I’m enjoying it, but it touches on abstract algebra as an alternative approach rather than leaning on it for everything; I’d kind of prefer the latter.
I was wondering if there was a function f(x, y, z) so that x and z represent the left and right sides of common mathematic operators and y represents the level of operation. So f(1, 2, 4) would be 1 + 4 and f(2, 2, 4) would be 2 * 4. Better versions of f(x, y, z) would have fewer end cases hardcoded into it.
The reason behind this is to handle operator levels greater than addition, multiplication, and exponents. The casual analysis from my grade school and undergrad level math shows the pattern that multiplication is repeated addition and exponents are repeated multiplication.
My quick attempts at coming up with such a function are spiraling into greater and greater complexities. I figured someone else has to have thought about this. Do you know of a place I can start reading up on ideas similar to this? Is what I am doing even plausible?
Quick thoughts based on me playing around:
Addition may be level 0, not level 1
The sequences never really look exactly like multiplication tables, but the patterns are similar enough to appease me
Ideally, everything can be reduced to the simple concept of X + 1 so as to walk along the number line
In practical terms, I have no idea how to express “negative” levels. Division and roots are unapproachable at this point in my playing around.
Cool, thanks. It seems like one of my first tries was producing numbers similar to the Ackermann function. Knuth’s arrow notation essentially takes over after multiplication. But those two articles will give me enough to read to keep moving on. :)
Do you know of any that go the other way into smaller and smaller numbers?
EDIT: I found the right subject name through links on your links. It is called hyperoperation.
Sure, that works, but it isn’t exactly what I am looking for. Is it possible to express the division operator in a manner similar to how multiplication can be expressed using addition? My instinct is telling me probably not.
You can have inverse operations for the higher operations as well. 4^4 is 256, so you can think of 4 as the “tetrated root” of 256. Also see this
(I’m using ‘tetrating’ as a term for the operation after exponentiation: in other words, 4 tetrated to the 4th is 4^(4^(4^4))).
Two problems: there may not be a clear way to define tetrating and higher operations to fractional amounts, and exponentiation and up aren’t associative, so you need a convention for what to do with the parentheses.
Hyper operators. You can represent even bigger numbers with Conway chained arrow notation. Eliezer’s 3^^^^3 is a form of hyper operator notation, where ^ is exponentiation, ^^ is tetration, ^^^ is pentation, etc.
If you’ve ever looked into really big numbers, you’ll find info about Ackermann’s function, which is trivially convertable to hyper notation. There’s also Busy Beaver numbers, which grow faster than any computable function.
I’m enjoying it, but it touches on abstract algebra as an alternative approach rather than leaning on it for everything; I’d kind of prefer the latter.
You may be a good person to ask this question:
I was wondering if there was a function f(x, y, z) so that x and z represent the left and right sides of common mathematic operators and y represents the level of operation. So f(1, 2, 4) would be 1 + 4 and f(2, 2, 4) would be 2 * 4. Better versions of f(x, y, z) would have fewer end cases hardcoded into it.
The reason behind this is to handle operator levels greater than addition, multiplication, and exponents. The casual analysis from my grade school and undergrad level math shows the pattern that multiplication is repeated addition and exponents are repeated multiplication.
My quick attempts at coming up with such a function are spiraling into greater and greater complexities. I figured someone else has to have thought about this. Do you know of a place I can start reading up on ideas similar to this? Is what I am doing even plausible?
Quick thoughts based on me playing around:
Addition may be level 0, not level 1
The sequences never really look exactly like multiplication tables, but the patterns are similar enough to appease me
Ideally, everything can be reduced to the simple concept of X + 1 so as to walk along the number line
In practical terms, I have no idea how to express “negative” levels. Division and roots are unapproachable at this point in my playing around.
Ackermann function
Knuth’s arrow notation
Cool, thanks. It seems like one of my first tries was producing numbers similar to the Ackermann function. Knuth’s arrow notation essentially takes over after multiplication. But those two articles will give me enough to read to keep moving on. :)
Do you know of any that go the other way into smaller and smaller numbers?
EDIT: I found the right subject name through links on your links. It is called hyperoperation.
1 / Ackermann function.
Sure, that works, but it isn’t exactly what I am looking for. Is it possible to express the division operator in a manner similar to how multiplication can be expressed using addition? My instinct is telling me probably not.
You can have inverse operations for the higher operations as well. 4^4 is 256, so you can think of 4 as the “tetrated root” of 256. Also see this
(I’m using ‘tetrating’ as a term for the operation after exponentiation: in other words, 4 tetrated to the 4th is 4^(4^(4^4))).
Two problems: there may not be a clear way to define tetrating and higher operations to fractional amounts, and exponentiation and up aren’t associative, so you need a convention for what to do with the parentheses.
Hyper operators. You can represent even bigger numbers with Conway chained arrow notation. Eliezer’s 3^^^^3 is a form of hyper operator notation, where ^ is exponentiation, ^^ is tetration, ^^^ is pentation, etc.
If you’ve ever looked into really big numbers, you’ll find info about Ackermann’s function, which is trivially convertable to hyper notation. There’s also Busy Beaver numbers, which grow faster than any computable function.
Yes, this is exactly what I was looking for. Thank you.