The Fourth Operation: What Comes After Exponentiation
1–10 of 54 posts
Re: The Fourth Operation: What Comes After Exponentiation
#2This 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.
Re: The Fourth Operation: What Comes After Exponentiation
#3Incremental games taught me.
Re: The Fourth Operation: What Comes After Exponentiation
#4There 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...
Re: The Fourth Operation: What Comes After Exponentiation
#5There 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...
[deleted]
Re: The Fourth Operation: What Comes After Exponentiation
#6There 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...
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.
Re: The Fourth Operation: What Comes After Exponentiation
#7There 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)
Re: The Fourth Operation: What Comes After Exponentiation
#8There 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...
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.
Re: The Fourth Operation: What Comes After Exponentiation
#9There 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, 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.
Re: The Fourth Operation: What Comes After Exponentiation
#10Earlier quoted context omitted.
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.
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.