Proving Rational Numbers and Irrational Numbers

[cragar]

Homework Statement

Show that if $a\in\mathbb{Q}$ and $t\in\mathbb{I}$
then $a+t\in\mathbb{I}$ and $at\in\mathbb{I}$
as long as a≠0

The Attempt at a Solution

Let $a=\frac{x}{y}$ where x and y are integers. and t is an irrational number
If I have a+t . since t cannot be written as a fraction, there's no way an integer times an irrational number will be an integer so this number will be irrational. and also at and a+t would be irrational.

 

[jbunniii]

I didn't understand your argument. I suggest starting by assuming the contrary. Suppose

$$a + t \not\in \mathbb{I}$$

Then

$$a + t \in \mathbb{Q}$$

So there is some rational $r$ such that $a + t = r$. Can you explain why this is impossible?

 

[cragar]

ok i see. So we assume that a+t is a rational number. let a=x/y
and let a+t=L/M=x/y+t=L/M
and when we subtract x/y from both sides and we get a common denominator and simplify the right hand side we get that t is a rational number. which is a contradiction, therefore a+t is an irrational number. does this work

 

[jbunniii]

ok i see. So we assume that a+t is a rational number. let a=x/y
and let a+t=L/M=x/y+t=L/M
and when we subtract x/y from both sides and we get a common denominator and simplify the right hand side we get that t is a rational number. which is a contradiction, therefore a+t is an irrational number. does this work

Looks good.

 

[cragar]

sweet thanks for the help