2. Basic Mathematical Concepts and Methods

Probably the most dramatic event in the life of the French mathematician Jean Le Rond d’Alembert (1717–1783) was that as a newborn he was left on the steps of a church. He was found and taken to a home for homeless children. Later, his father found him and provided for his son’s living and education. D’Alembert made contributions to mathematics, mechanics, and mathematical physics. The eighteenth century was the age of European enlightenment and nothing represented the spirit of that age better than the Encyclopédistes, a group of intellectuals gathered around Diderot including Voltaire, Condorcet, and d’Alembert. They published the 28-volume Encyclopedia that contained articles on all areas of human knowledge including political economy.

John Napier (1550–1617), a Scottish nobleman, conceived the idea of the logarithm. The first tables using base 10 were calculated by Henry Briggs (1561–1631), a professor of geometry at Gresham College.

Alternatively, the fact that \(p_{n + 1}\)is not divisible by any prime contradicts the fundamental theorem of arithmetic that states that any integer k > 1 has a unique factorization of the form

Therefore, \(p_{n + 1}\)must be a prime.

For a better idea of the problem and its solution you may want to check Appel and Haken’s article in Scientific American (October 1977) or their book Every Planar Map is Four Colorable (1989). A more technical understanding of the subject could be gained from textbooks on graph theory or discrete mathematics.

Consider the equation \(x^n + y^n = z^n \). If n = 2, we can find integers satisfying the equation \(3^2 + 4^2 = 5^2 \). But could the same be done for \(n \ge 3?\) French mathematician Pierre de Fermat (1601–1665) claimed that he could prove that no such solutions could be found. But because he was writing on the margin of a book, he said he could not write it out. In all likelihood he did not have such a proof. Over the years, many contributed to the solution of the problem. In 1993, the British mathematician Andrew John Wiles (1953) (now at Princeton University in the United States) announced that he had proved the theorem. But there was a significant gap in the proof that took Wiles and a co-worker one and a half years to fill. There are two books written for the public on this subject: Fermat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem (1996) by Amir Aczel, and Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem (1997) by Simon Singh. Both are available in paperback.

Other trigonometric functions exist, but we will not discuss them here because economists rarely if ever come across them and, therefore, we have no reason to clutter the subject with many unfamiliar notations.

The theorem is named after Thales de Miletos (624 B.C.–547 B.C.) although the germ of the idea dates back to 1650 B.C. and the building of the Pyramids.

This is the famous Pythagoras theorem that the square of hypotenuse is equal to the sum of the squares of the other two sides of a right-angle triangle. Egyptians who built the Pyramids clearly had an empirical understanding of this theorem. Pythagoras (569 B.C.–475 B.C.), for whom the theorem is named, is one of the great mathematicians of antiquity and pioneers of mathematics.

A function \(y = f\left( x \right)\) is called periodic if \(f(x) = f(x + c),\;\;c\not = 0\).

It is customary to introduce complex numbers in the context of the solution to quadratic equations involving the square root of a negative number. This practice has the unfortunate consequence that students may get the impression that somewhere among the real numbers or along the real line there are caves where complex numbers are hiding and once in a while show their faces.

Angles are measured in radians. If you use a calculator, you need to set it in the radian mode to get the same numbers as in the text. If your calculator is in the degree mode, then in order to get the same numbers as in the text, \(\theta = \tan ^{ - 1} (x/y)\) needs to be converted into radians by multiplying it by \(\pi /180\).

We shall provide a proof of these relationships in Chap.​ 10.

Abraham De Moivre (1667–1754), a French mathematician who spent most of his life in England, was a pioneer in the development of probability theory and analytic geometry. He was appointed to the commission set up to examine Newton’s and Leibnitz’s claims for the discovery of calculus.