The Fourth Operation: What Comes After Exponentiation(ariwatch.com)
ariwatch.com
The Fourth Operation: What Comes After Exponentiation
http://ariwatch.com/VS/Algorithms/TheFourthOperation.htm
53 comments
> So the sequence of operations is: adding 1 to a number, addition, multiplication, exponentiation,
As I was reading this, I was visualizing it as geometry. - Incrementation as a line, where each "step" moves you along that line - Addition as a 2d graph, where each point along the x axis increments by 1, and each point along the y axis indicates "how many times" - Multiplication as a 3d graph, in the same pattern - Exponentiation - it fell apart because I couldn't visualize it anymore.
Not particularly insightful, I guess, but I found it interesting that it seemed "automatic" to me to view it this way.
As I was reading this, I was visualizing it as geometry. - Incrementation as a line, where each "step" moves you along that line - Addition as a 2d graph, where each point along the x axis increments by 1, and each point along the y axis indicates "how many times" - Multiplication as a 3d graph, in the same pattern - Exponentiation - it fell apart because I couldn't visualize it anymore.
Not particularly insightful, I guess, but I found it interesting that it seemed "automatic" to me to view it this way.
I think this is very insightful. Math is designed to tackle problems which are challenging/not possible to visualize. So to get good at symbolic mathematics at some point you need to let go of needing to visualize everything. To get a bit more perspective one could contrast symbolic math with geometry which is built on visualization.
You can visualise incrementation as a single step along a line, addition as a step of any length along the line.
Multiplication: the formula for area of a rectangle is "length times height", so IMHO now this - "a * b" is best visualised as 2d. Or if you like, "m lines of length n", visualised as a stack of lines.
Exponentiation covers "to the power of 2" (area of a 2d object) "to the power of 3" (volume of a 3d cube) and beyond, so now we've run out of graph dimensions.
I think what we can take from this, is that when going from first operation to second, second to third etc, the rate of change is not constant, it is increasing. Since the sequence is "no extra dimensions, one extra dimension, all the extra dimensions"
Multiplication: the formula for area of a rectangle is "length times height", so IMHO now this - "a * b" is best visualised as 2d. Or if you like, "m lines of length n", visualised as a stack of lines.
Exponentiation covers "to the power of 2" (area of a 2d object) "to the power of 3" (volume of a 3d cube) and beyond, so now we've run out of graph dimensions.
I think what we can take from this, is that when going from first operation to second, second to third etc, the rate of change is not constant, it is increasing. Since the sequence is "no extra dimensions, one extra dimension, all the extra dimensions"
Actually all operations can be written as loops of their arguments, since the increment operation takes no arguments
Often in math the property (a + b) * c = ac + bc is more useful than the repeated addition[0]
So if you try to find the zeroth operation in this fashion you get (a ? b) * c = ac ? bc which is satisfied[1] by a ? b = max(a,b) or a ? b = min(a,b).
[0] (a + 1) * c = a*c + c
[1] I do not remember whether these are the only two solutions but with min you get tropical geometry[2]
[2] https://en.wikipedia.org/wiki/Tropical_geometry
So if you try to find the zeroth operation in this fashion you get (a ? b) * c = ac ? bc which is satisfied[1] by a ? b = max(a,b) or a ? b = min(a,b).
[0] (a + 1) * c = a*c + c
[1] I do not remember whether these are the only two solutions but with min you get tropical geometry[2]
[2] https://en.wikipedia.org/wiki/Tropical_geometry
The article discusses this. It calls it "succession", and raises arguments against it being the zeroth operation.
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....
This is more interesting than the title makes it sound. It is not about what is usually called tetration, but is mostly about fractional iterates and ways to compute them.
Related: https://en.wikipedia.org/wiki/Half-exponential_function
This actually shows up in complexity theory someplace.
Related: https://en.wikipedia.org/wiki/Half-exponential_function
This actually shows up in complexity theory someplace.
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.
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.
Well, if you’re allowed to use extra symbols like ^[], then the answer can be 999!!!!…!!!!. With as many factorials as you like.
Well, if you want that way, then bracket notation can be used like this too, which in the end will hold a higher number than factorial one.
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
[0] https://en.wikipedia.org/wiki/Graham%27s_number
[1] https://www.youtube.com/watch?v=GuigptwlVHo
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
[1] https://en.wikipedia.org/wiki/Operator_theory
One of my favorite explanations of some of these concepts is Wait-but-why's explanation of Grahm's Number [0]. It breaks this concept down in a way that's very accessible to people like me who have a pretty limited grasp of many of these mathematical concepts.
