2. Ordered FieldsJuly 9, 2019The crisis on irrational numbersTo study Calculus, one has to deal with “realnumbers”. There are interesting stories about real numbers.What are numbers?Mathematics was originated from many ancient civilizations such as Mesopotamian andEgyptian civilization, etc. It reached at its first peak in Greece between 600 to 300 B.C.Greek mathematicians established many remarkable achievements such as Pythagoras the-orem (a2+b2=c2for right triangles) and Euclidean geometry.During the period, integers, and rational numbers (i.e., a number in the form ofpqwherep, qare integers withq6= 0) are used.1Greek mathematicians believed that all the numbersin the world are rational numbers.However, Hippasus of Metapontum2, a disciple of Pythagoras, made an astonishingdiscovery:√2 is not a rational number.(1)In fact, suppose that√2 =pqfor some integerspandq6= 0. Assume thatpandqhaveno common factors (otherwise it can be divided).By taking square, we have 2 =p2/q2, and hence2q2=p2.We conclude thatpis an even number. Then we can writep= 2m. From above, we see 2q2= 4m2, i.e.,q2= 2m2.1The notion of negative number was accepted much later.2Hippasus of Metapontum , 500 B.C. in Magna Graecia, was a Greek philosopher.10

We also conclude thatqis an even number.Bothpandqhave common factor 2, which is a contradiction with the assumption thatpandqhave no common factor. Our claim (1) is proved.3The discovery of more and more irrationals made it necessary for the Greeks to facereality. Before finding a solution, Greeks hesitated to study numbers and equations. It wasa crisis in the history of mathematics.What the Greeks did to avoid the crisis was to use geometry to avoid irrational numbers.Eudoxus(480 BC - 355 BC) introduced a new theory of proportion:one that does notinvolve numbers. Instead he studied geometrical objects such as line segments, angles, etc.,while avoiding giving numerical values to lengths of line segments, sizes of angles, and othermagnitudes.The theory of proportions was so successful that it delayed the development of theoriesfor real numbers for 2000 years.As we said, the Greeks accepted√2 as the diagonal ofthe unit square, but any arithmetic approach to√2, whether by sequences, decimals, orcontinued fractions, is infinite and therefore less intuitive. Until the nineteenth century thisseemed a good reason for considering geometry to be a better foundation for mathematicsthan arithmetic.The modern real number theoryThe modern real number theory was established byW.R. Dedekind(1831-1916) in 1872. His idea is constructive: started from integers, rationalnumbers, and then to define real number by so-called “Dedekind curs”.

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