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.)

structure of decimal fractions

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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