The Fourth Operation: What Comes After Exponentiation • Hacker News Board

Re: The Fourth Operation: What Comes After Exponentiation

#11

Post by unnouinceput » Sat, Jul 30, 2022, 7:03 AM UTC

This article is not that good. I prefer the wiki pages about it (https://en.wikipedia.org/wiki/Tetration) or for a more general approach the best is wiki about hyperoperation (https://en.wikipedia.org/wiki/Hyperoperation). As for notation the square bracket notation is the simplest.

Story time: A while ago some other parent, trying to be smartass, asked the kids in one of those outside school activity (this was before COVID) "what is the highest number they can write using only 3 digits". Of course the kids, who barely understood multiplication and just learned in math the power of (a^b) operation, said "999". He said is "9^9^9 and started to explain to them how large that number is. After he was done, I said "you know, they are right, the highest number using only 3 digits is 999, but you used special notation. Now, if the rules say that we are allowed to use special notation then 9^9^9 is not the highest number, but 9[9]9 is. And then I had to explain to him what is that for the next 30 minutes. I lost him somewhere around pentation because he insisted how big that number is and I started to calculate it using previous base (power of -> tetration -> pentation -> etc). In the end I had to tell him, that using bracket notation his number is just 9[3]3, which is lower than 9[9]9.

Re: The Fourth Operation: What Comes After Exponentiation

#12

Post by aaaaaaaaaaab » Sat, Jul 30, 2022, 7:09 AM UTC

unnouinceput wrote: post #11

This article is not that good. I prefer the wiki pages about it ( https://en.wikipedia.org/wiki/Tetration ) or for a more general approach the best is wiki about hyperoperation ( https://en.wikipedia.org/wiki/Hyperoperation ). As for notation the square bracket notation is the simplest. Story time: A while ago some other parent, trying to be smartass, asked the kids in one of those outside school activity (this was b…

Well, if you’re allowed to use extra symbols like ^[], then the answer can be 999!!!!…!!!!. With as many factorials as you like.

Re: The Fourth Operation: What Comes After Exponentiation

#13

Post by isaac21259 » Sat, Jul 30, 2022, 7:10 AM UTC

HidyBush wrote: post #10

isaac21259 wrote: post #8

Earlier quoted context omitted.

What about irrational numbers? There's no neat way to view multiplication of two irrational numbers as repeated addition. And even if there were a way I don't think it's a useful way to think or teach after the first couple years because it makes obvious things like √2×√2 = 2 seem weird and mysterious.

Irrational numbers are limits of sequences of rational numbers. Multiplying two real numbers is simply taking the limit of a sequence of multiplications between rational numbers that converge to the two real ones.

That's a pretty far departure from the original "multiplication is just repeated addition". Regardless, I don't think any student would find it helpful to hear "Multiplying two real numbers is simply taking the limit of a sequence of multiplications between rational numbers that converge to the two real ones". In my country irrational numbers are introduced two or three years before limits so you couldn't teach it in schools effectively either.

Re: The Fourth Operation: What Comes After Exponentiation

#14

Post by HidyBush » Sat, Jul 30, 2022, 7:21 AM UTC

isaac21259 wrote: post #13

HidyBush wrote: post #10

Earlier quoted context omitted.

Irrational numbers are limits of sequences of rational numbers. Multiplying two real numbers is simply taking the limit of a sequence of multiplications between rational numbers that converge to the two real ones.

That's a pretty far departure from the original "multiplication is just repeated addition". Regardless, I don't think any student would find it helpful to hear "Multiplying two real numbers is simply taking the limit of a sequence of multiplications between rational numbers that converge to the two real ones". In my country irrational numbers are introduced two or three years before limits so you couldn't teach it in…

Multiplication outside of positive integers is not "repeated addition".

It took us thousands of years to properly define real numbers. High school students can live without a perfect explanation, or we can just teach limits before college since they are the fundamental concept if calculus.

