1988 The pecsent volume is the lursi paperback edition of the previously published hardcover version (ISBN I). Rather, we have tried to produce a faithful translation of the entire original, which can serve as a scholarly reference as well as casual reading.
Lamotke
In the gradual construction of the number system that followed, some themes constantly recur. Complex numbers were the starting point for one of the greatest creations of 19th century mathematics, the theory of complex functions.
PART C
Mainzer
- Addition, Multiplication and Ordering of the Natural Num-
- ifn<vnandm<n,thenm=n
- Ifn<mandm<l,thenn(l
The first documents date back to the earliest civilizations in the Nile, Euphrates and Tigris valleys. Then N and H' are canonically isomorphic, that is, there is only one bijective mapping. by swapping the roles of H and H' we get the corresponding mapping. identity mapping), we use the uniqueness claim of the recursion theorem for A = N, a = 0 and g = S. Both o and Id are mappings.
P1) OEN
The Order Relation in is defined by
In this case, the ratio 00 : a1 of the line segments is equal to the ratio m : n of two natural numbers. Due to symmetry, each side of a regular pentagon is parallel to one of the diagonals.
EUDOXUS and the Theory of Proportion. The Babylonians worked with rational approximations to irrational (incommensurable) ra-
Quantities of the same kind are ordered: a < 6 if and only if there exists a c such that. The relationships between geometric quantities of the same kind, which do not necessarily have to be comparable to each other (relationships of line segments, of surfaces, etc.) form the subject of the theory.
Irrational Numbers in Modern (that is, post-mediae'val) Math-
Many theorems in the theory of proportion can nowadays be interpreted simply as arithmetical laws governing calculations with real numbers. The goals of the theory of proportion were geometrical results, such as the precise justification of many formulas relating to surfaces and volumes.
2. DEDEKIND CUTS
CAUCHY's Criterion for Convergence. In accordahce with
A sequence (r,) of rational numbers is said to be a fundamental sequence or Cauchy sequence, if for every rational e > 0 there is an index k such that I?m — ri,I < e for all m ,n k. The rational series (ri,) is said to be rationally convergent if there is a rational number r such that for every e > 0 there exists an index k, with — r( < c for all n /c.
The Ring of Fundamental Sequences. The set F of all fundamental sequences becomes a ring when addition and multiplication are defined
It contains Q as a subset where we identify each rational r with the corresponding class of constant sequences modulo N.
The Completely Ordered Residue Class Field F/N. A rational
This procedure can be seen as an application of the claim that the geometric mean is between the harmonic mean and the arithmetic mean: < < in the special case b = 1. Determining the area of the circle that lies between those v- inscribed and bounded polygons are another example of interval nesting.
Nested Intervals and Completeness. The introduction of real num- bers by means of nested intervals is motivated by the following situation
We therefore have a well-defined mapping of the fundamental series modulo the zero series into the set of equivalence classes of nets of nested intervals. Again, the description of z as a cut (a, /3) can arise from a definition of the set a by means of statements that say nothing directly about the position of z.
The Natural Numbers, the Integers, and the Rational Numbers in the Real Number Field should all be recoverable once the latter has
13 of [13], where it is called "the theory of relations," following the tradition established by EUCLID in his Elements), but on the axioms (Rl)—(R3) §2. 1. Natural numbers, integers and rational numbers in the field of real numbers should be renewable when the latter .. is satisfied when K elements are taken instead of rational numbers), then a contains the largest element. We can now show that (ba) converges to s. c) (d): The Archimedean property of ordering K can be proved as follows. Between a — a and s, there is room for only one term of sequence number (no).
Existence and Uniqueness of the Real Numbers. We now show that the axiom system (R1)—(R3) for the real numbers characterizes them
To prove this, we start from the fact that K must contain the field Q of the rationals. The ordering relation in K can be defined based on the field structure alone. The problem of the consistency of set theory is treated in the last chapter.
