The Fourth Operation: What Comes After Exponentiation
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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.
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.
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"
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]
Related: https://en.wikipedia.org/wiki/Half-exponential_function
This actually shows up in complexity theory someplace.
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.
[0] https://waitbutwhy.com/2014/11/1000000-grahams-number.html
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.
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.
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.
> 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.
https://www.maa.org/external_archive/devlin/devlin_06_08.htm... https://www.maa.org/external_archive/devlin/devlin_0708_08.h...
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.
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 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.
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.*
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.
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)
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.
> 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.
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.
Once you approximate them with rationals you can also imagine adding a fraction of the second multiplicand.
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.
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)