In mathematics there are lots of different types of numbers. These numbers are categorised in several ways. For example:

Positive numbers: 1, 3, 5.5, 10 etc

Integers: -5, -1, 4, 50 etc

Real numbers: -1.5, 6, Pi

Irrational numbers, like Pi and the square root of 2 (irrational numbers can’t be expressed as a fraction of two integers)

Then there are numbers that are not real, like the square root of -1, which is called an imaginary number

Historically mathematicians have debated what category, if any, zero fits into. Does it fit with the integers? Is it rational? is it real? Some of these debates have been quite big disputes

Not sure what you mean specifically or what happened on that show, but historically (like, a long time ago) numbers were far more practical in use, there was not really a branch of mathematics for studying numbers or theory related to it. At one point the “first” math could be considered geometry. Now obviously 0 is useful and important, but if you are one of most of the population a couple thousand years ago you probably don’t have any reason to think about what it would look like to trade 0 sheep for 0 coins.

I’m not sure if “controversial” is the word I would use, but zero is a higher level mathematical concept than it seems, and isn’t immediately useful to people who only deal with tangible math concepts. You never count zero potatoes. You never mix bread dough with a zero-to-something ratio. You never build a house with a side that is zero feet long. If you ever need to describe something that has a quantity or size of zero, there are other words that already mean that. “None.”

Even philosophically, it’s kind of complex to describe the absence of something, especially if that something wasn’t there before (“never existed” vs “there was some and now it’s gone”).

It, and negative numbers, introduce uncomfortable paradoxes. All cultures basically understood zero could exist, but the axiom was ‘nothing can’t exist’. That is a whole ball of wax that includes ideas of things like ‘the void’. Zeno’s paradoxes can easily be broken if we accept the idea of zero, the void, nothing is a countable value. Even negative number made more sense even though they obviously didn’t exist. -1 + -2 = -3 and the like. 0 + 0 = …0. Any number times zero is magically turned into zero. I say it is paradoxical because to accept zero exists, then by definition it can’t be nothing. This isn’t new, all of computer science is based off of a self referential paradox, but that isn’t what we are talking about here.

Keep in mind that ancient peoples took math a lot more literally. Countable items existed because we needed to buy and sell, if you have zero transactions you don’t write anything down. In geometry you can accept you have zero subdivisions in a ratio (something like zero halfs) but you started out with *something* to create a half, you just don’t have any of them.

Due to the paradox of something that exists that doesn’t exist, people got religious about it, even calling those who considered zero to be ‘real’ atheists, or non believers. It was similar to the uproar against atomism, how can the void exist? Turns out the atomists were more right than wrong, but it kicked up a zeal in people. As Arabian mathematicians invented and worked with algebra, it became obvious that zero caused things to make sense. They started accepting the idea that math could include concepts that aren’t directly related to anything in our physical world. That evolution was important to ideas like ‘the imaginary plane’ (Euler) and powers higher than 3 (Descartes). Remember a power is a dimension, so 1 is a line, 2 is a ‘square’, and 3 is a ‘cube’. That last one gives us our three dimensional world. What the heck is the power of 4, well, it is a tesseract, and it does not exist in our world in anyway we can perceive.

It’s because of how we used to use numbers.

We used to use them to count things and we only counted things that there were more than one of. It makes sense to say, “How many bananas do I have.” It makes very little sense to say, “How many bananas do I not have?”

So any time ancient people used numbers there was always at least one of that thing.

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