The operations over time

Today, schoolchildren still learn to perform calculations "by hand" by setting out their arithmetic operations on paper. These well-known algorithms are not the only ones that exist. Several techniques have evolved over time; the ones we are taught result from choices that may seem arbitrary. In Antiquity, it was abacuses that were used to represent numbers and add them. In the 17th century, John Napier developed his famous rods for performing multiplications. All these tools had to solve the same major problem: handling carries!

What better than a polynomial to approximate, near a point, a complicated function? Better yet: if the latter is "sufficiently regular", the coefficients of its Taylor expansion are its successive derivatives; this is the famous Taylor formula that teaches us this. Yet we get surprises: the adequacy between Taylor expansion at any order and Taylor development is sometimes called into question. Marvels of mathematics, and in particular of analysis, nobody can do without Taylor expansions. They are indeed indispensable in limits of functions, in the study of particular points of a curve, but also in probability calculations.