The short answer is that it depends on what you mean by “big”. A better answer leads us to my area of study, Measure Theory. This is the mathematics of all kinds of measures—length, area, volume, weight, even probabilities—and it underpins much of mathematics. While the modern foundations go back to the late 19th century, throughout antiquity the various approaches to dealing with size and quantity have had an incalculable effect on world cultures speaking to both the foundational quality of the question and its importance to society.
In economics, having a standard to measure goods against (say gold) allows for currency. Motivated by the fairness in measuring and portioning out bread to workers, the Egyptians developed a unique way of dividing numbers that presaged mathematical limits. Architects and Engineers need to measure stresses on our buildings, bridges, and vehicles. Physicists quantify our world using measures. From our smallest scales of inquiry to our largest works as a species, measures are essential.
So, let’s take a circle like the one to the left with a radius of 1. What are some answers to our question? What are some ways of talking about how big this circle is?
Geometry tells us that a circle’s circumference, or the length of its edge, is 2π×(radius) = 2π ≈ 6.283. Similarly, Calculus or Geometry says that the area enclosed by a circle is π×(radius)2 = π ≈ 3.1415. In fact, over 2000 years ago Archimedes found these formulas and used the aptly named Method of Exhaustion to derive an estimate for π as being between 310/71 and 31/7 or about 3.1408 and 3.1429.
Yet another way of measuring our circle is given by thinking of it as the edge of the bullseye on a dart board. Striking in the bullseye is sufficiently difficult, but what about hitting the exact edge
of it? No matter how hard you try, the dart will hit inside or outside the bulleye; never on the exact edge because the edge (being a circle) has no thickness. In terms of probability, this means that the chance of hitting our circle is 0. And so in this probability measure, our circle is measure 0.
So, how big is our circle? In length 2π, in area π, in probability 0, and the list goes on. The point is, measures are a way for us to capture some of the information about an object using numbers. They provide a path from Geometry, aka circles, shapes, curves, spaces, etc. to numbers like 2π, π, and 0. For me that’s the draw of measures; they let me think about numbers using pictures…and for me, that feels more natural.
By Tyler Bryson, Private Tutor