1.1 Workshop 1 (Exploration)

1.1.1 Preparatory exercises (30 mins)

You may already be familiar with the representation of numbers in decimal notation (e.g. $0.5$ or $0.555\ldots$ ). You may also have come across numbers that can be written as fractions of integers (e.g. $\frac{1}{2}$ or $\frac{5}{9}$ ). In the Exploration workshop, we will explore the properties of each of these representations, and the connections between them.

Complete the following exercises before the Exploration Workshop on Tuesday 16th September 2025

Exercise 1.1.

Without using a calculator or computer, write each of the following real numbers as a fraction of integers.

(a)

$0.1$
(b)

$0.2$
(c)

$0.01$
(d)

$0.11$
(e)

$0.111$

Exercise 1.2.

Without using a calculator or computer, write each of the following real numbers in decimal notation.

(a)

$\frac{3}{10}$
(b)

$\frac{1}{5}$
(c)

$\frac{1}{10}+\frac{1}{100}$ [There are two ways you could start with this one.]
(d)

$\frac{1}{3}$

Exercise 1.3.

Use whatever process you know for division to write the following fractions of integers in decimal notation.

Hint: For example, you might have learned ‘short division’ (sometimes called the ‘bus stop method’) or ‘long division’ in your previous studies.

(a)

$\frac{11}{9}$
(b)

$\frac{1}{7}$

[Hint: for (a) $\frac{11}{9}$ is the same as $11.0000\ldots$ divided by $9$ .]

1.1.2 Workshop tasks

Task 1.4 (15 min).

In this task, we’ll explore the decimal notation of fractions $\frac{1}{n}$ where $n$ is a positive integer.

1.

In your group, show each other your “division algorithm” - the method you used to write $\frac{1}{7}$ in decimal notation in the preparatory exercises.
2.

Use a division algorithm to write these fractions in decimal notation:

(a) $\frac{1}{4}$ , (b) $\frac{1}{5}$ , (c) $\frac{1}{6}$ , (d) $\frac{1}{11}$ .
3.

Compare the decimals you’ve just calculated alongside the decimal notation for $\frac{1}{7}$ . What do they have in common? Where do they differ?
4.

EXTENSION: Use what you’ve observed to make a claim about the decimal representation of all fractions. Can you make a general argument to back up your claim?

Task 1.5 (20 min).

In this task, we’ll investigate decimal representations of the form ‘ $0.\overline{d}$ ’, where $d$ is a digit between $1$ and $9$ inclusive.

1.

Write down the sum that ‘ $0.\overline{d}$ ’ represents.
2.

Rewrite the sum using ‘sigma-notation’.
3.

Have you seen this sort of sum before? Can you evaluate it?
4.

What number does a decimal of the form ‘ $0.\overline{d_{1}d_{2}}$ ’ represent? Can you generalise what you’ve found for decimals of the form ‘ $0.\overline{d_{1}d_{2}\ldots d_{n}}$ ’ with $n$ digits repeating?
5.

EXTENSION: What number does a decimal of the form ‘ $0.m_{1}m_{2}\ldots m_{k}\overline{d_{1}d_{2}\ldots d_{n}}$ ’ where $m_{1},m_{2},\ldots,m_{k},d_{1},d_{2},\ldots,d_{n}$ are all single digits, represent?
6.

EXTENSION: We say the decimal $0.\overline{d_{1}d_{2}\ldots d_{n}}$ has period $n$ because the length of its repeating string consists of $n$ digits.

What’s the period of $0.\overline{001}$ ? What’s the period of $(0.\overline{001})^{2}$ ? Is it what you expected?

Definition (Irrational number).

A real number that is not rational is said to be irrational.

Task 1.6 (10 min).

Study the definition of an irrational number.

1.

What can you say about the decimal representation of an irrational number?
2.

If, for every integer $n\geqslant 1$ , $a_{n}$ is the remainder when $n$ is divided by 7, is $0.a_{1}a_{2}a_{3}\ldots$ rational or irrational?
3.

Think of three examples of your own of decimals that are irrational, and justify why.

Definition (Rational number).

A rational number is a real number that can be written as $p/q$ where $p$ and $q$ are integers, and $q>0$ .

Definition (Irrational number).

A real number that is not rational is said to be irrational.

Task 1.7 (15 min).

Investigate whether the following is rational or irrational. If you make any general claims, try to form a general argument to back them up.

1.

The product of two rational numbers.
2.

The sum of two rational numbers.
3.

The product of a rational number and an irrational number.
4.

EXTENSION: The sum of two irrational numbers.
5.

EXTENSION: The product of two irrational numbers.