i think this a really nice way of thinking of things, especially for regular everyday life.
as a mathematician though, i wanted to mention how utterly and terribly cursed square roots are. (mainly just to share some of the horrors that lurk beneath the surface.) they’ve been a problem for quite some time. even in ancient greece, people were running into trouble with √2. it was only fairly recently (around the 17th century) that they started looking at complex numbers in order to get a handle on √-1. square roots led to the invention of two different “extensions” of the standard number systems: the real numbers (e.g. for √2), and later, the complex numbers (e.g. for √-1).
at the heart of it, the problem is that there’s a fairly straightforward way to define exponentiation by whole numbers: 3n just means multiply 3 by itself a bunch of times. but square roots want us to exponentiate things by a fraction, and its not really clear what 31/2 is supposed to mean. it ends up being that 31/2 is just defined as 31/2 = x, where x is "“the number that satisfies x2 = 3"”. and so we’re in this weird situation where exponentiating by a fraction is somehow defined differently than exponentiating by a whole number.
but this is similar to how multiplication is defined: when you multiply something by a whole number, you just add a number to itself a bunch of times; but if you want to multiply by a fraction, then you have to get a bit creative. and in a very real sense, multiplication “is the exponentiation of addition”.
i think this a really nice way of thinking of things, especially for regular everyday life.
as a mathematician though, i wanted to mention how utterly and terribly cursed square roots are. (mainly just to share some of the horrors that lurk beneath the surface.) they’ve been a problem for quite some time. even in ancient greece, people were running into trouble with √2. it was only fairly recently (around the 17th century) that they started looking at complex numbers in order to get a handle on √-1. square roots led to the invention of two different “extensions” of the standard number systems: the real numbers (e.g. for √2), and later, the complex numbers (e.g. for √-1).
at the heart of it, the problem is that there’s a fairly straightforward way to define exponentiation by whole numbers: 3n just means multiply 3 by itself a bunch of times. but square roots want us to exponentiate things by a fraction, and its not really clear what 31/2 is supposed to mean. it ends up being that 31/2 is just defined as 31/2 = x, where x is "“the number that satisfies x2 = 3"”. and so we’re in this weird situation where exponentiating by a fraction is somehow defined differently than exponentiating by a whole number.
but this is similar to how multiplication is defined: when you multiply something by a whole number, you just add a number to itself a bunch of times; but if you want to multiply by a fraction, then you have to get a bit creative. and in a very real sense, multiplication “is the exponentiation of addition”.