• 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 multiplication. 0 is the natural "midpoint" of the reals under addition, and exp(0) = 1.
(Closely related to that last one: when dealing only with positive numbers (and their limits 0 and ∞), it's natural to think of their group operation, multiplication, for which 1 is the identity, creating a symmetry between the numbers smaller than 1 and the numbers larger than 1 under reciprocation.)
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
It implies that tan(π/8) would be "halfway" between 0 and 1 in this sense of what "halfway" means. In this sense, 0.5 is not halfway between 0 and 1, 0.414 is.
It's easier to visualize it. You're standing on the roof of a 1-meter-tall building on a flat earth with a perfectly clear atmosphere. If you look straight out (90°) you can see to infinity (tan 90°). If you look down (0°) you can see where you are (tan 0°). If you look halfway between those (45°), you can see 1 (tan 45°) meter straight in front of you. But if you look halfway down again, you won't see exactly 0.5 meters, will you?
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.
tan(π/8) = √2 - 1, not 1/2, but the value is indeed halfway between 0 and 1 if you take the distance function to be
d(a, b) = |b - a| / (√(1 + a²)√(1 + b²)) or
d(a, b) = |b - a| / |1 + ab|
(These are chordal distance or stereographic distance, respectively, when the real number line is used to represent points on a circle under stereographic projection.)
The halfway point is 0.5 when you use the distance function d(a, b) = |b - a|.
Traditionally one distinguishes between rational, algebraic, and transcendental numbers; e is transcendental. This tree is related to continued fraction expansions. Any number is a limit of a path in this tree, and one can talk about the pattern in the path from a CS automata theory point of view. Now e comes out one of the easier limits.
> In surreal numbers [1], the midpoint between 0 and infinity would be the simplest number greater than 0, which is { 0 | } = 1
But { 0 | ω } = 1, right?
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.
https://www.alamy.com/portrait-of-cardinal-flavio-chigii-163... https://www.pinterest.com/pin/462252349233672807/
tiny = infinitesimal
huge = infinity
N = number to be bucketed
tiny <= N < 10 --> bucket 1
10 <= N < 20 --> bucket 2
...
X <= N < huge --> last bucket
As for how common they are, we learn about them in any introductory calculus course when defining derivatives. You come across the idea whenever discussing limits, if somewhat obliquely.
If I learned about it in high school math, and again in "real" math courses at my university, I'd say it's pretty standard.
Also, you seem to be conflating "common" with "standard". "standard" is a mathematical term. Infinitesimal are handwavy in standard analysis (epsilon-delta are the rigorous alternative), but exist rigorously in nonstandard analysis.
There are other instances where I've had a need for such a smallest positive number, where logic is simplified as opposed to checking for 0 in a special way. Whether there's an agreed upon term for that, I know where I've found value in programming tasks.
When I need such a thing, it is almost invariably in comparisons, so I am not doing arithmetic with multiple instances of that smallest representable positive number.
Philosophical problems surrounding the perplexing concept of infinity were already hotly debated by the ancient Greeks. Aristotle made an ontological distinction between actual and potential infinities, and argued that actual infinity cannot exist but potential infinity can. This was also the consensus position of later scholars, and became a sticking point in the acceptance of calculus because infinitesimals (and infinite sums of them) were an example of the ontologically questionable actual infinities.
As I mentioned before, standard modern analysis is based on limits, not infinitesimals, and requires no extension of real numbers. Indeed the limit definition of calculus only requires the concept of potential infinities, so philosophers should be able to rest easy! But infinitesimals still occur in our notation which is largely inherited from Leibniz, however. We say that the derivative of y(x) is dy/dx, or the antiderivative of y(x) is ∫ y(x) dx, and while acknowledging that dy and dx are not actual mathematical objects, just syntax, we still do arithmetic on them whenever it's convenient to do so! For example, when we make a change of variables in an integral, we can substitute x = f(t) for some f, and then say dx/dt = f'(t) and "multiply by dt" to get dx = f'(t) dt to figure out what we should put in the place of the "dx" in the integral.
Actual infinitesimal numbers are not dead, either, they're used in a branch of analysis called nonstandard analysis which formalizes them in the logically rigorous manner that is now expected from mathematics.
________
¹ Not that they had a rigorous theory of real numbers, either, that came in the 19th and early 20th century. In fact what we now understand as formal, axiomatized math didn't really exist before the 19th century at all!
[1] https://en.wikipedia.org/wiki/Surreal_number