU]

Look at the hint in post #7 once more ...