# Rational and Irrational Numbers

1. Jan 25, 2005

### lokisapocalypse

I need to show that a rational + irrational number is irrational. I am trying to do a proof by contradiction.

So far I have:

Suppose a rational, b irrational.
Then a = p/q for p, q in Z.
Then a + b = p/q + b = (p + qb) / q
But I don't know where to go from here because I still have a rational plus an irrational and that is what I am trying to show.

Also, would a similar proof work to show that an irrational + irrational = irrational?

2. Jan 25, 2005

### Curious3141

When you're doing a contradiction, you should assume the opposite of the premise and show that it leads to an absurd conclusion.

For this problem, try this :

Let $a = \frac{p}{q}$ where p and q are coprime integers.

b is irrational.

Now let us say their sum is a rational number, which can be expressed as [itex]\frac{s}{t}[/tex], where s and t are coprime integers.

Then,

$$a + b = \frac{s}{t}$$

$$b = \frac{s}{t} - a = \frac{s}{t} - \frac{p}{q} = \frac{sq - pt}{qt}$$

We have just shown that b can be expressed as the ratio of two integers. But b is irrational.

This is a contradiction. Hence the assumption is false and the original premise is true.

This isn't even true in general. What can you say about the sum of $$\sqrt{2}$$ and $$(5 - \sqrt{2})$$ ?

3. Jan 25, 2005

### Hurkyl

Staff Emeritus
The proof is one line.

So, you're doing proof by contradiction, and have assumed that there are rational numbers p and q, and an irrational number z, such that:

p + z = q

right?

It is also fairly easy to construct counterexamples... but the method might be easier to find once you've done the first problem.

4. Sep 2, 2010

### Anjuyogi

I need to show that a rational - irrational number is irrational. I am trying to do a proof by contradiction.
Plz send me the related answer as I need it....

Last edited: Sep 2, 2010
5. Sep 2, 2010

### HallsofIvy

Staff Emeritus
Since this clearly has nothing to do with physics, I am moving it to "precalculus homework".

6. Sep 2, 2010

### Mentallic

Anjuyogi, take a look at the previous posts?