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We prove a complex polynomial of degree $n$ has at most $\lceil n/2 \rceil$ attractive fixed points lying on a line. We also consider the general case., 4 pages

The three Pauli matrices are normally given [1] as the 2 × 2 matrices:where ‘i’ is the usual complex number imaginary unit.These matrices obey the relations a2 = I = b2 = c2(where I is the 2 × 2 identity matrix), as well as the anticommutation relations:Within the quantities ia,ib and ic,i is a scalar multiplier of the 2 × 2 Pauli matrices and, of course, commutes with each of a, b, c.

Are there solutions of the equation x2 + 1 = 0 ? Carl Fredrich Gauss (1777–1855) conjectured that there was a solution and that it was the square root of - 1 . But since the squares of all real numbers, positive or negative, are positive, Gauss introduced a fanciful idea. His solution to this equation was , which he named i. He integrated i with the real numbers to form a set known as , the complex numbers, where each element in that set was of the form a + bi, where a, . Gauss illustrated this on a graph, the horizontal axis became the real axis and represented the real coefficient, while the vertical axis became the imaginary axis and represented the imaginary coefficient.

In this article a theorem similar to Napoleon’s theorem is established for the Fermat point configuration of a triangle. Areal coordinates are used throughout, with ABC as triangle of reference. For a full account of how to define and use these coordinates see Bradley [1]. Alternative synthetic proofs, generously supplied by the referee, are also included. The presentation of the paper with its emphasis on coordinates was designed in part to show that an algebraic treatment of the Fermat point configuration is possible, as well as the more familiar synthetic or complex number treatments.

Eperson's conjecture is that if n ⩾ 3 is odd then 3n2 can be written in at least two ways as a sum of three squares. We shall give a fairly elementary proof of this, from scratch. Let q be a real or complex number with Define for n ⩾ 1.

Recursively defined sequences (in other words difference equations) appear in modelling systems changing in discrete time steps. The asymptotic behaviour of such sequences tells us information about the state of the modelled system after a long time. The simplest difference equations are linear, but the standard solving method of even these simple equations involves eigenvalue and eigenvector computation, complex numbers and linear algebra.

“Find the modulus and the argument of each root of the equation:(University of London A level, part question.)We were investigating iterative methods of solving the equation, using a micro-package (in BASIC) which performs complex arithmetic and displays the results in an Argand diagram. The facility for “doodling” which this provides led to some interesting conjectures and results, of which this note is an account.

The Alexandrian Greek mathematician Heron—whom we associate with the formula √{s(s − a)(s − b)(s − c)} for the area of a triangle—got involved in a calculation about a pyramid design which led to the evaluation of √(81 − 144). This occurs in his book Stereometria (C. A.D. 75) and to ‘solve’ it the numbers are turned round thus: √(144 − 81), to give √63 which is taken to be . (Is this a reasonable approximation for √63?) It is not known whether Heron made this transpositional error or whether a copier of his work was responsible. This seems to be the first occasion in which the square root of a negative number was stumbled across—a concept not properly understood for another 1750 years!

The object of this article is to show how the problems of transformation geometry, and in particular the determination of the composition of two or more successive transformations, can be solved far more easily by Argand diagram methods leading to algebraic equations involving complex numbers, rather than by purely geometric considerations. It must be conceded, of course, that the methods advocated here cannot be extended to three dimensions.

The debate about how best to introduce complex numbers is no doubt perennial. (See for instance [1] for a brief survey of some of the possibilities.) One flaw which many presentations have is that they appear somewhat arbitrary. For example one can define complex numbers as polynomial residues to the modulus y2 +1: this definition ought to provoke the question as to why this particular modulus should be chosen.

The approach to plane mechanics by vector methods is often considered to be an unnecessarily sophisticated one and it is true that the preliminary work in vector algebra is lengthy and out of keeping with the elegance achieved. The natural tool, it has been said, is the complex number, and I shall obtain some of the fundamental results by using only complex number theory and vector addition.

A joint paper by Tranter and the present author concerning the heat flow in an infinite medium heated by a sphere has necessitated a proof that no complex zeros of the equation

The object of this address is to pass in review various theories of complex numbers and to scrutinise them from the standpoint of the teacher engaged in initiating his pupils into this subject. The material of the address is therefore very elementary and well known and roughly a century old, but it is arranged and combined in a way which may perhaps throw some new light on an old problem. It is important to realise at the outset the fundamental issues which are involved; and, although it is undesirable to intrude explicit philosophy into a first lesson on complex numbers, it is well for the teacher to have these metamathematical considerations clear in his own mind.

Contemporary writers of textbooks on electricity and magnetism and on hydrodynamics make considerable use of the complex variable in two-dimensional problems. It seems strange that two-dimensional dynamics is not taught in this way, even though the application of vectors to three-dimensional dynamics has now become respectable. Mr. Buckley's article preceding this has shown how the basic results may be established. The following examples are further indications of how this powerful technique may be profitably employed in two-dimensional kinematics. The methods are essentially labour-saving.