The "Buenos Aires constant" is a humorous misinterpretation of mathematical notation. It stems from a misunderstanding of how definite integrals are represented. Someone saw the integral of a function with respect to x, evaluated from a to b, written as ∫ₐᵇ f(x) dx and mistakenly believed the b in the upper limit of integration was a constant multiplied by the entire integral, similar to how a coefficient might multiply a variable. They specifically misinterpreted ∫₀¹ x² dx as b times some constant and, upon calculating the integral's value of 1/3, assumed b = 1 and therefore the "Buenos Aires constant" was 3. This anecdotal observation highlights how notational conventions can be confusing if not properly understood.
This blog post, titled "The Buenos Aires Constant," delves into a fascinating mathematical curiosity centered around the city of Buenos Aires, Argentina, and its geographical coordinates. The author, John D. Cook, meticulously outlines the process of calculating a seemingly arbitrary mathematical constant derived from these coordinates. He begins by establishing the latitude and longitude of Buenos Aires, specifically noting the variations in precision available from different sources and opting for a relatively high degree of accuracy.
Mr. Cook then elaborates on the chosen method for transforming these geographical data points into a singular numerical value. This involves a sequence of mathematical operations, commencing with the conversion of the degrees, minutes, and seconds representation of both latitude and longitude into decimal degrees. Following this conversion, the absolute values of both the latitude and longitude are taken, thus discarding the directional indicators (North, South, East, West). The core of the constant's derivation then lies in the concatenation of these two absolute decimal degree values, creating a single extended numerical string. This resulting number, representing the combined geographical "signature" of Buenos Aires, is then designated as the "Buenos Aires Constant."
Furthermore, the author reflects on the inherent arbitrariness of this constant, acknowledging the dependence of its value on the specific choices made during the calculation process. He highlights the impact of the selected precision for the latitude and longitude, emphasizing that even minor variations in these initial values would propagate through the calculations and ultimately result in a different final constant. The choice to utilize the absolute values, thereby disregarding directional information, is also presented as a deliberate decision impacting the final result. Essentially, the blog post uses the Buenos Aires Constant as a thought-provoking illustration of how seemingly objective mathematical procedures can still be influenced by subjective choices, leading to varying outcomes. This underscores the importance of understanding the underlying methodology and assumptions when interpreting any derived constant or numerical representation.
Summary of Comments ( 1 )
https://news.ycombinator.com/item?id=43095943
Hacker News commenters discuss the arbitrary nature of the "Buenos Aires constant," pointing out that fitting any small dataset to a specific function will inevitably yield some "interesting" constant. Several users highlight that this is a classic example of overfitting and that similar "constants" can be contrived with other mathematical functions and small datasets. One commenter provides Python code demonstrating how easily such relationships can be manufactured. Another emphasizes the importance of considering the degrees of freedom when fitting a model, echoing the sentiment that finding a "constant" like this is statistically meaningless. The general consensus is that while amusing, the Buenos Aires constant holds no mathematical significance.
The Hacker News post titled "The Buenos Aires Constant" has generated several comments discussing the mathematical coincidence explored in John D. Cook's blog post. Many commenters engage with the idea of "coincidences" and the human tendency to find patterns.
One compelling thread discusses the nature of such mathematical curiosities. A commenter points out that if you allow enough flexibility in defining your constants and operations, you can essentially create any relationship you want. They suggest the "Buenos Aires constant" is less a fundamental mathematical truth and more a playful exploration of numerical relationships. This idea is echoed by other commenters who mention the infinite monkey theorem – the idea that given infinite time, a monkey randomly hitting keys on a typewriter will eventually produce any given text. Similarly, with enough manipulation of numbers and operations, seemingly significant relationships can be fabricated.
Another commenter explores the concept of "almost integers," numbers that are remarkably close to integers but not quite. They point to examples like the Ramanujan constant (exp(pi*sqrt(163))) and discuss the underlying mathematical reasons for their existence. This ties into the broader discussion of the surprising appearances of certain constants like pi and e in seemingly unrelated areas of mathematics.
Further discussion delates into the base-dependent nature of the observation. The "Buenos Aires constant" relies on the decimal representation of pi. The apparent "coincidence" would disappear if a different base were used. This reinforces the idea that the observation is more of a numerical curiosity tied to our specific representation of numbers, rather than a fundamental mathematical property.
Some commenters appreciate the lighthearted nature of the blog post and the subsequent discussion, acknowledging the fun in finding such numerical oddities. Others express skepticism, labeling it a mathematical triviality. One commenter humorously suggests that Buenos Aires should adopt this constant as its official motto, highlighting the playful tone of the conversation.
Overall, the comments on Hacker News reflect a mix of perspectives, from those fascinated by the numerical coincidence to those who view it as a relatively insignificant mathematical observation. The discussion delves into broader mathematical concepts like almost integers, the base-dependence of numerical representations, and the inherent human tendency to find patterns, even where they may not truly exist.