The difference between P-adic number and Rational number
When used as nouns, p-adic number means an element of a completion of the field of rational numbers with respect to a p-adic ultrametric, whereas rational number means a number that can be expressed as the ratio of two integers.
check bellow for the other definitions of P-adic number and Rational number
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P-adic number as a noun (number theory):
An element of a completion of the field of rational numbers with respect to a p-adic ultrametric.
Examples:
"The expansion (21)2121<sub>''p''</sub> is equal to the rational <b>''p''-adic number</b> <math>\textstyle{2 p + 1 \over p^2 - 1}.</math>"
"In the set of [[p-adic number 3-adic numbers]], the closed ball of radius 1/3 "centered" at 1, call it ''B'', is the set <math>\textstyle\{x \exists n \in \mathbb{Z} . \, x = 3 n + 1 \}.</math> This closed ball partitions into exactly three smaller closed balls of radius 1/9: <math>\{x \exists n \in \mathbb{Z} . \, x = 1 + 9 n \},</math> <math>\{x \exists n \in \mathbb{Z} . \, x = 4 + 9 n \},</math> and <math>\{x \exists n \in \mathbb{Z} . \, x = 7 + 9 n \}.</math> Then each of those balls partitions into exactly 3 smaller closed balls of radius 1/27, and the sub-partitioning can be continued indefinitely, in a fractal manner.<br>Likewise, going upwards in the hierarchy, ''B'' is part of the closed ball of radius 1 centered at 1, namely, the set of integers. Two other closed balls of radius 1 are "centered" at 1/3 and 2/3, and all three closed balls of radius 1 form a closed ball of radius 3, <math>\{x \exists n \in \mathbb{Z} . \, x = 1 + {n\over 3} \},</math> which is one out of three closed balls forming a closed ball of radius 9, and so on."
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Rational number as a noun (mathematics):
A number that can be expressed as the ratio of two integers.