31–40 of 54 posts
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.
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...
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…
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.
Earlier quoted context omitted.
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…
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.
Once you approximate them with rationals you can also imagine adding a fraction of the second multiplicand.
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.
If 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 s…
Earlier quoted context omitted.
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…
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.
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)
Earlier quoted context omitted.
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 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.
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.
If 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 s…
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.
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.