You may have heard the saying "One rotten apple spoils the whole barrel". Well, with irrational numbers, it is similar.
Remember that we have different number types including rational numbers which can be written as fractions or terminating or repeating decimals like $3$3, $\frac{4}{5}$−45, $\sqrt{16}$√16, or $0.\overline{3}$0.3. We also have irrational numbers which are nonterminating, nonrepeating decimals like $\sqrt{2}$√2, $\pi$π or even $\sqrt{\pi}$√π.
Consider $\frac{1}{4}$14, $4.2$4.2, $\sqrt{3}$√3 and $\pi$π. We have two rational numbers $\frac{1}{4}$14 and $4.2$4.2, as well as two irrational numbers $\sqrt{3}$√3 and $\pi$π. What happens when we take the sum and product of various combinations of rational and irrational numbers?
Let's first look at the sum of two rational numbers and see if we get a rational or an irrational number.
$\frac{1}{4}+4.2$14+4.2 
Sum of two rational numbers 

$=$=  $0.25+4.2$0.25+4.2 
Convert the fraction to a decimal 
$=$=  $4.45$4.45 
Add the decimals 
We find that we get $4.45$4.45 which is a rational number. It will always be the case that the sum of two rational numbers will also be rational.
Now let's look at the product of two rational numbers and see if we get a rational or an irrational number.
$\frac{1}{4}\times4.2$14×4.2 
Sum of two rational numbers 

$=$=  $\frac{1}{4}\times\frac{42}{10}$14×4210 
Convert the decimal to a fraction 
$=$=  $\frac{42}{40}$4240 
Multiply the fractions 
$=$=  $\frac{21}{20}$2120 
Simplify the fractions 
We find that we get $\frac{21}{20}$2120 which is a rational number. It will always be the case that the product of two rational numbers will also be rational.
If we had two rational numbers $\frac{a}{b}$ab and $\frac{c}{d}$cd, where $a,b,c,d$a,b,c,d are integers, can you see how their sum and product would also be rational?
Let's first consider the sum of one rational and one irrational number and see if we get a rational or an irrational number.
Is $\sqrt{3}+\frac{1}{4}$√3+14 rational or irrational? Well, if we put $\sqrt{3}$√3 in our calculator, we will see that it is a nonterminating, nonrepeating decimal, so even if we add $0.25$0.25 to it, it will still be a nonterminating, nonrepeating decimal.
It will always be the case that the sum of a rational and an irrational number will be irrational.
Now let's look at the product of a rational and an irrational number.
Is $\frac{1}{4}\times\sqrt{3}$14×√3 rational or irrational? Well, $\sqrt{3}$√3 is a nonterminating, nonrepeating decimal, so even if we take a quarter of it, it will still be a nonterminating, nonrepeating decimal. But what happens if we multiply an irrational number by zero, which is a rational number? $0\times\sqrt{3}=0$0×√3=0 so for this trivial case, we have the product of a rational number and an irrational number resulting in a rational number, but this is the exception to the rule.
It will always be the case that the product of a nonzero rational and an irrational number will be irrational.
Let's first consider the sum of two irrational numbers and see if we get a rational or an irrational number.
Examples  Type of Number 

$\pi+\sqrt{3}$π+√3  Irrational 
$\pi+\left(\pi\right)$π+(−π)  Rational 
Is $\pi+\sqrt{3}$π+√3 rational or irrational? Well, $\sqrt{3}$√3 is a nonterminating, nonrepeating decimal and $\pi$π is a different nonterminating, nonrepeating decimal. Could these ever add to become a terminating decimal? The answer is not in the particular case, but two nonterminating, nonrepeating decimals can become rational. How? What do we know about $aa$a−a?
Is $\pi+\left(\pi\right)$π+(−π) rational or irrational? Well, $\pi+\left(\pi\right)=0$π+(−π)=0, which is rational, so it must be rational.
It will often be the case that the sum of two irrational numbers will be irrational, but not always.
Now let's consider the product of two irrational numbers and see if we get a rational or an irrational number.
Examples  Type of number 

$\pi\times\sqrt{3}$π×√3  Irrational 
$\sqrt{3}\times\sqrt{3}$√3×√3  Rational 
Is $\pi\times\sqrt{3}$π×√3 rational or irrational? We can see on our calculator, that this product is still irrational. However, this is not always the case for two irrational numbers. Let's look at square roots.
Is $\sqrt{3}\times\sqrt{3}$√3×√3 rational or irrational? Well, $\sqrt{3}\times\sqrt{3}=\left(\sqrt{3}\right)^2$√3×√3=(√3)2, which is just $3$3, so it is rational.
It will often be the case that the product of two irrational numbers will be irrational, but not always.
Is the result of the sum a rational or an irrational number?
$\frac{3}{10}+\frac{7}{8}$310+78
Rational
Irrational
Rational
Irrational
Is the result of the sum a rational or an irrational number?
$\frac{3}{11}+\sqrt{7}$311+√7
Rational
Irrational
Rational
Irrational
Is the result of the product a rational or an irrational number?
$\frac{3}{11}\times\sqrt{7}$311×√7
Rational
Irrational
Rational
Irrational
Describe the relationships between the subsets of the real number system