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The Fourth Operation: What Comes After Exponentiation
- throwaway81523 4y agoThis 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 https://en.wikipedia.org/wiki/Half-exponential_function This actually shows up in complexity theory someplace.
- angarg12 4y agoIncremental games taught me.
- paulmooreparks 4y agoThere 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.html https://www.maa.org/external_archive/devlin/devlin_06_08.htm... https://www.maa.org/external_archive/devlin/devlin_0708_08.html https://www.maa.org/external_archive/devlin/devlin_0708_08.h...
- deleted 4y ago[deleted]
- hervature 4y agoCould 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.
- isaac21259 4y agoWhat 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.
- HidyBush 4y agoIrrational 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.
- isaac21259 4y agoThat'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.
- HidyBush 4y agoMultiplication 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.
- isaac21259 4y agoTo clarify where I live limits are introduced in high school, irrational numbers just much earlier.
- naniwaduni 4y agoYou don't need a rigorous notion of limits to informally notice that irrationals have arbitrarily close rational approximations, e.g. by adding successive digits.
- alisonkisk 4y ago
- orangecat 4y agoI 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.
- hervature 4y agoThat'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.
- orangecat 4y agoSorry, I was trying to agree with you and apparently phrased it poorly :)
- HidyBush 4y agoI 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)
- vikingerik 4y agoThose 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.
- magnio 4y agoThe 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.
- Retric 4y agoThe 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.
- lixtra 4y agoHow 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.
- Retric 4y agoThat 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.
- anon291 4y ago
- unnouinceput 4y agoThis article is not that good. I prefer the wiki pages about it (https://en.wikipedia.org/wiki/Tetration https://en.wikipedia.org/wiki/Tetration) or for a more general approach the best is wiki about hyperoperation (https://en.wikipedia.org/wiki/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.
- aaaaaaaaaaab 4y agoWell, if you’re allowed to use extra symbols like ^[], then the answer can be 999!!!!…!!!!. With as many factorials as you like.
- unnouinceput 4y agoWell, 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.
- ObiWanFrijoles 4y agoGraham'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 https://en.wikipedia.org/wiki/Graham%27s_number [1] https://www.youtube.com/watch?v=GuigptwlVHo https://www.youtube.com/watch?v=GuigptwlVHo
- nonrandomstring 4y agoWhat 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 https://en.wikipedia.org/wiki/Operator_theory
- paurea 4y agoI 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.html?m=1 https://paureahack.blogspot.com/2016/06/supersum-subproduct....
- SideburnsOfDoom 4y agoIf 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.
- afiori 4y agoincrement is no longer a binary operation
- zelphirkalt 4y agoNotationwise 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.
- agumonkey 4y agoCan 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.
- JoeyBananas 4y agowait until you learn about Ackermann
- afiori 4y agoThe article is about extending the domain of the ackermann function
- russianGuy83829 4y agoI wonder why exponentials are so common in nature/physics but tetration is not
- alisonkisk 4y agoThe numbers get impossibly big almost immediately.
- isaac21259 4y agoExponentials 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.
- ufo 4y agoIn 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.
- 0x0203 4y agoOne 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 https://waitbutwhy.com/2014/11/1000000-grahams-number.html
- adrian_b 4y agoIf it is said that "Those are the first, second, and third operations: addition, multiplication, and exponentiation", then one should not forget that according to this numbering there is a "zeroth" operation: adding 1 to a number (incrementation). 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.
- anon291 4y agoActually all operations can be written as loops of their arguments, since the increment operation takes no arguments
- RHSeeger 4y ago> 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.
- TrainedMonkey 4y agoI 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.
- 4y ago
- teddyh 4y agoNever mind the fourth operation; what about all the positive reals? http://www.absurdnotions.org/page128.html http://www.absurdnotions.org/page128.html
- Syzygies 4y agoSeveral 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 https://en.wikipedia.org/wiki/Tropical_geometry
- lisper 4y agoFrom 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.
- freemint 4y agoAre you aware whether this function is analytic?