P-adic number

Number system extending the rational numbers

In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers that is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number 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 base 3 versus the 3-adic expansion, 1 5 = 0. 0121 ¯ ( base 3) = 0 ⋅ 3 0 + 0 ⋅ 3 − 1 + 1 ⋅ 3 − 2 + 2 ⋅ 3 − 3 + ⋯ 1 5 = 1210 ¯ 2 ( 3-adic) = ⋯ + 2 ⋅ 3 3 + 1 ⋅ 3 2 + 0 ⋅ 3 1 + 2 ⋅ 3 0 .

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