Earlier in class, we started to discuss different bases for mathematical systems. I brought up base 1 and we all got into a debate as to whether or not base 1 actually makes sense.
The answer is, it depends on who you ask. To understand why, we need to really know what it means for a number to be represented in base “n”. A way of rigorously defining this is to define a sequence of whole numbers a0, a1, a2,…, with each a being less than n, such that the number you want to represent is equal to a0+a1*n+a2*n2+… For example, representing 5 in base 2 would give us a0=1, a1=0 and a2=1, because 1 + 0*2 + 1*4 = 5. Then, it is a simple matter of writing the number as a0a1a2…, or in this specific case, 101.
We want to have 4 properties when doing this:
1. Each number has at least 1 representation.
2. Each number has no more than 1 representation.
3. Each representation corresponds to at least one n.
4. Each representation corresponds to no more than 1 n.
Some of these properties are more important than others. As it turns out, properties 1 and 4 are key. Property 1 guarantees that anything we want to write in our base, we can. It’d be silly if we had a numerical system that couldn’t list numbers over 1,000,000, for example. Property 4 is even more important: it guarantees that no two different numbers can look the same. It would be really silly if we looked at a number and had to guess whether or not it was 3 or 4.
As it turns out, to have ALL of these properties, we must specify that n is greater than or equal to 2. Why? If we set n equal to 1, then the restriction that each a must be less than n requires ALL a’s to be equal to 0. Meaning that 0 becomes the ONLY number we can represent. Not very useful.
Well, let’s see if we can salvage this. Let’s say we drop the restriction that each of the a’s has to be less than our n. Now our situation becomes more hopeless. In base 1, we could, for example, write 6 as 51, or 2211, or 312, or 501, and so on. While it may be clear what each of these numbers is, there’d be no point in being able to write it so many ways. And if we picked the simplest one to represent our system, why not just pick the number itself? And if we’re representing a number by simply itself, why complicate things and work in this odd base-1 system anyways?
This is why when mathematicians and computer scientists talk about base-1, they are mostly referring to the unary mathematical system.
If you want to represent a number in the unary system, say N, all you do is repeat the same symbol N times. So 10 becomes ||||||||||, and so on. This type of numerical representation is a lot like counting on your fingers or tallying a certain number of objects. It’s also useful for birthday cakes.
Now, let’s see if this satisfies the above 4 conditions:
1. Each number has at least 1 representation. Clearly true
2. Each number has no more than 1 representation. True
3. Each representation corresponds to a number. If a representation has N slashes, the number it represents is simply N. True
4. Each representation corresponds to no more than one number. If a number represents N, it can’t represent anything else in this system. True.
Alright, it looks like we have a good system. Now, let’s take a look at how one would add, subtract, multiply or divide. Addition and subtraction are quire easy in the unary system: For addition, just combine the two sets of slashes, and for subtraction take away a certain number of slashes from one of them. In fact, division and multiplication are also quite easy. Calculating N/M is as easy as crossing out M slashes from the number N, marking that as a 1, and repeating until you no longer have M slashes left. N*M is as simple as writing down N slashes M times.
The advantages of the unary system is that you do not need to memorize any multiplication tables to do multiplication. As long as you know the system, it’s incredibly simple. The obvious downside is that even for relatively small numbers multiplied by one another the result can become extremely large extremely quickly. You can think of it sort of as a trade-off: In base 10, for example, multiplication and division are much more difficult, but can be much, much quicker to calculate and much easier to understand.
How this actually relates to math history is through ancient Egyptian mathematics. What’s interesting about their style of mathematics is that it incorporates elements of both the unary system and the base 10 system, and isn’t truly one or the other, but a hybrid of both. In the ancient Egyptian system, a 1 is represented as |, 2 as ||, and so on. But 10 gets its own symbol, represented as an upside down U, and 100, 1,000, 10,000, 100,000 and 1,000,000 all have their own representation. This is somewhat “in-between” the two bases: on the one hand, the order in which you place the digits doesn’t matter, but on the other, numbers become quite a bit longer than they need to be.
This goes to show one of the cooler parts of mathematics: when you take an idea such as “dividing by 0” or “base-1” to its logical extreme, you start to understand deeper relationships between numbers, and can come up with some very interesting results.