Decimal fractions are numbers written in decimal notation. They are a very common way of displaying fractions in a compact manner. You will encounter them everywhere in your daily life: on price tags, sizes or on the display of your calculator.

Examples:

  • %%15.731%% is a decimal fraction. If written as a fraction it would look like this: %%\dfrac {15731}{1000}%%

  • %%7.50\, $%% are %%7\, $%% and %%50\, \mathrm {ct}%%; definitely no one would write it as %%\dfrac {15}{2} \,$%%.

  • A quarter inch written as a decimal fraction is %%0.25 \, \mathrm {in}%%.

Structure of decimal fractions

Every decimal fraction is constructed in the following way:
integer - decimal mark - decimal places:

  • The decimal mark - written like a full stop - separates the integer part from the decimal places.
  • In front of the decimal mark the integer part is written.
    (Ones, tens, hundreds etc.)
  • After the decimal mark the decimal places are written.
    (tenths, hundredths, thousandths etc.)

The decimal fraction shown in the figure will look like this when written in detail: $$863.267= 863+ \dfrac {2}{10} +\dfrac{6}{100} + \dfrac{7}{1000}$$

Decimal fractions in the place value table

Decimal fractions like %%863.267%% can be easily entered into the place value table if you expand it to include tenths, hundredths, thousandths etc.:

Th

H

T

O

.

t

h

th

0

8

6

3

.

2

6

7

Number of decimal places in a decimal fraction

Essentially, a decimal fraction can possess any number of decimal places.
You need to distinguish between three different cases:

1. Terminating decimal fractions

In this case there is a finite number of decimal places after the decimal mark.

Examples:

  • %%190.4%%
  • %%0.045000%%
    (Comment: Zeros at the end of a decimal fraction do not change its value; %%0.045000%% is the same as %%0.045%%)
  • %%235.678554467%%

2. Periodic decimal fractions

These decimal fractions possess an infinite number of decimal places after the decimal mark, but after a certain point there is an infinitely-repeated digit sequence.

This digit sequence is called the repetend of the decimal fraction and indicated by drawing a horizontal line above it.

Examples:

  • %%0.\overline {3}=0.3333333333…%%
  • %%0.1\overline {6}=0.16666666….%%
  • %%0.\overline {16}=0.1616161616….%%
  • %%245.0674\overline {45698}=245.0674456984569845698…%%

(Comment: A decimal fraction with a repetend "in between" (for example: %%8. \overline {0} 21%%) does not make any sense, since you would have to wait for the end of an infinite number of zeros before putting the %%21%%.)

3. Non-terminating non-periodic decimal fractions

If you tried to write an irrational number like %%\sqrt {2}%% or %%\pi%% as a decimal, you would end up with a number that has an infinite number of decimal places but no repeating sequence.

Examples:

  • %%\sqrt {2} =1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694… \approx1.41%%
  • %%\pi = 3.1415926…\approx3.14%%
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