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.