# The higher calculus. A history of real and complex analysis by Bottazini U. By Bottazini U.

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Exp(iO) cos (0) = for all 0 E R and so lexp(iO)1 can thus be written as = 1. + i sin (0) Every nonzero complex number z z= Izl (ir) = Izl exp(iO) = Izl (cos (0) + isin(O)). Here 0 is any number in arg(z). iY \ \ \ \ x Fig. 4 ir ' The function z -+ defined implicitly above, is used frequently. It is called the signum and is defined formally by sgn(z) = 1_o 1%%1 n {O}', z=o. 57) Exercise. Use Hamel bases to prove that the additive groups R and K are isomorphic. 58) Exercise. Define addition in K as usual and define multiplication "coordinatewise": (x + iy) (u + iv) = xu + iyv .

We write A ~ B to mean that such an order isomorphism exists. It is easy to see that the relation ~ is reflexive, symmetric, and transitive. With every linearly ordered set A we associate a symbol, called the order type 01 A, such that two linearly ordered sets A and B have the same symbol attached to them if and only 28 if A Chapter 1. Set theory and algebra ~ B. If A ~ B, we say that A and B are order isomorphic or have the same order type. We write ordA to denote the order type of A. If, in particular, A is well ordered, we call ordA an ordinal number.

A set 5 is called countable if either 5 is finite or 'B = N = No. Any set that is not countable is uncountable. A set 5 is countably infinite [or denumerable] if 'B = No. If 5 is countably infinite and I is a one-to-one function from N onto 5, then the sequence (xn) where Xn = I (n) is called an enumeration of 5. Note that Xn =l= Xm if n =l= m. 15) Theorem. Every infinite set has a countably infinite subset. Proof. Let A be any infinite set. We show by induction that for each n EN there exists a set An C A such that An = n.