Everybody, help me with this petty, borderline frivolous (and frankly, snarky) request!
What’s a mathematical adverb, similar to “exponentially”, to describe a fast-growing function that increases even vastly faster than exponentially. I’m looking for something that I can use in a colloquial sense, specifically:
“Making the arrangements for lunch with friends gets exponentially more difficult with each additional included friend, but <_________> more so if our cousin ‘J’ is included.”
(The intended audience of this proposed remark is a professional (retired) mathematician, so he’ll understand any mathematical jargon.)
Note that I’m specifically looking for an adverb, or at least an adjective that can be adverbed.
The intended usage is to explain that I’d like to do lunch with someone (actually, my brother) and we’ve already agreed to include Cousin ‘J’ as well, but I’m not going to be the one who has to arrange that.
Okay, I suppose – there’s a word I haven’t heard before. Looking it up, it seems to convey about what I had in mind – some kind of hyper-exponentiation. I doubt that many people would know this word, but I’m sure my brother would.
That wiki page mentions that tetration is not an elementary function, in the sense that addition, multiplication, and exponentiation are. Why is that? Is it simply by arbitrary definition?
That wiki page also has some serious formatting problem. It’s full of little embedded images, and mathematical and scientific pages often are, but the images don’t load – leaving blank spots in the page where a formula should be. I haven’t noticed this problem with other Wiki pages.
But still, I have a word I can use. I wonder if “hyper-exponentially” also works?
Doesn’t “geometrically” grow less fast than exponentially? Or is that the same?
If hyper-4-exponentiation (“tetration”) is just the next level of a mathematical for-next loop, why are the earlier ones called “elementary” functions but this one is not?
I could work with an adjective as well as an adverb – either just re-phrase the original remark a bit or adverbify the adjective. It might grow Ackermannishly. :rolleyes: (Knuthfully?)
“Tetration” sounds more like something that happens at an industrial scale inside a chemical synthesis factory, resulting in a chemical that Derek Lowe won’t work with. So I might stick with “hyper-exponentially”.
Among Computer Science types, we just use “super-exponential” for a vague beyond-exponential rate. But generally we are more precise: double-exponential, etc.
Since it’s unlikely you would need to refer to any growth greater than elementary, let alone Ackermann style growth, the question is how hyperbolic do you want your language to be?
You can skip to something like the Busy Beaver function or just “non-computable”. Of course, there are layers of non-computable beyond that, but you’d have to explain the meaning of various Sigma/Pi/Delta notations.
Yeah, in physics, “super-exponential” is the word we would use, too. Of course, there are many different types of super-exponential, but all of them are so rare that it’s not too useful to have words for more specific ones. If you need to be more specific, just give the mathematical function.
I don’t see that in the Wikipedia discussion of hyperbolic functions. For instance sinh = (x^x - e^-x)/2 – which is, in practice, a typical exponential function (the e^-x) dwindles very rapidly.
I may be wrong, but all my reading shows that hyperbolic functions approach exponential functions as asymptotes. They never actually become infinite.
“Asymptotically” could mean the dependent variable goes to infinity as the independent variable approaches a value, e.g. speed vs. relativistic mass. But I think it more often means the dependent variable gets closer to closer to a certain value (but never quite reaches it) as the independent value goes to infinity. E.g. y=1/x approaches y=0 asymptotically.
Functions can have “slant asymptotes” too, in which the function asymptotically approaches a line that is neither vertical nor horizontal. Commonly seen in functions that are the quotient of two polynomials, where the numerator is a higher degree than the denominator.
