2.1 Real numbers and their decimal representations

The numbers $0$, $1$, $-1$, $-\frac{1}{7}$, $\sqrt{2}$, $\pi$, $e$, $0.1$ and $0.333\ldots$ are examples of real numbers. There are numbers that are not real. For example, you will study complex numbers later in your degree programme. We begin by discussing different types of real number and different ways of representing them.

The most common way of writing integers is the decimal numeral system. This is also known as the Hindu–Arabic numeral system due to its origins in those civilisations. We will now define terms. Please check that you understand each definition and that it agrees with what you already know.

In the decimal numeral system, a non-negative real number $x$ (meaning $x\geq 0$) is written as a sum

where each of $a_{0},a_{1},a_{2}\ldots,a_{n}$ and $b_{1},b_{2},b_{3}\ldots$ is a digit (meaning one of $0,1,2,\ldots,9$).¹¹ 1 You may find this definition circular, since we use an integer written in decimal (namely $10$) to define what it means to write an integer in decimal. However, the ‘$10$’ here really means the integer obtained by adding together ten lots of $1$.

We abbreviate the sum above to ‘$a_{n}a_{n-1}\ldots a_{2}a_{1}a_{0}.b_{1}b_{2}\ldots$’, which is called a decimal representation of $x$. We also say that $x$ is written in decimal notation. Observe that we allow infinitely many digits to the right of the decimal point.

The integer $\sum_{k=0}^{n}a_{k}10^{k}$ (denoted by ‘$a_{n}a_{n-1}\ldots a_{1}a_{0}$’) is called the integer part of $x$. The real number $\sum_{k=1}^{\infty}b_{k}\frac{1}{10^{k}}$ (denoted by ‘0.$b_{1}b_{2}\ldots$’) is called the fractional part of $x$.

For example, the decimal representation $1478$ means $8\times 10^{0}+7\times 10^{1}+4\times 10^{2}+1\times 10^{3}$. The integer part of $1478$ is $1478$ and the fractional part is $0$.

The following examples demonstrate different properties that decimal representations can have.

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  $\frac{1}{2}$ has decimal representation $0.5000\ldots$, but usually we do not write the tail of $0$s and instead write $0.5$. We say that the decimal representation terminates.

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  $\frac{1}{3}$ has decimal representation $0.333\ldots$ (with $3$ repeating indefinitely to the right).

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  $\frac{1}{11}$ has decimal representation $0.090909\ldots$, and we shall write $0.\overline{09}$ to mean that the block "09" of two digits repeats indefinitely to the right.

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  $\pi$ has decimal representation $3.141592653589793$, to 15 decimal places. $\pi$ has no repeating pattern in the fractional part. This requires proof and is not obvious.

Note that above we only defined the decimal representation of a non-negative real number $x$. For completeness, if $x$ is a negative real number then $-x$ is positive and so has some decimal representation $-x=\text{`$a_{n}a_{n-1}\ldots a_{2}a_{1}a_{0}.b_{1}b_{2}\ldots$'}$. Then $x$ has integer part $-\sum_{k=0}^{n}a_{k}10^{k}$ and fractional part $-\sum_{k=1}^{\infty}b_{k}\frac{1}{10^{k}}$.