Decimal representation

A decimal representation of a <a href="/facts/Non-negative/z0xollg1">non-negative</a> <a href="/facts/Real_number/R02gw5Pb">real number</a> r is its expression as a <a href="/facts/Sequence/gV2S4uqp">sequence</a> of symbols consisting of <a href="/facts/Decimal_digit/JJdLBoGT">decimal digits</a> traditionally written with a single separator:

r
        =
        
          b
          
            k
          
        
        
          b
          
            k
            −
            1
          
        
        ⋯
        
          b
          
            0
          
        
        .
        
          a
          
            1
          
        
        
          a
          
            2
          
        
        ⋯
      
    
    {\displaystyle r=b_{k}b_{k-1}\cdots b_{0}.a_{1}a_{2}\cdots }

Here . is the <a href="/facts/Decimal_separator/t6ocVouk">decimal separator</a>, k is a <a href="/facts/Nonnegative_integer/u65og8JE">nonnegative integer</a>, and 
 
 
 
 
 b
 
 0
 
 
 ,
 ⋯
 ,
 
 b
 
 k
 
 
 ,
 
 a
 
 1
 
 
 ,
 
 a
 
 2
 
 
 ,
 ⋯
 
 
 {\displaystyle b_{0},\cdots ,b_{k},a_{1},a_{2},\cdots }
 
 are digits, which are symbols representing integers in the range 0, ..., 9.
Commonly, 
 
 
 
 
 b
 
 k
 
 
 ≠
 0
 
 
 {\displaystyle b_{k}\neq 0}
 
 if 
 
 
 
 k
 ≥
 1.
 
 
 {\displaystyle k\geq 1.}
 
 The sequence of the 
 
 
 
 
 a
 
 i
 
 
 
 
 {\displaystyle a_{i}}
 
—the digits after the dot—is generally <a href="/facts/Finite_sequence/gV2S4uqp">infinite</a>. If it is finite, the lacking digits are assumed to be 0. If all 
 
 
 
 
 a
 
 i
 
 
 
 
 {\displaystyle a_{i}}
 
 are 0, the separator is also omitted, resulting in a finite sequence of digits, which represents a <a href="/facts/Natural_number/u65og8JE">natural number</a>.
The decimal representation represents the <a href="/facts/Series_(mathematics)/yDnlsgIe">infinite sum</a>:

r
        =
        
          ∑
          
            i
            =
            0
          
          
            k
          
        
        
          b
          
            i
          
        
        
          10
          
            i
          
        
        +
        
          ∑
          
            i
            =
            1
          
          
            ∞
          
        
        
          
            
              a
              
                i
              
            
            
              10
              
                i
              
            
          
        
        .
      
    
    {\displaystyle r=\sum _{i=0}^{k}b_{i}10^{i}+\sum _{i=1}^{\infty }{\frac {a_{i}}{10^{i}}}.}

Every nonnegative real number has at least one such representation; it has two such representations (with 
 
 
 
 
 b
 
 k
 
 
 ≠
 0
 
 
 {\displaystyle b_{k}\neq 0}
 
 if 
 
 
 
 k
 >
 0
 
 
 {\displaystyle k>0}
 
) <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> one has a trailing infinite sequence of 0, and the other has a trailing infinite sequence of 9. For having a one-to-one correspondence between nonnegative real numbers and decimal representations, decimal representations with a trailing infinite sequence of 9 are sometimes excluded.

Decimal representation open-in-new

Decimal representation