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In surreal numbers [1], the midpoint between 0 and infinity would be the simplest number greater than 0, which is { 0 | } = 1 [1] https://en.wikipedia.org/wiki/Surreal_number
In surreal numbers [1], the midpoint between 0 and infinity would be the simplest number greater than 0, which is { 0 | } = 1 [1] https://en.wikipedia.org/wiki/Surreal_number
TIL: apart from infinity (number, larger than any real number in absolute value) there are infinitesimal (number, less than any real number in absolute value and not a zero). Edit: also, those infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632.
tiny = infinitesimal
huge = infinity
N = number to be bucketed
tiny bucket 1
10 bucket 2
...
X last bucket
I just use bigints wherever it's possible to transform the algorithm to work with bigints, and I "render to decimal" in views
In surreal numbers [1], the midpoint between 0 and infinity would be the simplest number greater than 0, which is { 0 | } = 1 [1] https://en.wikipedia.org/wiki/Surreal_number
Some other senses in which 1 can be thought of as the "midpoint" between 0 and ∞ come from the many common ways to invertibly map an interval onto (0, ∞), in which the midpoint of that interval maps onto 1. For example: • tan maps (0, π/2) onto (0, ∞), and tan(π/4) = 1. • f(x) = x/(1 - x) maps (0, 1) onto (0, ∞), and f(1/2) = 1. • exp is an isomorphism from the reals under addition to the positive reals under multipl…
If this was true:
tan maps (0, π/2) onto (0, ∞), and tan(π/4) = 1
Wouldnt it imply that
tan(π/8) would be halfway between 0 and 1 ie .5?
By my calculations it is 0.414 or √2 - 1
ALso with:
• f(x) = x/(1 - x) maps (0, 1) onto (0, ∞), and f(1/2) = 1.
wouldnt this mean that f(0.25) is supposed to be half way between 0 and 1 or .5. However f(0.25) = 0.25/0.75 = 1/3
Clearly we'll need to some work in TruffleRuby to be fully compatible. I really hope nobody's code depends on this. Infinity 8.988465674311579e+307 4.4942328371557893e+307 2.2471164185778946e+307 1.1235582092889473e+307 5.6177910464447366e+306 ... 32767.999999999996 16383.999999999998 8191.999999999999 4095.9999999999995 2047.9999999999998 1023.9999999999999 511.99999999999994 255.99999999999997 127.99999999999999 63…
Earlier quoted context omitted.
Some other senses in which 1 can be thought of as the "midpoint" between 0 and ∞ come from the many common ways to invertibly map an interval onto (0, ∞), in which the midpoint of that interval maps onto 1. For example: • tan maps (0, π/2) onto (0, ∞), and tan(π/4) = 1. • f(x) = x/(1 - x) maps (0, 1) onto (0, ∞), and f(1/2) = 1. • exp is an isomorphism from the reals under addition to the positive reals under multipl…
I have a question If this was true: tan maps (0, π/2) onto (0, ∞), and tan(π/4) = 1 Wouldnt it imply that tan(π/8) would be halfway between 0 and 1 ie .5? By my calculations it is 0.414 or √2 - 1 ALso with: • f(x) = x/(1 - x) maps (0, 1) onto (0, ∞), and f(1/2) = 1. wouldnt this mean that f(0.25) is supposed to be half way between 0 and 1 or .5. However f(0.25) = 0.25/0.75 = 1/3
Isn't it -1/12 actually?
I couldn't for the life of me remember the set of conditions and proof that shows this but it was interesting when reading through it a few years ago.
This all makes sense. It seems like there would be some special edge cases around the finite/NaN boundary that would be rarely exercised. I wonder if/how those are handled.
>>> import numpy as np
>>> 1 + np.Inf
inf
>>> (0 + np.Inf)/2
inf
>>>Earlier quoted context omitted.
Some other senses in which 1 can be thought of as the "midpoint" between 0 and ∞ come from the many common ways to invertibly map an interval onto (0, ∞), in which the midpoint of that interval maps onto 1. For example: • tan maps (0, π/2) onto (0, ∞), and tan(π/4) = 1. • f(x) = x/(1 - x) maps (0, 1) onto (0, ∞), and f(1/2) = 1. • exp is an isomorphism from the reals under addition to the positive reals under multipl…
I have a question If this was true: tan maps (0, π/2) onto (0, ∞), and tan(π/4) = 1 Wouldnt it imply that tan(π/8) would be halfway between 0 and 1 ie .5? By my calculations it is 0.414 or √2 - 1 ALso with: • f(x) = x/(1 - x) maps (0, 1) onto (0, ∞), and f(1/2) = 1. wouldnt this mean that f(0.25) is supposed to be half way between 0 and 1 or .5. However f(0.25) = 0.25/0.75 = 1/3
The original issue at hand is to talk sensibly about “half way” between 0 and infinity. But in the standard way we think about distance between numbers, there’s obviously no way to do that — you can’t add and subtract real numbers to infinity!
So, implicitly what’s happening here is that by talking about a map between a finite interval and (0,inf), we are equipping (0,inf) with a new, special definition of distance between numbers. This is called a metric.
Usually, when we talk about the distance between x and y, we mean `d(x,y) = |x-y|` — this is called the Euclidean metric (in 1 dimension).
Here, we’ve introduced a new metric on (0,inf): `d2(x,y) = |tan^-1(x) - tan^-1(y)|/(pi/8)`
The half way point between 0 and 1 under the Euclidean metric is 0.5. The half way point between 0 and 1 under our fancy new metric d2 is ~0.414.
TL;DR: You can generalize the notion of “distance between two numbers”, using distance functions called metrics. Under the typical metric, the half way point between 0 and 1 is 0.5. Under our cool new infinite-tan metric, the half way point between 0 and 1 is ~0.414.