Irrational numbers

Numbers that are not fractions

The ancient Greeks found something that troubled them. A square has a side of $1$. Its diagonal, by the Pythagoras theorem, is $\sqrt{1^2 + 1^2} = \sqrt{2}$. They tried to write this length as a fraction $\dfrac{p}{q}$, and they could not. It is not possible.

A number that cannot be written as $\dfrac{p}{q}$ is an irrational number. The word "ir-rational" means "not a ratio". The Greeks of the school of Pythagoras were so shocked by this that, according to a story, they tried to keep it secret.

The decimal of an irrational number

The decimal of a rational number ends or repeats. The decimal of an irrational number does neither. It goes on forever, and no block of digits repeats.

Other examples are $\sqrt{3}$, $\sqrt{5}$, $\sqrt{7}$ and $0.101001000100001\ldots$. In the last one, the number of zeros between the ones keeps growing, so there is no repeating block.

Be careful. $0.101010\ldots$ does repeat the block $10$. It is rational: it equals $\dfrac{10}{99}$.

Not every square root is irrational

• $\sqrt{4} = 2$ and $\sqrt{9} = 3$. These are rational.
• $\sqrt{0.25} = 0.5$. This is rational.
• $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{7}$ are irrational.

The square root of a whole number is irrational if the number is not a perfect square.

Set $n = 2$. The dot for a bottom of $12$ is close to $0$, since $\dfrac{17}{12}$ squared is $2.0069$. Close, but never exactly $0$, whatever bottom you try.

Showing $\sqrt{2}$ on the number line

An irrational number is a real point on the number line, and we can find it exactly. Draw the number line. At the point $1$, draw a line of length $1$, at a right angle to the number line. Join the starting point $0$ to the top of that line. By the Pythagoras theorem, the length is $\sqrt{1^2 + 1^2} = \sqrt{2}$. Now put the point of a compass at $0$, and draw an arc of this length to the number line. The arc meets the line at the point $\sqrt{2}$. In the same way, we can find $\sqrt{3}$, $\sqrt{5}$ and the others, one after the other.

About pi

We often use $\dfrac{22}{7}$ for $\pi$. But $\dfrac{22}{7} = 3.142857142857\ldots$ repeats, and $\pi = 3.14159265\ldots$ does not. So $\dfrac{22}{7}$ is only close to $\pi$. They are different numbers. The number $\pi$ is the ratio of the circumference of a circle to its diameter, and it is irrational.

It may seem strange that a ratio is irrational. The reason is that in the ratio $\dfrac{c}{d}$, the numbers $c$ and $d$ cannot both be whole numbers (or rational). If the diameter is rational, the circumference is not.

Why is the square root of 2 irrational?

We use a proof by contradiction. Suppose $\sqrt{2} = \dfrac{p}{q}$, with $p$ and $q$ having no common factor.

1. Then $2q^2 = p^2$. So $p^2$ is even, and so $p$ is even. Write $p = 2k$.
2. Then $2q^2 = 4k^2$, so $q^2 = 2k^2$. So $q^2$ is even, and $q$ is even.
3. Now $p$ and $q$ are both even. They have a common factor $2$. This contradicts what we started with.

So $\sqrt{2}$ cannot be written as $\dfrac{p}{q}$. It is irrational.

The same proof works for $\sqrt{3}$. If $3q^2 = p^2$, then $3$ divides $p^2$, and so $3$ divides $p$. Write $p = 3k$, and then $q^2 = 3k^2$, so $3$ divides $q$ as well. Again $p$ and $q$ have a common factor. So $\sqrt{3}$ is irrational.

Many irrational numbers

Between any two numbers, however close, there are both rational and irrational numbers. For example, between $0.1$ and $0.2$ the number $0.15$ is rational, and $0.1010010001\ldots$ is irrational. Also $\sqrt{2}$ and $\sqrt{3}$ are irrational numbers between $1.4$ and $1.8$.

The rational numbers and the irrational numbers together make the real numbers. Every point on the number line is a real number.

Practise with new numbers

Try it in Python

This program finds $60$ digits of $\sqrt{2}$, and it shows the digits of $\dfrac{1}{7}$.

Common mistakes

• Saying that a long decimal is irrational. $0.123123123\ldots$ is long, but it repeats, so it is rational.
• Saying that every square root is irrational. $\sqrt{25} = 5$ is rational.
• Taking $\dfrac{22}{7}$ as the exact value of $\pi$. It is only close.
• Thinking that an irrational number has no place on the number line. It has an exact place.

Remember these

• An irrational number cannot be written as $\dfrac{p}{q}$. Its decimal never ends and never repeats.
• $\sqrt{2}$, $\sqrt{3}$ and $\pi$ are irrational. $\sqrt{4}$ and $\dfrac{22}{7}$ are rational.
• The square root of a number that is not a perfect square is irrational.
• Rational and irrational numbers together are the real numbers.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Is $\sqrt{49}$ rational or irrational?

2. Is $0.272727\ldots$ rational or irrational?

3. Is $\dfrac{22}{7}$ equal to $\pi$?

4. Write an irrational number between $0.3$ and $0.4$.