The Building Blocks of Mathematics and their Beauty

Zeno's paradoxes caused the creation of thought behind the infinite summation. The Greeks were, undeniably, stumped! The concept of the limit solved this problem by showing that if the length of the summation grows, the precision of the number it defines will increase, up to the point that if we have a series (infinite summation), the number we are trying to be define will be obtained.

Without this limit concept, we can't possibly make the link between the decimal system and fractions. A non-terminating decimal describes some number with a sum of infinitely smaller numbers. We can't define 0.33333.... as 1/3 without it.

I have created a new discussion based on these questions. My intent was not to overshadow this thread, but to increase the longevity of this discussion. I think this more concise form will allow more users to respond more easily than in this thread, and that, of course, the discussion will last longer.

I'd like to point out that the concept of 0 innately existed long before it was mathematically developed. I think it is universal across all organisms of the concept of having nothing, due to the sense of homeostasis and knowing what perfect balance is within an organism's self and its environment. Additionally, many remote civilizations, before they were discovered, had no concept of numbers either. I'd like to point you to the Piraha tribe in the Amazon (I only know of them due to a recent Smithsonian documentary I saw), which have no number system, and see 1 as very little, 3 as few, 5 as a bit more, and 10 as a lot.

I know very well that infinity is not a complex number; it is, for the most part, an undefined concept. But if we assume that we use our operations like we do on complex numbers, it is easy to show that $n^2>2n$, so we could very well start an entire branch of mathematics based on infinity if we begin to treat it like a complex number (which is currently abhorred elsewhere in all of mathematics).

Could someone please explain to me why this is a bad comment?

I think people disagree with the fact that defining multiplication using addition makes the definition only valid for integer values. As I stated above, we defined multiplication for non integral values by considering two integers $k$ and $m$, with $k$ not divisible by $m$ and considering the value that, when multiplied by $k$ produces $m$. And the transcendental numbers $e$ and $\pi$ can be defined with finite multiplication and addition, as someone else pointed out in "Thinking like a Theorist" by defining $e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}...$ $\pi=\sqrt{6(\frac{1}{1}+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}...)}$

I think people also disagree with your last statement because by the way it is built (a set of simple axioms and then a buildup upon them) there are incredibly few "holes" as you say. The few holes that have ever been found were things like Zeno's paradoxes and Russell's paradox.

There are many properties, for example, we define the absolute value of $\frac{1}{0}$ as infinity. However, infinity is a concept our minds are seemingly only able to grasp when things become infinitely smaller, not larger, because dealing with infinitely large quantities is beyond the scope of our imagination. I think the next huge advancement in pure mathematics will be a rigorously interconnected branch of studies behind infinity and its properties--currently our closest reach towards this is the limit in calculus.