The area (S) of a regular polygon of side l circumscribed to a circle of radius r is S = nlr / 2, where n is the number of sides, since we can divide the polygon into n triangles of base l and height r (the radius of the circle is the apothem of the circumscribed polygon and therefore the height of the triangles); Now, nl is the perimeter (p) of the polygon, so the formula can be written like this: S = pr / 2.
If we are increasing the number of sides of the circumscribed polygon, its area will get closer and closer to that of the circle and its perimeter to the length of the circumference; in the limit p = 2πr, then S = 2πr.r / 2 = πrtwo. In this way we have found the area of the circle in the manner of Archimedes, just as we thought about last week.
The numbers 187 and 2019 are cousins to each other; but my wise readers have not found any other notable characteristic, so I will pose another small challenge related to them.
Sums of powers
On December 23, the great American mathematician of Belgian origin, Elias Stein, one of the greatest promoters, along with his teacher Antoni Zygmund, died of harmonic analysis, a powerful and versatile tool applicable to different fields of mathematics and physics (such as says Antonio Córdoba in an interesting article published in these same pages); for example, the theory of numbers, and problems such as Waring.
At the end of the eighteenth century, the English mathematician Edward Waring conjectured that, given a whole and positive exponent n, every natural number can be expressed as the sum of a limited number of nth powers, and that the maximum number of necessary addends is bounded and bound to the exponent in question. It is easier to see with a simple example: in the case of exponents 2 and 3, all natural numbers can be expressed as the sum of, at most, 4 squares and 9 cubes. Thus, 63 = 7two + 3two + 2two + 1two (Decomposition is not necessarily unique, can you find another, maybe one with less addends?). Lagrange showed that Waring was right in the case n = 2, and in 1909 Hilbert found a general demonstration, with which the conjecture ceased to be so to become certainty.
I invite my wise readers to express the numbers 187 and 2019 as the sum of 4 or less squares (and of 9 or less cubes that do not comply with an approved). And speaking of curious relationships between numbers, here is a sequence inspired by a very recent event: 16 1 2 3 5 1 … What is the next number?
Carlo Frabetti He is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 scientific dissemination works for adults, children and young people, among them Damn physics,Damn mathematics or The big game. He was a screenwriter The Cristal ball.