The story goes that Hippasos of Metapontum discovered the existence of irrational numbers like √2 while at sea, and his fellow Pythagoreans threw him overboard, because it proved that there were aspects of the world that could not be represented with rational numbers. That said, the notion of a square root had been established for at least a thousand years by that point (by the Egyptians and Sumerians), so the freaking out was less about their representation and more about their properties.
> For "0" I would have made an empty box...
Your representations are parasitic on the fact that you already have a deeply ingrained representation for those concepts and are comfortable manipulating those concepts. It's like suggesting that you'd reinvent the wheel if you lived in a civilization with no wheels merely because it seems obvious to you looking at it today. Really it's not at all obvious, as no New World civilization ever developed the wheel, and most Old World civilizations borrowed it rather than inventing it independently.
Think of it this way: if you understood fractions, but nobody else did, how would you represent them symbolically such that everyone else would? I'm of the opinion that our typical notation for calculus is spectacular (especially compared to Newton's original notation—the d/dx notation used today comes via Leibniz) but if you don't understand calculus, how does that help you? You understand negative numbers, but to a person with no concept of debt, a person for whom numbers represent 'how many sheep you have', what does that mean, and how do you symbolize it so that they do understand?
That's 4. Because you used four strings. Oh, you do it with one string? It looks like 1. You do it with N strings? That's N. Oh, you're trying to talk about what's inside? Nothing's inside. That's silly. We both know there's nothing in there. Why are you trying to show me nothing?
Are you trying to multiply by the inside? That doesn't make any sense. Why would you multiply by nothing, or add by nothing? What's the point? No, I don't know what it does. Why should I care about your nonsense?
> For the -2 I would have placed two of the strings in a different location.
Now do "square root of negative nine".
> For the Roman numeral conundrum I would have placed IV and then a line and then a V below that.
Oh, so there's 4 on the top and 5 on the bottom? So, 9 total, right? IX?
> There was a time when numbers like "4/5", "-2", "0", and "√5" made people freak out because we didn't have the symbols to represent them
Who freaked out, and how could they freak out if they could not behold these numbers in the first place (i.e. no symbolic representations)?
> without drawing a shape that somehow encoded that number
Recommend you use "represented that number" because encoding has a strong, separate type of meaning to me.
> Now, pop quiz: how do you represent something like "0" or "-2" in this world? If you were a Roman using Roman numerals, how would you represent "4/5"?
For "0" I would have made an empty box, out of strings if necessary. For the -2 I would have placed two of the strings in a different location. For the Roman numeral conundrum I would have placed IV and then a line and then a V below that. Now, I just solved your conundrums. Or didn't I? To really speak to beginners it's important to delve into this stuff. So far I have not discovered why our number system is so great, and I don't feel like I'm freaking out about anything in particular. :-)
Thanks for your writeup, though. I hope you can turn it into a book for people like me.
--Designer guy who is not super great at math