How to write zero in the unary numeral system

• In the decimal system: $...-3, -2, -1, 0, 1, 2, 3...$
• In the binary system: $...-11, -10, -1, 0, 1, 10, 11...$
• In the unary system: $-111, -11, -1, \text{ ??? }, 1, 11, 111...$

I was only able to find a related question but none with an answer so far:

unary numbers: base-1 numeral system

Solution 1:

You do something like this:

Sneaky, no?

"Unary" is just a tally-mark system, not a really full-fledged numeral system. You represent Zero by not representing any quantity at all.

One Apple: |

Two Apples: ||

No Apples:

And that's basically it. Putting any other symbol on the system to represent a Zero makes it a binary system.

In the comments, Joshua talks about head-tail delimiters for unary systems. You can use head-tail delimiters to make the zero-quantity evident or to separate groups of numbers.

For example, one could write "||", "|||", and "" to represent 2, 3 and 0. In this case, the quotes aren't part of the numbers but they help us visualize where the number starts or ends and they make the 0-value visible.

Keep in mind that "unary" isn't really a numeral system with the same representative capabilities of binary, hex, etc. For it to be used properly as a mathematical tool you end up needing to hack it a bit adding some other symbols. At that point you can just use something else that can represent your problem on a more practical way.

Solution 2:

In fact, there is no "unary" number system that is the analog of fixed-radix systems (e.g., base $2$, base $10$, etc). A radix, by definition, is a number greater than $1$.

Your "unary" system is really nothing more than tally marks. So zero would be represented by the absence of any symbol. These are really strings, not numbers. You can't add them using column addition. So you really have

$$-3_{10} = “-111_1”$$ $$-2_{10} = “-11_1”$$ $$-1_{10} = “-1_1”$$ $$0_{10} = “”$$ $$1_{10} = “1_1”$$ $$2_{10} = “11_1”$$ $$3_{10} = “111_1”$$