[0] https://waitbutwhy.com/2014/11/1000000-grahams-number.html
[0] https://waitbutwhy.com/2014/11/1000000-grahams-number.html
Can anybody confirm that Ackermann's function was related to the idea of hyperoperation ? Article say it encodes addition, multiplication, .. as a numeric parameter but hyperoperations are never mentionned.
Notationwise this is not so great. Yet another special syntax, the subscript, is used for expressing an operation. And subscript is already used for multiple purposes, even with my limited math knowledge: base of logarithm and labels for selecting a part of something bigger, like a matrix.
Notationwise it would be good to do as some computer languages do and just use a name and wrap everything: (+ (* (^ (_ something D) C) B) A) = imagine this one graphically: ((((something_D)^C) * B) + A). No ambiguity, no question in what order to apply operations.
Notationwise it would be good to do as some computer languages do and just use a name and wrap everything: (+ (* (^ (_ something D) C) B) A) = imagine this one graphically: ((((something_D)^C) * B) + A). No ambiguity, no question in what order to apply operations.
Several comments independently propose incrementation as coming before addition, multiplication.
One can only make sense of progressions like this by finding relationships between the levels. There are at least two ways to see a connection between addition and multiplication: The logarithm, and polynomials. Polynomials are well-studied expressions combining addition and multiplication.
Algebraic geometry studies sets defined by polynomial equations, as linear algebra studies sets defined by linear equations. Tropical geometry [1] is a version of algebraic geometry that replaces addition by minimization, and multiplication by addition.
Pure mathematicians like to hear music in their choices; they want these choices to arise naturally and support deep theories. Tropical geometry passes this test. One wants an inevitability to one's choices, a belief that alien life would reach the same conclusions.
If there are two answers to "what's zeroth? What's fourth?" that itself is interesting. However, one wants extensive evidence to believe that there's branching. Otherwise, we decide that we've simply stepped onto the wrong trail and need to backtrack. Pure mathematicians live in profound fear of just playing house, making stuff up because it sounds good, even if it looks to everyone else like that's what we do.
[1] https://en.wikipedia.org/wiki/Tropical_geometry
One can only make sense of progressions like this by finding relationships between the levels. There are at least two ways to see a connection between addition and multiplication: The logarithm, and polynomials. Polynomials are well-studied expressions combining addition and multiplication.
Algebraic geometry studies sets defined by polynomial equations, as linear algebra studies sets defined by linear equations. Tropical geometry [1] is a version of algebraic geometry that replaces addition by minimization, and multiplication by addition.
Pure mathematicians like to hear music in their choices; they want these choices to arise naturally and support deep theories. Tropical geometry passes this test. One wants an inevitability to one's choices, a belief that alien life would reach the same conclusions.
If there are two answers to "what's zeroth? What's fourth?" that itself is interesting. However, one wants extensive evidence to believe that there's branching. Otherwise, we decide that we've simply stepped onto the wrong trail and need to backtrack. Pure mathematicians live in profound fear of just playing house, making stuff up because it sounds good, even if it looks to everyone else like that's what we do.
[1] https://en.wikipedia.org/wiki/Tropical_geometry
If addition is "the first operation", then "increment" is the zeroth operation. You know, counting 1,2,3,4.
If we define multiplication as repeated addition, then we define addition as repeated increment, where A + B is: start at zero, increment A times then B times.
If we define multiplication as repeated addition, then we define addition as repeated increment, where A + B is: start at zero, increment A times then B times.
increment is no longer a binary operation
I wonder why exponentials are so common in nature/physics but tetration is not
Exponentials come up quite naturally from differential equations because it's often suprisingly useful to talk about something's rate of change in terms of itself. As far as I know there's no similar connection with tetration.
In other words: if it grows or decays by a rate that's proportional to the current amount, over time it'll follow an exponential curve.
The numbers get impossibly big almost immediately.
From the afterword:
> Multiplication can be considered iterated addition only when one of the numbers – the number measuring the degree of iteration – is pure. Adding five apples to itself three times makes 15 apples. But three apples or three oranges as a count for adding makes no sense.
