More than Ten numbers in ten?

Question

(I am going to be using the written out form of numbers seeing as the extra number im going to be talking about does not exist.)

Ok. To start... What if there where Eleven numbers in Ten?

For example (for the sake of not sounding stupid) it would look something like this: 1,2,3,4,5,6,7,8,9,A,10. "A" representing a new number.

Just to avoid nay confusion, "A" is not an existing number. It would be like instead of saying "One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Ten." it would be "One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Nate, Ten."

A simple math problem would be done a bit differently. example 1: One + Ten = Eleven

So if you where to count one at a time till you reach the answer: One, Two, Three, Four, Five, Six, Seven, Eight, Nine, A, Ten, Eleven. (12 numbers)

I have only done a few light problems with this method so far.

(the only way i have to do the calculation at the moment is paper and pencil because standard calculators aren't programmed for this type of thing)

Hoping to get some feedback on this, I'm sure there has been several people who have thought of this but i could not find and papers or articles about it.

In conclusion, does anyone think that there could be any real world applications for this or if re-creating the mathematics system all together would open up a whole new world for us?

(really don't know what to tag this)

Answer 1

You are just referring to non-decimal numerations. Let $b$ your preferred integer, called the base (for you $11$ rather than $10$). Any number can be uniquely represented as

$$d_kb^k+\cdots +d_2b^2+d_1b+d_0$$ where the $d_k$ are the digits, from $0$ to $b-1$.

The smallest base is $2$, corresponding to the binary notation (only with digits $01$). Historically, base $8$ was used in the early days of computers (digits $01234567$, called the octal notation). It was superseded by base $16$, giving the hexadecimal representation ($0123456789ABCDEF$), still much in use.

Other bases have less interest. Anyway, it is worth to note the the Babylonians were using a mixed $6/10$ numeration system equivalent to base $60$ (!), that left a trace in our daily life with hours of sixty minutes and minutes of sixty seconds. Such a numeration system is terrible.

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In your base, one would for instance have

$$8A3_{11}=8_{10}\cdot11_{10}^2+10_{10}\cdot11_{10}+3_{10}=1009_{10}.$$

For non-decimal basis, there is no particular convention for reading. One would just pronounce "eight a three".

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Also notice that smaller bases are more "economical" in terms of the size of the addition/multiplication tables. This is why the binary numeration has been universally adopted in digital devices (computers), for its extreme simplicity.

So the "revolution" you evoke already occurred, but with base $2$.

On the opposite, larger bases are unattractive. Our base $10$ system is obviously related to our number of fingers, and has no particularly interesting property. Even ess for $11$ I am afraid.

Answer 2

What you are proposing is a positional number system with a base 11 (where usually base 10 is used). Positional number systems are commonly applied, particularly in computer systems, that almost universally use binary (base 2, so 0 and 1 are the only digits used). Octal and hexadecimal (bases 8 and 16, the latter using letters A-F as digits after 9) are also commonly used among programmers because they are easy to translate to and from binary and are more human-readable than plain binary.

However, note that in your example, A is the ten of our decimal system, and 10 is eleven, and so on. Positional system only touches the notation, but doesn't really create "new numbers". Also, base 11 math wouldn't be really convenient for most uses, as it'd invalidate some convenient properties of base 10 like even numbers being easy to recognize by them ending in 0, 2, 4, 6, or 8.