In my last post I highlighted the crushing superiority of the long-scale over the short-scale name-system for numbers. I’ve always had a preference for the long-scale, and maybe part of this is because it is closer to how I originally assumed large-number names worked. As a little nipper I made “false” assumptions, which would have associated even the most accessible large number names (e.g. Trillion, Quadrillion, Quintillion) with significantly larger numbers than their official values.
The Longerscale
Here is how I assumed large numbers worked:
- A Million is a Thousand Thousands (10^6)
- A Billion is a Million Millions (10^12)
- A Trillion is a Billion Billions (10^24)
- A Quadrillion is a Trillion Trillions (10^48)
- A Quintillion is a Quadrillion Quadrillions (10^96) And so on.
I was undeniably correct in my definition of a Million, I was correct in my definition of a Billion assuming the use of the long-scale, but with my definitions of Trillions we’re going into unexplored territory.
Notice how, by these definitions, a Quintillion is already of a comparable size to a Googol? (10^100).
The general N-illion number here is 10^(3*2^N). So a Centillion would be approximately 10^(3.8*10^30), unimaginably greater than the shortscale 10^303 and the longscale 10^600.
Extended Th H T U
Under the Longerscale, my names for the different digits would be (in ascending order):
Units, Tens, Hundreds, Thousands, Tens of Thousands, Hundreds of Thousands, Millions,
Millions, Tens of Millions, Hundreds of Millions, Thousands of Millions, Tens of Thousands of Millions, Hundreds of Thousands of Millions, and then…
Billions, Tens of Billions, Hundreds of Billions, Thousands of Billions, Tens of Thousands of Billions, Hundreds of Thousands of Billions, Millions of Billions, Tens of Millions of Billions, Hundreds of Millions of Billions, Thousands of Millions of Billions, Tens of Thousands of Millions of Billions, Hundreds of Thousands of Millions of Billions, and then…
Trillions, Tens of Trillions, Hundreds of Trillions, Thousands of Trillions, Tens of Thousands of Trillions, Hundreds of Thousands of Trillions, Millions of Trillions, Tens of Millions of Trillions, Hundreds of Millions of Trillions, Thousands of Millions of Trillions, Tens of Thousands of Millions of Trillions, Hundreds of Thousands of Millions of Trillions, Billions of Trillions, Tens of Billions of Trillions, Hundreds of Billions of Trillions, Thousands of Billions of Trillions, Tens of Thousands of Billions of Trillions, Hundreds of Thousands of Billions of Trillions, Millions of Billions of Trillions, Tens of Millions of Billions of Trillions, Hundreds of Millions of Billions of Trillions, Thousands of Millions of Billions of Trillions, Tens of Thousands of Milions of Billions of Trillions, Hundreds of Thousands of Millions of Billions of Trillions, and so on.
Suddenly the names of numbers feel less like convenient labels and more like combinatorics problems or towers of Hanoi or counting-in-binary. While this system does squeeze the most out of each new name, it is completely understandable that this way would not be adopted for general use.
I’m not sure how old I was when I first thought “a trillion is a billion billions”, but I remember being baffled by the random order of letters on a QWERTY keyboard, and finding the fact that the letters were all capitals rather tricky and confusing around that time. So presumably I was really young. This is probably a fairly common misconception among young people born into the aristocratic longscale castes.
The Evenlongerscale
A hundred is ten tens. A million is a thousand thousands. Why shouldn’t a thousand be a hundred hundreds, as opposed to the normie ten hundreds? That would make this whole system so much more consistent, and make our numbers so much bigger! Let’s call this the Evenlongerscale.
- A Hundred is Ten Tens (10^2)
- A Thousand is a Hundred Hundreds (10^4)
- A Million is a Thousand Thousands (10^8)
- A Billion is a Million Millions (10^16)
- A Trillion is a Billion Billions (10^32)
- A Quadrillion is a Trillion Trillions (10^64)
- A Quintillion is a Quadrillion Quadrillions (10^128) And so on.
Now our Quintillion exceeds a Googol! Also, look at these neat powers of two! Our N-illion is now 10^(2^(N+2)), pushing our Centillion to approximately 10^(5.1*10^30)
It’d take a bit of getting used to, but here’s what the digit names would look like up to the millions: Units, Tens, Hundreds, Tens of Hundreds, Thousands, Tens of Thousands, Hundreds of Thousands, Tens of Hundreds of Thousands, Millions.
To keep things neat, it may be worth renaming the Millions here to Trillions, and the Billions to Quadrillions, and the old Trillions to Quintillions, and so on. This would give us a formula 10^(2^N) for the N-illion, preserving the words “Million” and “Billion” as similies for “Hundred” and “Thousand” Respectively. N-illion is not to be confused with the Nillion (0-illion) which would be a similie for Ten.