
Topology of Numbers - espeed
http://pi.math.cornell.edu/~hatcher/TN/TNpage.html
======
heinrichf
Hatcher is the author of famous freely available and beautifully typeset
topology books
[http://pi.math.cornell.edu/~hatcher/#anchor1772800](http://pi.math.cornell.edu/~hatcher/#anchor1772800)
(most notably "Algebraic topology").

"By special arrangement with the publisher, an online version will continue to
be available for free download here, subject to the terms in the copyright
notice."

Some details about how he typesets his books here:
[http://pi.math.cornell.edu/~hatcher/AT/typography.html](http://pi.math.cornell.edu/~hatcher/AT/typography.html)

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a-nikolaev
Indeed, his work is spectacular. I also think that his geometric approach is
quite unique and instructive (although I did not get into his books that far).

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strmpnk
It’s a shame they only mention Farey when introducing the binary tree
construction. It’s called a Stern-Brocot tree (after two independent
discoveries). Farey sequences end up being a specific enumeration over this
tree.

It’s interesting to see the range of applications this structure has. While it
helped me understand a few ideas like how a Cauchy sequence might work,
practical applications included things like finding approximate ratios for
floating points with various limits on the scale of the denominator and single
update list sorting systems that don’t rely on midpoints (they run out of
precision and require relabeling rather early). The history of the concept is
also worth looking at.

A good overview is available in Graham, Knuth, and Patashnik‘s book “Concrete
Mathematics.”

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dbranes
What a confusing name in an era where homotopy-theoretic methods in arithmetic
geometry are flourishing. Looks Hatcher covers non of that and use "topology"
to mean "geometrically motivated".

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gnulinux
I agree, I can't stop scratching my head. Maybe "Geometry of Numbers" is a
better name.

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yesenadam
_Geography_ maybe?

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bbeonx
Oh cool, this is from Allen Hatcher. I've worked through some of his algebraic
topology and it was very well written. I'm excited to look through this

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jhinra
I'm impressed how many people are here to say, "Oh, Allen Hatcher!" I did the
same. It certainly says a lot about Hatcher as a textbook writer.

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james_s_tayler
"elementary number theory from a geometric point of view"

It's stunning how intuitive geometric explanations can feel. I like the
geometric approach.

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ilyagr
The Farey sequence that is central to this book also appears in the functional
pearl on "Enumerating the Rationals" that recently appeared on HN.

[https://news.ycombinator.com/item?id=18515413](https://news.ycombinator.com/item?id=18515413)

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markhollis
I can imagine this text will be popular among Project Euler problem solvers,
given the shared themes of continued fraction, Farey Sequences etc.

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Cobord
There is also the Zariski topology of Spec Z that is the topology of numbers
in the sense of algebraic geometry. Or of Z[x]

