Decimals that never end: irrational numbers

So far every dec­i­mal you have met was one of two kinds. Either it stopped, like $0.625$, or it went on for ever with a block of dig­its com­ing round again, like $0.\overline{3}$. Both kinds can be writ­ten as a frac­tion $\displaystyle \frac{p}{q}$, so both are ratio­nal num­bers. But is there a third kind? Can a dec­i­mal go on for ever and never repeat? Yes, it can. Num­bers like that are called irra­tional num­bers, and this les­son is your first meet­ing with them.

The two kinds you know#

Here is a quick reminder.

Remem­ber. A ter­mi­nat­ing dec­i­mal is ratio­nal. A non-ter­mi­nat­ing recur­ring dec­i­mal is ratio­nal too.

A dec­i­mal that never repeats#

Look at this dec­i­mal. After the point, we write $1$, then one $0$ and a $1$, then two 0s and a $1$, then three 0s and a $1$, and so on.

$$0.1010010001000010000010\ldots$$

There is a pat­tern, so you can always say what comes next. But there is no block of dig­its that repeats. The runs of zeros keep get­ting longer, so no fixed block can ever come round again. This dec­i­mal never stops and never repeats.

A dec­i­mal like this is called non-ter­mi­nat­ing non-recur­ring. It can­not be writ­ten as a frac­tion $\displaystyle \frac{p}{q}$.

The num­ber whose square is 2#

Here is a famous exam­ple. Is there a num­ber which, mul­ti­plied by itself, gives $2$?

Try whole num­bers first. $1 \times 1 = 1$, which is too small. $2 \times 2 = 4$, which is too big. So the num­ber is between $1$ and $2$.

Now try one dec­i­mal place.

$$\begin{aligned} 1.4 \times 1.4 &= 1.96 \\ 1.5 \times 1.5 &= 2.25 \end{aligned}$$

$1.96$ is a lit­tle less than $2$ and $2.25$ is more. So the num­ber is between $1.4$ and $1.5$.

Now try two dec­i­mal places.

$$\begin{aligned} 1.41 \times 1.41 &= 1.9881 \\ 1.42 \times 1.42 &= 2.0164 \end{aligned}$$

So the num­ber is between $1.41$ and $1.42$. One more step:

$$\begin{aligned} 1.414 \times 1.414 &= 1.999396 \\ 1.415 \times 1.415 &= 2.002225 \end{aligned}$$

So it is between $1.414$ and $1.415$.

You can keep going for as long as you like. Every step gives one more digit, and the squares get closer and closer to $2$, but they never land on it exactly. This num­ber is writ­ten $\sqrt{2}$ and read as "the square root of 2".

$$\sqrt{2} = 1.41421356\ldots$$

Math­e­mati­cians have proved that the dig­its of $\sqrt{2}$ never stop and never repeat. So $\sqrt{2}$ can­not be writ­ten as $\displaystyle \frac{p}{q}$. It is an irra­tional num­ber.

You will learn much more about squares and square roots in the next chap­ter.

What an irra­tional num­ber is#

Remem­ber. An irra­tional num­ber is a num­ber that can­not be writ­ten as $\displaystyle \frac{p}{q}$, where $p$ and $q$ are whole num­bers and $q$ is not $0$. Its dec­i­mal is non-ter­mi­nat­ing and non-recur­ring.

Some exam­ples:

• $\sqrt{2} = 1.41421\ldots$
• $\sqrt{3} = 1.73205\ldots$
• $\sqrt{5} = 2.23606\ldots$
• $\pi = 3.14159\ldots$

Not every square root is irra­tional. $\sqrt{4} = 2$, because $2 \times 2 = 4$. $\sqrt{9} = 3$, because $3 \times 3 = 9$. These are whole num­bers, so they are ratio­nal.

Pi and 22 over 7#

The num­ber $\pi$ (say "pie") tells you how many times the dis­tance across a cir­cle fits around it. Its dec­i­mal starts $3.14159265\ldots$ and never stops or repeats. So $\pi$ is irra­tional.

In sums we often use $\displaystyle \frac{22}{7}$ in place of $\pi$. But look at its dec­i­mal from an ear­lier les­son.

$$\displaystyle \frac{22}{7} = 3.\overline{142857} = 3.142857142857\ldots$$

It repeats, so $\displaystyle \frac{22}{7}$ is ratio­nal. It agrees with $\pi$ only in the first two dig­its after the point. We say $\pi$ is approx­i­mately equal to $\displaystyle \frac{22}{7}$, and write $\displaystyle \pi \approx \frac{22}{7}$.

Sort­ing dec­i­mals#

To decide whether a dec­i­mal is ratio­nal or irra­tional, ask two ques­tions.

1. Does it stop? If yes, it is ratio­nal.
2. If it does not stop, does a block of dig­its repeat? If yes, it is ratio­nal. If no, it is irra­tional.

Exam­ple 1#

Is $2.\overline{5}$ ratio­nal or irra­tional?

It does not stop, but the $5$ repeats. So it is ratio­nal. In fact $\displaystyle 2.\overline{5} = \frac{25 - 2}{9} = \frac{23}{9}$.

Exam­ple 2#

Is $0.202002000200002\ldots$ ratio­nal or irra­tional?

It does not stop. The runs of zeros between the 2s keep get­ting longer, so no block repeats. It is irra­tional.

Exam­ple 3#

Between which two dec­i­mals with one place after the point does $\sqrt{3}$ lie?

$1.7 \times 1.7 = 2.89$, which is less than $3$. $1.8 \times 1.8 = 3.24$, which is more than $3$. So $\sqrt{3}$ lies between $1.7$ and $1.8$.

Real num­bers#

Put all the ratio­nal num­bers and all the irra­tional num­bers together and you get the real num­bers. Every point on the num­ber line is a real num­ber. You will study the real num­bers and their rules in a later chap­ter.

Com­mon mis­takes#

• Think­ing every long dec­i­mal is irra­tional. $0.\overline{142857}$ is long, but it repeats, so it is ratio­nal.
• Think­ing a dec­i­mal with a pat­tern must repeat. $0.101001000\ldots$ has a pat­tern but no repeat­ing block.
• Writ­ing $\displaystyle \pi = \frac{22}{7}$. They are close, but not equal.
• Think­ing every square root is irra­tional. $\sqrt{9} = 3$ is ratio­nal.

Prac­tice#

1. Is $0.75$ ratio­nal or irra­tional?
2. Is $0.\overline{12}$ ratio­nal or irra­tional?
3. Is $0.303003000300003\ldots$ ratio­nal or irra­tional?
4. Is $\sqrt{9}$ ratio­nal or irra­tional?
5. Is $\sqrt{7}$ ratio­nal or irra­tional?
6. Is $\displaystyle \frac{22}{7}$ ratio­nal or irra­tional?
7. Find $1.73 \times 1.73$ and $1.74 \times 1.74$. Between which two of these dec­i­mals does $\sqrt{3}$ lie?
8. Between which two dec­i­mals with one place after the point does $\sqrt{5}$ lie?
9. Between which two dec­i­mals with two places after the point does $\sqrt{7}$ lie? Use $2.64$ and $2.65$.
10. A cal­cu­la­tor shows $\sqrt{2}$ as $1.4142$. Is the num­ber $1.4142$ ratio­nal or irra­tional?

Answers#