No, I’m saying that when you focus down to the level of each digit that gets derived by the algorithm, it’s incredible easy to understand how that digit is derived. Given that a rational number is just a finite series of such digits (or a repeating pattern of such digits), understanding the derivation of each digit amounts to an understanding of the derivation of the entire series.
What is 25 divided by 4?
Well, I know that you can multiple 4 six times to get 24, and the remainder is 1. So I understand how I derive the first digit: 6.
Then when we divide the remainder 1 by 4, I know that we’ll get 4 quarters. So I understand how we get the next two digits: 6.25.
^ There’s nothing complicated here.
Division never gives us irrational numbers. Irrational numbers are, by definition, not representable as a ratio. So I’m not sure what process you’re talking about that sometimes gives us rational numbers and sometimes gives us irrational numbers. But I jumped into this discussion in response to your comment: “nobody on earth actually knows why some rational numbers numbers do or don’t terminate…” ← If you’re talking about only rational numbers, then I assume you’re talking about the difference between, for example, 25/4=6.25 vs. 10/3=3.333… both of which we can understand why they do or don’t terminate. But why some quantities can only be represented as irrational numbers in our base 10 system, I agree that we (or I) don’t understand.
Here’s a theory though: viewtopic.php?t=162173
What is the argument again?