# Sums and Products of Rational and Irrational Numbers

#### paulmdrdo

##### Active member
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?

#### Prove It

##### Well-known member
MHB Math Helper
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*}?

#### paulmdrdo

##### Active member
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.

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#### MarkFL

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

$$\displaystyle \alpha=\pi$$

$$\displaystyle \beta=1-\pi$$

$$\displaystyle \alpha+\beta=1$$

#### paulmdrdo

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

#### MarkFL

Staff member
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

Staff member
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.