1.05 Patterns with rational and irrational numbers

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 non-terminating, non-repeating decimals like $\sqrt{2}$√2, $\pi$π or even $\sqrt{\pi}$√π.

 

Exploration

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?

Two rational 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?

 

One rational and one irrational number

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 non-terminating, non-repeating decimal, so even if we add $0.25$0.25 to it, it will still be a non-terminating, non-repeating 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 non-terminating, non-repeating decimal, so even if we take a quarter of it, it will still be a non-terminating, non-repeating 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 non-zero rational and an irrational number will be irrational.

 

Two irrational numbers

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 non-terminating, non-repeating decimal and $\pi$π is a different non-terminating, non-repeating decimal. Could these ever add to become a terminating decimal? The answer is not in the particular case, but two non-terminating, non-repeating decimals can become rational. How? What do we know about $a-a$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.

 

Practice questions

Question 1

Question 2

Question 3