Leon P Smith on Nostr: npub1zdp33…2vqv8 Yeah, you have the positive integers n down the right spine of the ...
npub1zdp33shl69xr0uq3x8n5gsjykq9upycwh6nqm02c3f6x0frrn0dq42vqv8 (npub1zdp…vqv8) Yeah, you have the positive integers n down the right spine of the Stern-Brocot Tree, and all the unit fractions 1/n down the left spine. My idea is to use the unit fractions in an analogy to units of measurement to teach how to add fractions the right way, by adding like units.
This contrasts nicely with the mediant operation at the heart of the Stern-Brocot Tree, also known as "adding fractions the wrong way", which I hope will help students remember which operation is which.
Moreover, the mediant is not a well-defined function of fractions, which sets up a very nice future lesson, as fully owning and utilizing well-definedness is the key to mastering modular arithmetic.
In addition, the Stern-Brocot tree is intimately connected to the extended Euclidean Algorithm, and as such it encompasses like, 85% of the algorithmic content of elementary number theory, and all of the trickiest bits. Want to compute a modular multiplicative inverse, or the trickier direction of the chinese remainder theorem? The Stern-Brocot tree can do it!
Kevin Bacon and the Stern-Brocot Tree mentions a lot of connections such as these, though I honestly I've come across several more connections that need to be elaborated upon, including Christoffel Words, Knot Theory, Rational Tangles, the 3-strand braid group, the moduli space of isosceles triangles, elliptic curve cryptography, and modular forms.
It's the gift that keeps on giving, and while there are a number of concepts in math that are this way, I don't think anything is as well-connected to cutting-edge research mathematics, *and* also approachable by children.
In terms of an undergraduate math curriculum, the Stern-Brocot Tree is highly relevant to Discrete Math, Abstract Algebra, and Number Theory, and seems well worth mentioning in Linear Algebra, Real Analysis, Complex Analysis, and Numerical Analysis.
This contrasts nicely with the mediant operation at the heart of the Stern-Brocot Tree, also known as "adding fractions the wrong way", which I hope will help students remember which operation is which.
Moreover, the mediant is not a well-defined function of fractions, which sets up a very nice future lesson, as fully owning and utilizing well-definedness is the key to mastering modular arithmetic.
In addition, the Stern-Brocot tree is intimately connected to the extended Euclidean Algorithm, and as such it encompasses like, 85% of the algorithmic content of elementary number theory, and all of the trickiest bits. Want to compute a modular multiplicative inverse, or the trickier direction of the chinese remainder theorem? The Stern-Brocot tree can do it!
Kevin Bacon and the Stern-Brocot Tree mentions a lot of connections such as these, though I honestly I've come across several more connections that need to be elaborated upon, including Christoffel Words, Knot Theory, Rational Tangles, the 3-strand braid group, the moduli space of isosceles triangles, elliptic curve cryptography, and modular forms.
It's the gift that keeps on giving, and while there are a number of concepts in math that are this way, I don't think anything is as well-connected to cutting-edge research mathematics, *and* also approachable by children.
In terms of an undergraduate math curriculum, the Stern-Brocot Tree is highly relevant to Discrete Math, Abstract Algebra, and Number Theory, and seems well worth mentioning in Linear Algebra, Real Analysis, Complex Analysis, and Numerical Analysis.