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In mathematics, the dozadic number system (or the 10-adic number system) extends the ordinary arithmetic of the rational numbers in a different way 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, Dozadic numbers are considered to be close when their difference is divisible by a high power of 10: the higher the power, the closer they are. This property enables dozadic 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.

Introduction[]

This section is an informal introduction to dozadic numbers. The dozadic numbers are generally not used in mathematics: since 10 is not prime or prime power, the dozadics are not a field. More formal constructions and properties are given below.

In the standard dozenal representation, almost all[note 1] real numbers do not have a terminating dozenal representation.

For example, 1/5 is represented as a non-terminating dozenal as follows

and 1/7 is represented as a non-terminating dozenal as follows

Informally, non-terminating dozenal are easily understood, because it is clear that a real number can be approximated to any required degree of precision by a terminating dozenal. If two dozenal expansions differ only after the 10th decimal place, they are quite close to one another; and if they differ only after the 20th decimal place, they are even closer.

Dozadic numbers (10-adic numbers) use a similar non-terminating expansion, but with a different concept of "closeness". Whereas two dozenal expansions are close to one another if their difference is a large negative power of 10, two dozadic expansions are close if their difference is a large positive power of 10. Thus 4739 and 5739, which differ by 103, are close in the 10-adic world, and 72694473 and 82694473 are even closer, differing by 107.

More precisely, a positive rational number Template:Mvar can be uniquely expressed as Template:Math, where Template:Mvar and Template:Mvar are positive integers and Template:Mvar, Template:Mvar, and 10 are pairwise relatively prime. Let the Template:Nowrap "absolute value"[note 2] of 

be

 .

Additionally, we define

 .

Now, taking Template:Math and Template:Math we have

Template:Math, Template:Math, Template:Math,

with the consequence that we have

 .

Closeness in any number system is defined by a metric. Using the 10-adic metric the distance between numbers Template:Mvar and Template:Mvar is given by Template:Math. An interesting consequence of the 10-adic metric (or of a Template:Mvar-adic metric) is that there is no longer a need for the negative sign. (In fact, there is no order relation which is compatible with the ring operations and this metric.) As an example, by examining the following sequence we can see how unsigned 10-adics can get progressively closer and closer to the number βˆ’1:

       so .
       so .
       so .
       so .

and taking this sequence to its limit, we can deduce the 10-adic expansion of βˆ’1

 ,

thus

 ,

an expansion which clearly is a ten's complement representation.

In this notation, 10-adic expansions can be extended indefinitely to the left, in contrast to decimal expansions, which can be extended indefinitely to the right. Note that this is not the only way to write Template:Mvar-adic numbers – for alternatives see the Notation section below.

More formally, a 10-adic number can be defined as

where each of the Template:Math is a digit taken from the set {0, 1, … , 9} and the initial index Template:Mvar may be positive, negative or 0, but must be finite. From this definition, it is clear that positive integers and positive rational numbers with terminating decimal expansions will have terminating 10-adic expansions that are identical to their decimal expansions. Other numbers may have non-terminating 10-adic expansions.

It is possible to define addition, subtraction, and multiplication on 10-adic numbers in a consistent way, so that the 10-adic numbers form a commutative ring.

We can create 10-adic expansions for "negative" numbers[note 3] as follows

and fractions which have non-terminating decimal expansions also have non-terminating 10-adic expansions. For example

Generalizing the last example, we can find a 10-adic expansion with no digits to the right of the decimal point for any rational number Template:Math such that Template:Math is co-prime to 10; Euler's theorem guarantees that if Template:Mvar is co-prime to 10, then there is an Template:Mvar such that Template:Math is a multiple of Template:Math. The other rational numbers can be expressed as 10-adic numbers with some digits after the decimal point.

As noted above, 10-adic numbers have a major drawback. It is possible to find pairs of non-zero 10-adic numbers (which are not rational, thus having an infinite number of digits) whose product is 0.[1][note 4] This means that 10-adic numbers do not always have multiplicative inverses i.e. valid reciprocals, which in turn implies that though 10-adic numbers form a ring they do not form a field, a deficiency that makes them much less useful as an analytical tool. Another way of saying this is that the ring of 10-adic numbers is not an integral domain because they contain zero divisors.[note 4] The reason for this property turns out to be that 10 is a composite number which is not a power of a prime. This problem is simply avoided by using a prime number Template:Mvar or a prime power Template:Mvar as the base of the number system instead of 10 and indeed for this reason Template:Mvar in Template:Mvar-adic is usually taken to be prime.


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