Remmert1
In particular, we need x, the ratio of the circumference to the diameter of the circle. 34; On the Analytical Representation of Direction - An Essay', which can be found in the Transactions of the Danish Academy for the year 1798. CAUCHY was still dissatisfied with the interpretation of the symbol i in 1847, and thus long after HAMILTON (see next paragraph).
2. THE FIELD C
With each complex number c = a+ ib we associate the C-linear mapping. the so-called left regular representation as defined in Algebra). So, for example, the linear transformation z iz corresponding to i is a counterclockwise rotation through a right angle, which sends I to i,i to —1, and so on (see also 1.5) . The linear transformation defined by c = a+ ib is thus described by the matrix One is thus led to consider the following map.
C—iMat(2,R),
3. ALGEBRAIC PROPERTIES OF THE FIELD C
The Natural Scalar Product and Euclidean Length Izi
When z is real, Izi coincides with the absolute value, as defined in the usual way for real numbers. The vectors iz and z are always perpendicular to each other because Re(izi) = IzI2Re(i) = 0. The reader would like to use this for a simple proof of the theorem that the heights of a triangle converge at a common point , the orthocenter (see the image above where the orthocenter is located —vi).
BEC,
- Square Root8 and nth Roots. Let n 1 be a natural number, and
- Cosine Theorem and the Triangle Inequality. Just as for every
- IzI0,
- Cyclic Quadrilaterals and Cross-Ratio. Any four distinct points
- Multiplication of Complex Numbers in Polar Coordinates. Since
S1 is represented in the Gaussian plane by the circumference of the unit circle centered on the origin. One can get rid of the exceptional role of —1 and —E in the equations (2) and (3) if one replaces A by A/sc and simplifies. As one of the most important applications of polar coordinates, we will demonstrate the following.
Remmert
Let's take a closer look at the problem involved: the complex zeros of the real polynomial. A careful and balanced critique, together with a completion of the first Gaussian proof, was first given in 1920 by A. What may well be the simplest of all the proofs of the Fundamental Theorem of Algebra was published in 1814 by ft.
ARGAND
Proof of the Fundamental Theorem. In addition to the minimum theorem we need
Proof of ARGAND's Inequality. The decisive role in the proof is played by the following
These consequences of the fundamental theorem are completely elementary and result from the simple fact that a polynomial with zero in c always has the factor z —c. Every complex polynomial I E C[Z] of degree n 1 is, regardless of the order of the factors, uniquely represented in the form. Every complex polynomial of degree nih has exactly n zeros where each of the c zeros is counted according to its multiplicity.
Factorization of Real Polynomials. Every real polynomial I =
Each of the last two theorems is equivalent to the Fundamental theorem of algebra. For ease of using the statement (1.3), we will write the coefficients of the given polynomial with alternating signs. The key to solving the problem of squaring the circle lies in the fundamental ratio = 1.
Definition by Measuring a Circle. In any circle the ratio of the circumference C to the diameter, and the ratio of the area A to the square
In any circle, the ratio of the circumference C to the diameter is the ratio of the area A to the square.
This faction was rediscovered by the Dutchman Valentin OTilo towards the end of the 16th century. This value also appears in the works of al-HwARIzM1 (Baghdad, early 9th century AD). The first analytical representation of s was found by VIETA in 1579 in the form of the infinite product.
Analytical Formulae. The first analytical representation of s was found by VIETA in 1579 in the form of the infinite product
These equations can be elevated to the status of definitions of It is worth pointing out here that already in 1841 WEIERSTRASS in his function-theoretic proof of the expansion theorem, now commonly known as the LAURENT series theorem, proposed the idea of defining s by the improper integral. In lectures and books on the infinitesimal calculus, integrals are not normally used to define *, because the integral calculus is generally treated only after the differential calculus, while it must be introduced at an early stage as zeros of the sine and cosine functions respectively . It is therefore more common to define as the smallest positive zero of the cosine function defined by its power series; the existence of such a zero is proven using the intermediate value theorem.
