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Sums and Products of Rational and Irrational Numbers

paulmdrdo

Active member
May 13, 2013
386
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

Well-known member
MHB Math Helper
Jan 26, 2012
1,404
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 [tex]\displaystyle \begin{align*} \sqrt{2} \cdot \sqrt{2} \end{align*}[/tex]?
 

paulmdrdo

Active member
May 13, 2013
386
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.
 
Last edited:

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
The sum of two irrational numbers can be rational, consider:

\(\displaystyle \alpha=\pi\)

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

\(\displaystyle \alpha+\beta=1\)
 

paulmdrdo

Active member
May 13, 2013
386
does that mean when we add two irrational number we always obtain rational numbers?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
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

Administrator
Staff member
Jan 26, 2012
4,052
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.
 

paulmdrdo

Active member
May 13, 2013
386
can you give me general rule about this.
 

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,404
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.