p-adic number

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, given a <a href="/facts/Prime_number/L20FLkgn">prime number</a> p, the p-adic numbers form an extension of the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> which is distinct from the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, though with some similar properties; p-adic numbers can be written in a form similar to (possibly <a href="/facts/Infinity_(mathematics)/cF506uD7">infinite</a>) <a href="/facts/Decimal_representation/HCLBt4UQ">decimals</a>, but with digits based on a <a href="/facts/Prime_number/L20FLkgn">prime number</a> p rather than ten, and extending to the left rather than to the right.
For example, comparing the expansion of the rational number 
 
 
 
 
 
 
 1
 5
 
 
 
 
 
 {\displaystyle {\tfrac {1}{5}}}
 
 in <a href="/facts/Ternary_numeral_system/bMUfR74P">base 3</a> vs. the 3-adic expansion,

1
                      5

=
                0.01210121
                …
                 
                (
                
                  base 
                
                3
                )

=
                0
                ⋅
                
                  3
                  
                    0
                  
                
                +
                0
                ⋅
                
                  3
                  
                    −
                    1
                  
                
                +
                1
                ⋅
                
                  3
                  
                    −
                    2
                  
                
                +
                2
                ⋅
                
                  3
                  
                    −
                    3
                  
                
                +
                ⋯
              
            
            
              
                
                  
                    
                      1
                      5

=
                …
                121012102
                 
                 
                (
                
                  3-adic
                
                )

=
                ⋯
                +
                2
                ⋅
                
                  3
                  
                    3
                  
                
                +
                1
                ⋅
                
                  3
                  
                    2
                  
                
                +
                0
                ⋅
                
                  3
                  
                    1
                  
                
                +
                2
                ⋅
                
                  3
                  
                    0
                  
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{alignedat}{3}{\tfrac {1}{5}}&{}=0.01210121\ldots \ ({\text{base }}3)&&{}=0\cdot 3^{0}+0\cdot 3^{-1}+1\cdot 3^{-2}+2\cdot 3^{-3}+\cdots \\[5mu]{\tfrac {1}{5}}&{}=\dots 121012102\ \ ({\text{3-adic}})&&{}=\cdots +2\cdot 3^{3}+1\cdot 3^{2}+0\cdot 3^{1}+2\cdot 3^{0}.\end{alignedat}}}

Formally, given a prime number p, a p-adic number can be defined as a <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a>

s
        =
        
          ∑
          
            i
            =
            k
          
          
            ∞
          
        
        
          a
          
            i
          
        
        
          p
          
            i
          
        
        =
        
          a
          
            k
          
        
        
          p
          
            k
          
        
        +
        
          a
          
            k
            +
            1
          
        
        
          p
          
            k
            +
            1
          
        
        +
        
          a
          
            k
            +
            2
          
        
        
          p
          
            k
            +
            2
          
        
        +
        ⋯
      
    
    {\displaystyle s=\sum _{i=k}^{\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\cdots }

where k is an <a href="/facts/Integer/PUwwrawx">integer</a> (possibly negative), and each 
 
 
 
 
 a
 
 i
 
 
 
 
 {\displaystyle a_{i}}
 
 is an integer such that 
 
 
 
 0
 ≤
 
 a
 
 i
 
 
 <
 p
 .
 
 
 {\displaystyle 0\leq a_{i}<p.}
 
 A p-adic integer is a p-adic number such that 
 
 
 
 k
 ≥
 0.
 
 
 {\displaystyle k\geq 0.}

In general the series that represents a p-adic number is not <a href="/facts/Convergent_series/DTok170z">convergent</a> in the usual sense, but it is convergent for the <a href="/facts/P-adic_absolute_value/wktvC3UP">p-adic absolute value</a> 
 
 
 
 
 |
 
 s
 
 
 |
 
 
 p
 
 
 =
 
 p
 
 −
 k
 
 
 ,
 
 
 {\displaystyle |s|_{p}=p^{-k},}
 
 where k is the least integer i such that 
 
 
 
 
 a
 
 i
 
 
 ≠
 0
 
 
 {\displaystyle a_{i}\neq 0}
 
 (if all 
 
 
 
 
 a
 
 i
 
 
 
 
 {\displaystyle a_{i}}
 
 are zero, one has the zero p-adic number, which has 0 as its p-adic absolute value).
Every rational number can be uniquely expressed as the sum of a series as above, with respect to the p-adic absolute value. This allows considering rational numbers as special p-adic numbers, and alternatively defining the p-adic numbers as the <a href="/facts/Completion_(metric_space)/lsckmU8G">completion</a> of the rational numbers for the p-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.
p-adic numbers were first described by <a href="/facts/Kurt_Hensel/xoKnoBfv">Kurt Hensel</a> in 1897, though, with hindsight, some of <a href="/facts/Ernst_Kummer/dGnaTQyG">Ernst Kummer's</a> earlier work can be interpreted as implicitly using p-adic numbers.

p-adic number open-in-new

p-adic number