Hm, I wonder what happens if you consider "times" or "iterations" a unit? I haven't thought this through but it feels like the edge of a deep insight. Playing fast and loose with units and iterations feels like Lisp. Being strict about it feels like Haskell.
> Multiplication can be considered iterated addition only when one of the numbers – the number measuring the degree of iteration – is pure. Adding five apples to itself three times makes 15 apples. But three apples or three oranges as a count for adding makes no sense.
Hm, I wonder what happens if you consider "times" or "iterations" a unit? I haven't thought this through but it feels like the edge of a deep insight. Playing fast and loose with units and iterations feels like Lisp. Being strict about it feels like Haskell.
Never mind the fourth operation; what about all the positive reals?
http://www.absurdnotions.org/page128.html
http://www.absurdnotions.org/page128.html
wait until you learn about Ackermann
The article is about extending the domain of the ackermann function
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...
https://www.maa.org/external_archive/devlin/devlin_06_08.htm... https://www.maa.org/external_archive/devlin/devlin_0708_08.h...
The article you linked argues for a change in the way multiplication is explained to children, not the way it is defined.
> Telling students falsehoods on the assumption that they can be corrected later is rarely a good idea. And telling them that multiplication is repeated addition definitely requires undoing later.
I disagree. Understanding multiplication as repeated addition has always been an invaluable intuition, especially in the beginning, where explicit calculations are important. The biggest hurdle when introducing multiplication is getting them to understand the multiplication table. The fact that it is defined as a separate operation in the definition of ring/field is almost irrelevant in the pedagogical context, just as we don't start teaching real numbers with Dedekind cuts.
> Telling students falsehoods on the assumption that they can be corrected later is rarely a good idea. And telling them that multiplication is repeated addition definitely requires undoing later.
I disagree. Understanding multiplication as repeated addition has always been an invaluable intuition, especially in the beginning, where explicit calculations are important. The biggest hurdle when introducing multiplication is getting them to understand the multiplication table. The fact that it is defined as a separate operation in the definition of ring/field is almost irrelevant in the pedagogical context, just as we don't start teaching real numbers with Dedekind cuts.
The issue with that repeated addition intuition is it keeps breaking down.
How do you define pi * e, with addition? How do you define 2m * 2m, with addition?
Those feel completely intuitive now, but students can get the first far more easily than the second.
How do you define pi * e, with addition? How do you define 2m * 2m, with addition?
Those feel completely intuitive now, but students can get the first far more easily than the second.
How do you define pi or e but as a limit? At least for practical purposes.
Once you approximate them with rationals you can also imagine adding a fraction of the second multiplicand.
Once you approximate them with rationals you can also imagine adding a fraction of the second multiplicand.
That feels really intuitive now, but less so for a student working their way up to it.
Even more basic is the 2.7 * 3.1 = 27 * 31 and what do I do with the decimal place question. Kids first intuition is often 83.7 because it was one from the right in the numbers they started with.
In that context pi * e exposes several different challenges to peoples mental models of multiplication. Granted most people are just going to plug it into a calculator and trust the answer without much thought, but such is life.
Even more basic is the 2.7 * 3.1 = 27 * 31 and what do I do with the decimal place question. Kids first intuition is often 83.7 because it was one from the right in the numbers they started with.
In that context pi * e exposes several different challenges to peoples mental models of multiplication. Granted most people are just going to plug it into a calculator and trust the answer without much thought, but such is life.
How do you explain pi * e?
It's clear the multiplication as repeated addition holds for the natural numbers, which form a closed ring anyway. Pi and E are sufficiently advanced that by the time you get there you must understand that the operation being described by multiplication is not the same operation at all (depending on how one constructs the real numbers, multiplication of reals is multiplication of sets or functions)
It's clear the multiplication as repeated addition holds for the natural numbers, which form a closed ring anyway. Pi and E are sufficiently advanced that by the time you get there you must understand that the operation being described by multiplication is not the same operation at all (depending on how one constructs the real numbers, multiplication of reals is multiplication of sets or functions)
Could not disagree more with this take. Multiplication of fractions is simply the division of two whole number multiplications. Which, if you are teaching fractions, division has already been taught. Seems like a contrarian take for the sake of being contrarian rather than based on pedagogy. Glad he wasn’t my teacher as he would have confused me.
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.
Others have handled the appropriate construction of irrational numbers. The claim here is not that every student needs to know irrational numbers are a limiting sequence but this is exactly how mathematicians think of it. So it is strange to hear another mathematician claim that repeated addition is somehow lying. At some point, people kind of just accept that it is just something you punch into your calculator and don't even think about it anymore.
