> 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)? The story goes th

> 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)? 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?