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Theory of Functions, Parts I and II

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Two volumes of a classic 5-volume work in 1 handy edition. Part I considers general foundations of the theory of functions; Part II stresses special functions and characteristic, important types of functions, selected from single-valued and multiple-valued classes. Demonstrations are full and proofs given in detail. Section I. Fundamental Concepts Chapter 1. Numbers and Points 1.

Grounding Mathematics Education

Prerequisites 2. The Plane and Sphere of Complex Numbers 3. Point Sets and Sets of Numbers 4. Paths, Regions, Continua Chapter 2. Functions of a Complex Variable 5.

Continuity and Differentiability 7. Integral Theorems Chapter 3. The Integral of a Continuous Function 8.

Definition of the Definite Integral 9. This is a PDF-only article. The first page of the PDF of this article appears above. NOTE: We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. We do not capture any email address.

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Arnulfo Perez rated it it was amazing Sep 25, Jon rated it liked it Feb 11, Michael Lloyd-Billington rated it really liked it Dec 31, Pat Muchmore marked it as to-read Nov 19, Ryan marked it as to-read Mar 25, A base of 12 would seem a more likely candidate if this were the reason, yet no major civilisation seems to have come up with that base. On the other hand many measures do involve 12, for example it occurs frequently in weights, money and length subdivisions. For example in old British measures there were twelve inches in a foot, twelve pennies in a shilling etc.

Neugebauer proposed a theory based on the weights and measures that the Sumerians used. His idea basically is that a decimal counting system was modified to base 60 to allow for dividing weights and measures into thirds. However although Neugebauer may be correct, the counter argument would be that the system of weights and measures was a consequence of the number system rather than visa versa.

Several theories have been based on astronomical events. The suggestion that 60 is the product of the number of months in the year moons per year with the number of planets Mercury, Venus, Mars, Jupiter, Saturn again seems far fetched as a reason for base That the year was thought to have days was suggested as a reason for the number base of 60 by the historian of mathematics Moritz Cantor.

Again the idea is not that convincing since the Sumerians certainly knew that the year was longer than days. Another hypothesis concerns the fact that the sun moves through its diameter times during a day and, with 12 Sumerian hours in a day, one can come up with Some theories are based on geometry.

For example one theory is that an equilateral triangle was considered the fundamental geometrical building block by the Sumerians. Now there are sixty of these basic units in a circle so again we have the proposed reason for choosing 60 as a base. Notice this argument almost contradicts itself since it assumes 10 as the basic unit for division!

I [EFR] feel that all of these reasons are really not worth considering seriously. I just do not believe that anyone ever chose a number base for any civilisation. Can you imagine the Sumerians setting set up a committee to decide on their number base — no things just did not happen in that way. The reason has to involve the way that counting arose in the Sumerian civilisation, just as 10 became a base in other civilisations who began counting on their fingers, and twenty became a base for those who counted on both their fingers and toes.