II.1 Are Lengths Numbers?, Decimal Place Value, People Want a Number, Equal Status to All Numbers, Real, False and Imaginary, Number Systems Old and New, II.2 Geometry, Naive Geometry and The Greek Formulation, Pages 78 – 85

The Greeks figured out that lengths may not be commensurable.  They discovered incommensurable magnitudes.  The common example was the side and diagonal of a square.  They demonstrated that there is no common unit that could be used to represent the side and the diagonal as a product of this unit and a whole number.  Of course, they discovered that √2 and 1 were incommensurable or that √2 is irrational.

They did not like to work with lengths of this sort and used ratios instead.  They did not consider lengths to be numbers.  That may not seem reasonable to us today but it made sense to them back then (fourth century B.C.).  Eventually this system of ratios gave way to a generalized notion of numbers.

In the 5th century A.D. in India the system of 9 symbols with base 10 was created and used.  A place market to indicate an unused decimal space was also used.  This place marker or “nothing” eventually morphed into the use of a zero.

By the 9th century the decimal system had made its way to Baghdad of the Islamic world.  It was used in a popular book about Indian numeration by Al-Khwarizmi.  A few centuries later the decimal system became very popular in Europe.

Al-Khwarizmi in another book on algebra states “When I considered what people generally want in calculating, I found that it is always a number.”

In 1585 a Flemish mathematician and engineer, Stevin, popularized the decimal fractions in a booklet that translates to “The Tenth”.  He used his numbers for all positive numbers (and lengths), often realizing that they were only terminating decimal approximations to more precise numbers.  After the invention of the logarithm, sine and cosine tables appeared in decimal form and these cemented the use of the decimal representation of numbers.

In 1572 book “Algebra” a mathematician and engineer, Bombelli, used radicals to solve cubic equations.  About 50 year s later Albert Girard and Descartes said equations may have three types of roots: true (positive), false (negative) and imaginary (complex).  The understanding culminated by Gauss establishing the Fundamental Theorem of Algebra that every polynomial equation had a complete set of roots in the complex number system.

It became clear that one muse be concerned with what number system one was using.  a + bi with a and b both integer was explored by Gauss and Kummer as similar to the integers.  Galois defined and used rational numbers.  Johann Lambert in the 18th century established that e and pi are irrational and conjectured that they are not roots of a polynomial equation and hence transcendental numbers.  Subsequently Cantor showed that the vast majority of real numbers are transcendental. 

Hamilton developed the 4 dimensional quaternions in 1843 which are not commutative.

Galois introduced finite fields and function theorists worked with fields of functions.

Early in the 20-th century Kurt Hensel introduced the p-adic numbers, these were built from rational numbers with a special role given to the prime number p. 

Eventually these theories were brought together by Noether and the subject was known as “abstract algebra”.  Cayley created octonions.

Modern geometry was influenced by Hilbert and Einstein in the early 1900’s.  For several thousand years the ideas of Euclid had defined geometry layed out in his book “Elements”.

 Naive geometry is that of length, width and depth of our everyday experience.  This experience can be axiomatized into a mathematical subject.  The ideas of length, straightness and angle had to be understood before developing the mathematical theory. 

Egypt and Babylonia had ideas about geometry as evidenced by the large cities they built.  Euclid of Alexandria, around 300 B.C., wrote the definitive text on algebra entitled “Elements”. 

The elements consists of a sequence of books.  There are 13 books in all.  They cover, the study of plane figures: triangles, quadrilaterals and circles, the Pythagorean theorem, theory of ration and proportion, theory of similar figures, whole numbers, elementary number theory, lengths of the form √(a ±√b), three dimensional geometry.  Included here is the proof that there are an infinite number of prime numbers the construction of the five regular solids and proof that there are no more.

The logic used for the proofs in the Elements is exquisite.  It does not use circularity, has clear and acceptable inference rules and adequate definitions.

There is discussion of parallel lines and how they fit into the axiomatic system.

The mathematical space defined by the Elements is in fact infinite and so one may question whether or not it was intended as a simple idealization of the physical world.

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