LANDAU and His Contemporary Critics. BALTZER'S method of introducing ir is not geometrical, but it is probably the most convenient way
The mapping exp:C — CX is a homomorphism of the additive group C into the multiplicative group CX. In the proof of the above lemma, the identity exp A(z) = 1 + z plays a decisive role. In the appendix to this section, we will give another elementary proof of the lemma.
The Kernel of the Exponential Homomorphism. Definition of
However, by an elementary theorem of the general theory of topological groups, a connected group G has no open subgroups other than G. It is possible to prove this identity in an elementary way without using the differential calculus. We use Lemma 2 and . the following theorem, which is a consequence of the Intermediate value theorem of the infinitesimal calculus.
Definitions of cos z and sin z. We define the complex cosine and sine fsnctiona throughout C by
In this section we show that the number defined in 2.4 has all the properties one normally learns in real analysis. The characterization of and as the least positive zero of the sine and cosine functions respectively is a simple matter using the results of the preceding sections, if one makes use of the relationship between the exponential function and the trigonometric functions. It also immediately follows from (1), that the cosine function is even and the sine function is odd, that is to say that.
The Number T and the zeros of cos z and sin z. In contrast to
The Number w and the Periods of exp z, cos z and sin z. A
There is a fundamental difference between the behavior of the exponential function in the real and complex domains. In the real domain it assumes, since Ker(exp) fl = (0), every positive real value once and only once, while in the complex domain it possesses the purely imaginary "minimum period" 2iri (not seen in the real case) and assumes every value c 0—. It must be clearly stated that without resorting to the intermediate value theorem, the minus sign cannot be excluded in the formula = ±i.
The Number s and the Circumference and Area of a Circle. A
Using the Intermediate value theorem and cosO = 1, it can be shown, as above, that. Since the semicircle around the origin of radius r > 0 is represented by the function sir2 —x2, x E [—r,rJ, the area I of the whole circle is given by. The tangent function tan x is a monotonic strictly increasing function in the interval (— since its derivative.
VIETA's Product Formula for The halving-formula sin z =
EULER's Product for the Sine and WALLIS's Product for ir
The convergence of and thus with error terms of the order of 1/n and 1/n2 respectively. His method was to compare the coefficients of the powers of z after expanding the product on the right. The convergence of the series is very poor: about 100 million terms from the first series of EULER are needed to correctly represent 72/6 to the first seven decimal places.
The WEIERSTR.ASS Definition of '. The integral formula (1)
The Irrationality of and Its Continued Fraction Expansion
The extension to the left was found in 1656 by Lord BROUNCKER, the first President of the Royal Society, transforming WALLIS' product. In this way, the thousand-year-old question about the square of the circle was definitively answered in the negative. We seek to prove this in the following pages.] The propositions of HERMIT and LINDEMANN are included in the following general theorem.
Neukirch
In the case of polynomials 1(x) E C[z] the highest derivatives at the point z = a are (almost) given by the expansion coefficients. We are now led to the concept of p.adic number by observing that any rational number f Q can be given an analogous expansion with respect to any prime p of 7L. If we now wish to find a p-adic expansion for negative and even fractional numbers, then we are forced to consider infinite series of the form.
The residue classes a modp" E Z/p'tZ are expressible
The modified representation of the p-adic numbers that we referred to earlier is now achieved through the following. Let the sequence of digits (as) be periodic, that is, of the form. Despite this non-convergence, however, the field Q of the p-adic numbers can be constructed from the field Q in much the same way as the field R is constructed from Q.
The set
The identification of the new analytic definition of Z1 (and therefore of Q,) with the older Hensel definition can now be made. Given the transitivity of the norm, we may assume that K = Iftherefore. The residue class field = 0/p is a finite extension of the residue class field ic(p) = Z,/pZ, = IF, and therefore a finite field W,.
G(LIK) K/NLIK(L)
Koecher, R. Remmert
