- Concepts Explored
- Nomenclature of Large Numbers
- Mystery or Mistake of Irrational Numbers Giving Rational Results
- Prime Roots of Numbers (which are integers raised the same number of times, or otherwise)
- Roots of Irrational Numbers, Prime Roots of Irrational Numbers, Roots of Prime Roots, Prime Roots of Prime Roots
I was thinking about telling my friend that a game requires 10 billion bytes!! (i.e. 10 GB, and “giga” is the unit prefix denoting a factor of a billion). Just for a dramatic effect. Then I thought, I’ll find the number of binary digits, and tell him that the game requires 80 billion binary digits!! All stored without mistakes on your computer!!. Then I thought, I’d just tell him that the game requires so and so number of decimal digits, and with , that would have around 24 billion digits.
Then, I wanted to explain that in words, and after a bit of thinking, I figured out that the number is , which is a simple form of writing , and you know it lies between and for me to look it up on the googology wiki.
Aside
As I then diverted into thinking about what I described in the rest of this article, I forgot this original goal, and turns out the closest answer to this would be , and is a “Trialogue”, or , which means the result is “a trialogue raised to 2.4”. And is a “Googolthrong”.
But why stop there? Let’s dig deeper. is a “Guppythrong”. We just want to then multiply it by (Because a “hundred thousand” is a “hundred” times a “thousand”). Now is a “Maximusbillion”, or a “Billionplex”, or a “Billiplexion” (the name “Billiplexion” seems to be preferred, and “Maximusbillion” is not preferred). And is a “Disekatommyrillion” or “Trlastillion”.
Now we should know that a “Guppythrong” is just the square of a “Trialogue”, or a “Trialogue Trialogue”. Similarly, a “Disekatommyrillion” is a “Billiplexion Billiplexion Billiplexion”. By that logic, would be a “Billiplexion Disekatommyrillion”
==So the final answer would be a “Billiplexion Disekatommyrillion Googolthrong”.==
But then I wondered, = = = .
So can be or .
Now we know that the result is 1 followed by 24 billion 0s. However, it becomes difficult to believe that , an irrational number (251.19), when raised to a billion times will form a perfect number like 1 followed by 24 billion 0s.
A simpler version would be , which would be or . It’s hard to imagine that when 251.19 when multiplied by itself 10 times would lead to 1 followed by 24 zeroes. Now I didn’t perform this manually, and used an automatic reasoning model to , which is very close to the actual number!
[ Another example: So I wanted to try this out with a simpler number to see the steps myself (although I could’ve maybe used Python to see a long form calculation). In my naivety, thinking that since is twice the power of , I thought that would be (i.e. thinking = , because it is ) and decided to compute it. But that came out to be , because unlike my wrong assumption, was , not . The result was thus , which was . So this only happens for numbers at the scale of ten billion and above, since that’s when you get . ]
Now the math works, but with a loss of precision. This made me wonder, should these kinds of calculations even be valid? [This is in relation to my studies about making mathematics purely discrete, like 1) how I wrote on my local notes about how circles do not exist, and hence neither does pi, and 1.1.) the appearance of a circle only occurs to us because we are not zooming in enough to render the minor lines at the surface of what looks like a circle, all of which should have discrete lengths, and therefore the best approximations of circles, even in the ideal sense are uniform polygons of a high enough and finite number of sides, and 1.2) therefore there is no meaningful relation between the line joining the center of the polygon and a side or corner, and the perimeter of the polygon, except that 1.3) when the number of sides increases, this ratio tends to converge on a certain value, which we call pi, and also 2) how I tried to come up with a number system that removes 0 from the number system, effectively not seeing it as a number, and hence solves the paradox of division by 0, since such a division won't be logical - but that turned out to have already been a thing called the bijective base-10 number system, and had been the number system used in Japan and Korea.] For example, is . The part is easy, but is . That made me wonder how roots work. That is, if there could ever be an n-th root of 10 far down the line, where we would get a rational number. Because the entire idea of computing as involves taking an irrational number like , and then raising the result first to 12, and then the result of that to “10 billion”, that is, effectively raising it to “120 billion”. Or for the simpler case, computing when written as involves taking , raising it to 12 and then by 10, that is, effectively by “120”.
So, you find that if when you multiply this irrational number by itself 120 times, you get a simple number like , which is a “thousand billion billion”, or a “quadrillion”. However, thinking about it, multiplying by itself 5 times gives you 10, because that’s the definition of it being the fifth root, so multiplying it by itself 120 times is simply multiplying 10 by itself 24 times, because 120 divided by 5 is 24. So ultimately, this is no different from how the irrational square root of any rational number squares itself to give back the rational number.
Except, I wanted to look deeper into n-th roots, in which case, the properties of normal natural numbers become relevant too, because although the 4th root is the square root of the square root, the same cannot be said for the third root, or any other prime root. I guess prime roots are what are interesting to us. But again, the fourth root would then be interesting both as an “n-th root of an irrational number”, a “prime root of an irrational number”, a “prime root of a root”, and a “prime root of a prime root”.
My main point of wonder about these values is that, using them in calculations is like going into meaningless territory to make a leap of faith and get to the result safely. But then, we only get the perfect results because we stick to the realm of abstractions, and never expand the numbers really.