# Rational and Irrational Numbers

A number is rational if it can be written as a fraction with integer numerator and denominator, and irrational if it cannot. In this lesson, we will look at examples of rational and irrational numbers, and some basic ways of determining whether a number is rational or irrational.

## Rational numbers

A number is rational if it is equal to a ratio of two integers: that is, if it can be written as a fraction $$m/n$$ where $m$ and $n$ are integers.

 Is $$3/5$$ a rational number?

Notice that the number 2 can be written as $$2/1$$.

 Is 2 a rational number?
 Find a fraction that is equal to 3.
 Is 3 a rational number?
 Is every integer a rational number?

Remember that we can add and subtract fractions by putting them over a common denominator, and the result will still be a fraction. Similarly, if you multiply or divide two fractions, the result will again be a fraction.

 Is $$3/43 + 7/12$$ a rational number? (You don’t have to compute it.)
 Is $$7/22(355/113)$$ a rational number?

If $q$ and $r$ are rational numbers, then so are $q+r$, $q-r$, and $qr$. If $r ≠ 0$, $$q/r$$ is also a rational number.

(The symbol ‘$≠$’ means “is not equal to.”)

One common way of writing rational numbers is as repeating decimals. Remember that when we write a decimal with a bar over some of its numbers, like $1.\ov{27}$, what we mean is that those numbers are repeated infinitely:

$$1.\ov{27}=1.27272727...$$

You can use algebra to convert any repeating decimal into a fraction. For example, you can convert $1.\ov{27}$ to a fraction as follows:

$$\cl"tight"{\table x, , , =, 1.\ov{27}; x, , , =, 1.27\ov{27}, , , \; \cl"hint"\text"pulling out the first repetition"; 100x, , , =, 127.\ov{27}, , , \; \cl"hint"\text"multiplying both sides by 100"; 100x, -, x, =, 127.\ov{27}, -, x, \; \cl"hint"{\text"subtracting"\,x\,\text"from both sides..."}; \colspan 3 99x, =, 127.\ov{27}, -, 1.\ov{27}, \; \cl"hint"{\text"...and using the fact that"\,x= 1.\ov{27}\,\text"..."}; \colspan 3 99x, =, \colspan 3 126, \; \cl"hint"\text"...to make the repeated part cancel out"; \colspan 3 {99 x} / 99, =, \colspan 3 126/99; \colspan 3 x, =, \colspan 3 14/11 }$$

We multiplied by $100=10^2$ in order to shift the repeating decimal over by 2 decimal places, because there were 2 repeated digits. In general, if there are $n$ repeated digits, then we multiply by $10^n$ to shift the repeating decimal over by $n$ decimal places.

Convert the following repeating decimals into fractions.

## Irrational numbers

Having seen a large number of examples of rational numbers, you might wonder whether every number is rational. We’ll look at whether $√2$ is rational.

Finding out whether $√2$ is rational is the same as finding out whether there is a fraction whose square is 2. Using your calculator or the grid on the left, compute the square of each of the following fractions as a decimal, rounded to as many decimal places as fit in the answer box.

Fraction and its squareIs the square
equal to 2?
 As you can see, the squares of the numbers in the table get very close to 2. Are any of them exactly equal to 2?
 You can quickly square any rational number by entering it in the input boxes below the grid to the left. Can you find any rational number whose square is exactly equal to 2?

In fact, there is no such rational number. If you would like to see a proof of this, click here.

A number which is not rational is called irrational.

So $√2$ is an irrational number. In fact:

If $n$ is a positive integer and $√n$ is not an integer, then $√n$ is an irrational number.

This is also proved at the link above.

## Arithmetic with irrational numbers

In the last question, we learned that square roots such as $√2$ or $√37$ are irrational. We’ll now look at more complicated expressions formed from those square roots, such as $√2+1$ or $5√37$.

 We’ll start by trying to figure out if the number $√2+1$ is rational or irrational. What is a simpler way of writing the sum $(√2+1)+(-1)$?
 Is that sum rational or irrational?
 As you saw in ratDef, the sum of two rational numbers is rational. If $q$ is a rational number, is the sum $q+(-1)$ rational or irrational?
 If $q=√2+1$, the sum $q+(-1)$ is irrational. Can $√2+1$ be a rational number?
 Is $√2+1$ rational or irrational?
 Now, we’ll try to determine whether or not $5√37$ is rational. What is a simpler way of writing $$1/5(5√37)$$?
 Is that product rational or irrational?
 If $q$ is a rational number, is the product $$1/5 q$$ rational or irrational?
 If $q=5√37$, the product $$1/5 q$$ is irrational. Can $5√37$ be a rational number?
 Is $5√37$ rational or irrational?

In this way, we can see that:

If $q$ is a rational number and $s$ is an irrational number, then $s+q$ is irrational. If $q ≠ 0$, $qs$ is also irrational.

Now, we’ll look at what happens when you divide a rational number by an irrational number, by considering the specific example $$2/√3$$.

 We’re trying to figure out if the number $$2/√3$$ is rational or irrational. What is a simpler way of writing the product $$2/√3 (√3)$$?
 Is that product rational or irrational?
 If $q$ is a nonzero rational number, is the product $q√3$ rational or irrational? (Remember that the product of a nonzero rational number and an irrational number is irrational.)
 If $$q=2/√3$$, the product $q√3$ is rational. Can $$q=2/√3$$ be a rational number?
 Is $$q=2/√3$$ rational or irrational?

In this way, we can also see that:

If $q$ is a nonzero rational number and $s$ is an irrational number, then $$q/s$$ is irrational.

What can we say about the sum or product of two irrational numbers?

 We’ll start by looking at whether the sum $√3+(2-√3)$ is rational or irrational. What is a simpler way of writing this number?
 Is $√3+(2-√3)$ rational or irrational?
 Is $√3$ rational or irrational?
 Is $2-√3$ rational or irrational?
 Is it possible for the sum of two irrational numbers to be rational?
 Now, we’ll look at whether $√5+√5$ is rational or irrational. What is a simpler way of writing this number?
 Is $√5+√5$ rational or irrational?
 Is $√5$ rational or irrational?
 Is it possible for the sum of two irrational numbers to be irrational?

As you can see:

The sum of two irrational numbers may be either rational or irrational.

In sumQuestion, you saw that the sum of two irrational numbers could be either rational or irrational. In this question, we’ll examine the product of two irrational numbers.

 We’ll start by looking at whether the product $√3(√3)$ is rational or irrational. What is a simpler way of writing this number?
 Is $√3(√3)$ rational or irrational?
 Is $√3$ rational or irrational?
 Is it possible for the product of two irrational numbers to be rational?
 Now, we’ll look at whether $√2(√5)$ is rational or irrational. What is a simpler way of writing this number? (Remember that $√a(√b)=√{ab}$.)
 Is $√2(√5)$ rational or irrational?
 Is $√2$ rational or irrational?
 Is $√5$ rational or irrational?
 Is it possible for the product of two irrational numbers to be irrational?

Just as in sumQuestion:

The product of two irrational numbers may be either rational or irrational.

For each number in the table below, use the rules you have learned to say whether it is rational or irrational. (You don’t have to compute the actual numbers.)

NumberRational or
irrational?