Re: The Fourth Operation: What Comes After Exponentiation

#15

Post by ObiWanFrijoles » Sat, Jul 30, 2022, 7:56 AM UTC

unnouinceput wrote: post #11

This article is not that good. I prefer the wiki pages about it ( https://en.wikipedia.org/wiki/Tetration ) or for a more general approach the best is wiki about hyperoperation ( https://en.wikipedia.org/wiki/Hyperoperation ). As for notation the square bracket notation is the simplest. Story time: A while ago some other parent, trying to be smartass, asked the kids in one of those outside school activity (this was b…

Graham's number [0] can be defined using hyperoperations. There's an awesome video with Ron Graham himself on Numberphile (YouTube) [1]

[0] https://en.wikipedia.org/wiki/Graham%27s_number

[1] https://www.youtube.com/watch?v=GuigptwlVHo

Re: The Fourth Operation: What Comes After Exponentiation

#16

Post by isaac21259 » Sat, Jul 30, 2022, 7:57 AM UTC

HidyBush wrote: post #14

isaac21259 wrote: post #13

Earlier quoted context omitted.

That's a pretty far departure from the original "multiplication is just repeated addition". Regardless, I don't think any student would find it helpful to hear "Multiplying two real numbers is simply taking the limit of a sequence of multiplications between rational numbers that converge to the two real ones". In my country irrational numbers are introduced two or three years before limits so you couldn't teach it in…

Multiplication outside of positive integers is not "repeated addition". It took us thousands of years to properly define real numbers. High school students can live without a perfect explanation, or we can just teach limits before college since they are the fundamental concept if calculus.

To clarify where I live limits are introduced in high school, irrational numbers just much earlier.

Re: The Fourth Operation: What Comes After Exponentiation

#17

Post by vikingerik » Sat, Jul 30, 2022, 8:14 AM UTC

HidyBush wrote: post #7

paulmooreparks wrote: post #4

There is a bit of a debate about whether or not multiplication should be defined as repeated addition: https://www.maa.org/external_archive/devlin/devlin_06_08.htm... https://www.maa.org/external_archive/devlin/devlin_0708_08.h...

I mean, this is true for every operation once you extend it to a new domain. What is exponentiation? is 5^6 multiplying 5 for 6 times? sure, but how about 5^(-6)? what's up with that? and 5^(1/2)? and don't get me started on 5^(2/3)

Those follow. 5^(1/2) is multiplying by 5 one-halfth of a time. It is half the operation of multiplying by 5. Applying that operation twice results in 5.

5^(2/3) is two-thirds of the operation of multiplying by 5. Applying that operation three times results in multiplying by 5 for six-thirds times, or twice, and the result is 25.

5^-6 is multiplying by 5 negative-six times. What is multiplying a negative number of times? Dividing. You divide by 5 six times.

Re: The Fourth Operation: What Comes After Exponentiation

#18

Post by nonrandomstring » Sat, Jul 30, 2022, 8:21 AM UTC

What an amazing ride. I fell off when it got bumpy talking about generalising operators. I once read a chapter on "operator theory" [1] in a bid to understand transforms like FFT in a fresh way, but alas I don't have the skill to enjoy or use it well.

[1] https://en.wikipedia.org/wiki/Operator_theory

Re: The Fourth Operation: What Comes After Exponentiation

#19

Post by naniwaduni » Sat, Jul 30, 2022, 8:46 AM UTC

isaac21259 wrote: post #16

HidyBush wrote: post #14

Earlier quoted context omitted.

Multiplication outside of positive integers is not "repeated addition". It took us thousands of years to properly define real numbers. High school students can live without a perfect explanation, or we can just teach limits before college since they are the fundamental concept if calculus.

To clarify where I live limits are introduced in high school, irrational numbers just much earlier.

You don't need a rigorous notion of limits to informally notice that irrationals have arbitrarily close rational approximations, e.g. by adding successive digits.

Re: The Fourth Operation: What Comes After Exponentiation

#20

Post by paurea » Sat, Jul 30, 2022, 9:24 AM UTC

I attacked the same problem somewhat differently and found some interesting groups. You can switch the exponential map you use, use instead a Mittag-Leffler function and find a whole family of groups going from sum to multiplication. Being groups you can define generalized Fourier/Mellin transforms with them. I wrote a blog post explaining the approach (there is a link to the paper at the end for more details). https://paureahack.blogspot.com/2016/06/supersum-subproduct....