Types of decimal numbers

The exact decimal numbers have a finite number of decimal numbers. For example: $$$0,345$$$ $$$-1,78993$$$ $$$123434,001$$$

The pure periodic numbers have a decimal part that is repeated infinitely. $$$0,\widehat{3}=0,33333333333\ldots$$$

$$0,\widehat{126}=0,126126126126\ldots$$ $$0,\widehat{62}=0,626262626262\ldots$$

The ultimately periodic numbers have a non periodic part and then a periodic part. $$$0,54\widehat{3}=0,5433333333\ldots$$$

$$2,17\widehat{23}=2,172323232323\ldots$$ $$13,1\widehat{789}=13,1789789789\ldots$$

Non exact and non periodic numbers cannot be expressed as fractions. $$$\pi=3,141592653589793238\ldots$$$

$$e=2,718281828459045235\ldots$$ $$\sqrt{2}=1,414213562373095048\ldots$$

It is possible to find which type of decimal will be obtained from its equivalent fraction. It is enough to break down the denominator into fractions:

  • If it is formed only by $$2$$ and/or $$5$$ factors, it will be an exact decimal.
  • If it does not contain any $$2$$ and any $$5$$, it will be a pure periodic decimal.
  • If it contains $$2$$ or $$5$$ and other factors, it will be an ultimately periodic decimal.