Your example is quite bad because sqrt(2)*sqrt(2) = sqrt(2*2) = sqrt(4) = 2. So repeated addition works fine. Let's focus instead on pi*pi. The way calculators do this is precisely as some type of limiting sequence depending on how much precision you want. Because, one cannot "calculate" pi*pi exactly because it is irrational. So, you have 3*3, then 3.1*3.1, then 3.14*3.14, etc. which are all repeated additions with some division (e.g. 314*314/(100*100)). In reality, when multiplying two irrational numbers, we just use enough decimal points for floating point precision and then chop off any potentially erroneous digits after the multiplication.*
Your example is quite bad because sqrt(2)*sqrt(2) = sqrt(2*2) = sqrt(4) = 2. So repeated addition works fine. Let's focus instead on pi*pi. The way calculators do this is precisely as some type of limiting sequence depending on how much precision you want. Because, one cannot "calculate" pi*pi exactly because it is irrational. So, you have 3*3, then 3.1*3.1, then 3.14*3.14, etc. which are all repeated additions with some division (e.g. 314*314/(100*100)). In reality, when multiplying two irrational numbers, we just use enough decimal points for floating point precision and then chop off any potentially erroneous digits after the multiplication.*
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.
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.
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.
Multiplication is repeated additions is the informal way of stating the distributive property of multiplication and addition.
Probably you were taught how to multiply irrational by the property of powers (a^b * c^b = (a*c)^b etc.).
You were not taught a grand unifying theory of multiplication, you were taught how to manipulate operations to turn them into more useful operations.
Teaching these laws also prepares you for when a and b are just symbolic reals with no structure and those laws are the only thing you can use to manipulate them.
Probably you were taught how to multiply irrational by the property of powers (a^b * c^b = (a*c)^b etc.).
You were not taught a grand unifying theory of multiplication, you were taught how to manipulate operations to turn them into more useful operations.
Teaching these laws also prepares you for when a and b are just symbolic reals with no structure and those laws are the only thing you can use to manipulate them.
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.
It's a kind of generalization of repeated addition. Once you've been taught about π, you know that 2π lies between 23 and 24, without being told about limits. It may not be rigorous, but it's a good start.
You can't teach "the truth" (whatever you hold that to be). It would set back education instead of advancing it. In this case too, perfect is the enemy of good.
You can't teach "the truth" (whatever you hold that to be). It would set back education instead of advancing it. In this case too, perfect is the enemy of good.
I agree with your disagreement. By this standard we shouldn't teach F=ma in introductory physics, and we should require kindergartners to understand the ZFC axioms before we can tell them what "3" is.
That's actually the exact opposite of what I'm saying and the exact approach the author is saying. It is unclear what they are proposing, but it smells awfully similar to jumping to modern understanding mathematics in one shot to avoid "repeatedly lying" to students.
As others have pointed out, repeated addition as multiplication readily extends to rational numbers, then to irrational numbers as limits of rational sequences. This is exactly the progression that is taught in Rudin's analysis book and the way to construct the real numbers. At no point in time do you need to backtrack on repeated addition but you need to introduce new concepts division and limits. This is exactly teaching F=ma and then introducing relativity and quantum as the students gain more depth and break past the classical setting.
As others have pointed out, repeated addition as multiplication readily extends to rational numbers, then to irrational numbers as limits of rational sequences. This is exactly the progression that is taught in Rudin's analysis book and the way to construct the real numbers. At no point in time do you need to backtrack on repeated addition but you need to introduce new concepts division and limits. This is exactly teaching F=ma and then introducing relativity and quantum as the students gain more depth and break past the classical setting.
Sorry, I was trying to agree with you and apparently phrased it poorly :)
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)
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.
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.
[deleted]
Are you aware whether this function is analytic?
Incremental games taught me.
Addition is derived from adding 1 to a number in the same way as multiplication is derived from adding a number to zero (if instead of starting with zero the operation is started from an arbitrary number, like in the derivation of addition, then the multiply-add operation is obtained, which is implemented frequently as a single operation in hardware), or exponentiation from the multiplication of 1 with a number.
So the sequence of operations is: adding 1 to a number, addition, multiplication, exponentiation, ..., where any operation but the first in this sequence can be implemented as a loop using the previous operation.