In mathematics the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, -adic numbers have the interesting property that they are said to be close when their difference is divisible by a high power of – the higher the power the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.