As a disclaimer, this post can be considered a sort of mathematical rant. In daily life, it is a more or less agreed upon fact that we communicate natural numbers (eg. $1,2,3,…$) in what we call “base-10”. This is more or less ubiquitous around the world. If one thinks about this “base-10” phenomenon, it starts to seem weird and even paradoxial. Let me explain why.

Numerical symbols

Let’s think of our notion of numbers as symbolisms to represent quantities. For a moment, forget what numbers mean to you, and try to only associate numbers with quantities, as shown below:

If we restrict ourselves to only associating numbers as symbols, we start to put our attention on the actual quantities that these numbers represent. As shown above, we can essentially go on with our representations. If you think deeply about it, the final unique symbol that we have for such quantities is 9.

Birth of 10

So what comes after 9? Well, I would argue 9+1. But we don’t have a (layman/ubiquitous) symbol for such a quantity. So we use (what we know as) 10.

But what is 10 really? Well 10 can be broken down into a 1 and 0 (reading from left to right). 10 decodes itself to 1 aggregated quantity and 0 basic quantities. What an aggregated quantitiy is; is open to our definition.

Let’s say we use base-2, then 2 is our aggregated quantity and 10 represents 2. If we use base-3, 10 in base-3 represents 3.

Base 9+1

So we can see that 10, rather than being an actual symbolic number (like 0 to 9), is actually a referential number. 10 requires a clearly defined base and it represents aggregated quantities of that base.

With that idea in mind, it makes no sense to me to say that we use base-10. 10 is not a symbolic number, it is a referential number. So in my opinion, we should not refer to our layman number system as base-10. A more correct representation would be base 9+1, since 9+1 best represents what 10 means to us in daily life. Either that, or come up with a widely accepted symbol which represents the quantity encoded by 9+1.