Sums and Products of Rational and Irrational Numbers

[paulmdrdo1]

Explain why the sum, the difference, and the product of the
rational numbers are rational numbers. Is the product of the
irrational numbers necessarily irrational? What about
the sum?

Combining Rational Numbers with Irrational Numbers
In general, what can you say about the sum of a rational
and an irrational number? What about the product?

please explain with examples.

 

[Prove It]

Have you tried anything yourself yet? Here are a couple of hints...

What do you know about the closure of the rational numbers under addition and multiplication?

As for irrationals, what is $$\displaystyle \begin{align*} \sqrt{2} \cdot \sqrt{2} \end{align*}$$?

 

[paulmdrdo1]

this is what i tried,

When we add two rational numbers say 1/3+2/3 = 3/3 = 1. 1 is an element of the set of rational numbers. therefore whenever addition is performed on the elements of the set of rational numbers, an element of the set is obtained. performing multiplication on the set is analogous to that of addition.

 

[MarkFL]

The sum of two irrational numbers can be rational, consider:

\(\displaystyle \alpha=\pi\)

\(\displaystyle \beta=1-\pi\)

\(\displaystyle \alpha+\beta=1\)

 

[paulmdrdo1]

does that mean when we add two irrational number we always obtain rational numbers?

 

[MarkFL]

does that mean when we add two irrational number we always obtain rational numbers?

No, not by any means, I just demonstrated a case where the sum is rational. How about:

\(\displaystyle \alpha=\sqrt{2}\)

\(\displaystyle \beta=e\)

\(\displaystyle \alpha+\beta=\sqrt{2}+e\)

This sum is irrational.

 

[Jameson]

does that mean when we add two irrational number we always obtain rational numbers?

Nope, not always. Consider $\pi + \pi=2\pi$. Two irrational numbers that add up to another irrational.

 

[paulmdrdo1]

can you give me general rule about this.

 

[Prove It]

Do you know anything about existentials and universals?

If you want to prove something is always (universally) true, you need to make an argument that shows that there is does not exist a situation where it is not true.

If you want to prove something is not always (not universally) true, you just need to show that there exists a situation where it is not true.