Is it allowed to define a number system where a number has more than 1 representation?

Question

I was just curious;

Is it allowed for a number system to allow more than one representation for a number?

For example, if I define a number system as follows:

The 1st digit (from right) is worth 1.

The 2nd digit is worth 2.

The 3rd is worth 3.

The 4th is worth 5.

The 5th is worth 7.

The 6th is worth 11.

And so on.... for all primes.

Now $9_{10}=10010_P$ for instance as $9=7+2$

But $8_{10}=1100_P=10001_P$.

Is that allowed?

P. S.

Is there any practical use of the number system mentioned above? Or not, since even operations like addition are almost impossible in it.

Answer 1

It is certainly allowed for a number to have multiple representations in a system of representations. In fact, the decimal system which we use all the time has this property. The well known fact that $$1=0.999...=0.\overline{9},$$ is an example of this. Whether there is any practical use to the system you mention I don't know, but it seems unlikely. In the decimal number system every number has at most two representations, and it is easy to see whether two of them represent the same number. In your number system the number of ways to represent a number grows very quickly as the numbers grow, making it a nontrivial exercise to even figure out whether two representations represent the same number!

Answer 2

It is possible. The usual decimal representation of the the reals numbers has two representations for any number with a finite number of decimal places. For example. $1=0.\overline 9$.

Answer 3

Sure, you can define anything you care to. In fact many commonly used systems, like decimal numbers, have elements with multiple representations like $0.999\ldots=1.$ Another example would be fractions, where $1/3=2/6.$

See A066352 in the OEIS for an example of an early use (by S. S. Pillai) of this particular system, and see also the related sequence A007924.

Answer 4

You won't be thrown in jail for defining a bad number system, but you can certainly earn the scorn of professional mathematicians and wannabes, which in the big picture of things is not such a terrible thing.

In my opinion, it's better for a number to have more than one representation than it is for one representation to correspond to more than one number. But it's best for each integer to have unique representation.

Answer 5

You can certainly do this, and I can imagine situations where it might be useful. For instance, suppose you take the normal decimal valuation of each place (ones, tens, hundreds, etc.), but allow "digits" from $0$ through $A$. Then operations like addition and multiplication can be carried out with the usual algorithm, but with some flexibility about when/if to carry. If you give me $999999 + 1$, the result could be $1000000$ or $99999A$... the latter requires fewer operations, and if I'm just going to subtract something from it in the future, might save a few steps. As others have pointed out, comparing two numbers is no longer trivial; but it can be done by first coercing both numbers into a comparable form, e.g., performing all optional carries. The relative efficiency has to depend on how often you're going to be comparing things vs. performing arithmetic operations. (One thing you can say for sure, though, is that a non-unique representation takes more space, in bits, on average than a unique representation. From an information-theoretic perspective, this is because you're encoding some useless extra data -- which representation you chose -- along with the actual value.)