Decimal Numbers

Decimal numbers are numbers written in comma notation. They are essentially the same thing as a (usual) fraction, but written in a different way. For this reason, decimal numbers are sometimes also called decimal fractions.

In contrast to usual fractions, decimal numbers are used much more often (e.g., in computers or industrial applications).

Example

$15.731$ is an example for a decimal number. As a usual fraction, it would read $\dfrac {15731}{1000}$ or $15\dfrac {731}{1000}$ .
$\$ 7.50$ is an example, where a decimal number is used, e.g., for a price. For money, everyone uses decimal numbers. Written as a usual fraction, this amount would be $\$ \dfrac {15}{2}$ , but probably nobody would write it this way.
"Three quarters of a meter" is an example for a length as a usual fraction. As a decimal number, this length is $0.75 \; \mathrm {m}$ (which are $75 \, \mathrm {cm}$ ). Here, the decimal number notation is again much more common than the usual fraction.

Structure a decimal number

Each decimal number is built by the same principle:

Number - Decimal point - Decimal places.

The decimal point (.) separates the integer from the decimal places. In some countries, a comma (,) is used instead of a period (.) for this job.
In front of the decimal point, you put the integer part of the decimal number. This is the largest integer number, which is still smaller than the decimal number.
After the decimal separator you put the decimal places:
* The first digit after the separator denotes tenths (how often you add $1/10$ to the integer part)
* The second digit after the separator denotes hundredths (how often you add $1/100$ )
* The third digit after the separator denotes thousandths (how often you add $1/1000$ )
... and so on. You may put as many decimal places there, as you like.

The decimal number $863.267$ shown above is thus equivalent to:

\displaystyle 863.267=863+\frac{2}{10}+\frac{6}{100}+\frac{7}{1000}

Decimal numbers in the place value chart

Decimal numbers as $863.267$ can be put into a place value chart: Essentially, you are just labelling all digits: Those of the integer part - right to left - are ones (O), tens (T), hundreds (H) and the decimal places - left to right - are tenths (t), hundredths (h), thousandths (th), ten-thousandths (tth), and so on.

...	H	T	O	.	t	h	th	...
	8	6	3	.	2	6	7

Number of decimal places in a decimal number

A decimal number can have as many decimal places as you like. Even infinitely many of them! In fact, one may distinguish three cases:

1. Finite decimal numbers

In this case, there are finitely many digits after the decimal point, so the number will terminate at some point.

Example

$190.4$
$0.045000$
$4\, 504\, 473{.}27$
$235.678554467$ and similar numbers.

2. Repeating decimals (or Periodic decimal numbers)

Repeating decimals are decimal numbers with infinitely many digits after the decimal point, where the digits start repeating after a while.

The repeating block of numbers must be marked, e.g., with a horizontal bar above. (Depending on your country, there may also be other notations for the repeating block)

Example

$0,.3=0.3333333333\dots$
$0.1\overline{6}=0.16666666\dots.$
$0.\overline{16}=0.1616161616\dots.$
$245.0674\overline{45698}=245.0674456984569845698\dots$

(Note: A decimal number with a "repeating block in the middle", e.g. $8.\overline{0}21$ or something like that, does not exist. After all, when do you want to write the 21, if you have to wait for infinitely (!) many zeros being written before? )

3. Infinite, non-periodic decimal numbers

If you try to write an irrational number like $\sqrt {2}$ or $\pi$ as a decimal number, you may obtain infinitely many digits after the decimal point, which do never repeat periodically.

Example

$\sqrt{2}=1.41421356237309504880168872420969807856967187537694\dots\approx1.41$ (You can find more about it here: Square root of 2)
$\pi=3.1415926\dots\approx3.14$ (You can find more about it here: The number pi)

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