Complex Numbers in Geometry

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It was not until that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology. The English mathematician G.

Complex Numbers: Introduction

Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his treatise. Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around in Cauchy's case.

The common terms used in the theory are chiefly due to the founders. Two complex numbers are equal if and only if both their real and imaginary parts are equal.

How to Find Simple Treasures in Complex Numbers

If the complex numbers are written in polar form, they are equal if and only if they have the same argument and the same magnitude. Since complex numbers are naturally thought of as existing on a two-dimensional plane, there is no natural linear ordering on the set of complex numbers. In fact, there is no linear ordering on the complex numbers that is compatible with addition and multiplication — the complex numbers cannot have the structure of an ordered field.

Conjugating twice gives the original complex number. This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. This process is sometimes called " rationalization " of the denominator although the denominator in the final expression might be an irrational real number , because it resembles the method to remove roots from simple expressions in a denominator.

The real and imaginary parts of a complex number z can be extracted using the conjugation:. Conjugation is also employed in inversive geometry , a branch of geometry studying reflections more general than ones about a line. In the network analysis of electrical circuits , the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for.

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That is to say:. A visualization of the subtraction can be achieved by considering addition of the negative subtrahend. Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates.

Standards in this domain:

In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. The picture at the right illustrates the multiplication of. Thus, the formula. Euler's formula states that, for any real number x ,. This can be proved through induction by observing that. The rearrangement of terms is justified because each series is absolutely convergent.

Because cosine and sine are periodic functions, other possible values may be obtained. To deal with the existence of more than one possible value for a given input, the complex logarithm may be considered a multi-valued function, with. Alternatively, a branch cut can be used to define a single-valued "branch" of the complex logarithm.

When n is an integer, this simplifies to de Moivre's formula :. The n th roots of z are given by. Therefore, the n th root of z is considered as a multivalued function in z , as opposed to a usual function f , for which f z is a uniquely defined number. Formulas such as. The set C of complex numbers is a field. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers z 1 and z 2 :. These two laws and the other requirements on a field can be proven by the formulas given above, using the fact that the real numbers themselves form a field.

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When the underlying field for a mathematical topic or construct is the field of complex numbers, the topic's name is usually modified to reflect that fact. For example: complex analysis , complex matrix , complex polynomial , and complex Lie algebra. Given any complex numbers called coefficients a 0 , Because of this fact, C is called an algebraically closed field. There are various proofs of this theorem, either by analytic methods such as Liouville's theorem , or topological ones such as the winding number , or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root.

Because of this fact, theorems that hold for any algebraically closed field , apply to C.

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For example, any non-empty complex square matrix has at least one complex eigenvalue. The field C has the following three properties: first, it has characteristic 0. Second, its transcendence degree over Q , the prime field of C , is the cardinality of the continuum. Third, it is algebraically closed see above. It can be shown that any field having these properties is isomorphic as a field to C.

For example, the algebraic closure of Q p also satisfies these three properties, so these two fields are isomorphic as fields, but not as topological fields. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that C contains many proper subfields that are isomorphic to C.

Chapters 5 and 6: Complex Numbers

The preceding characterization of C describes only the algebraic aspects of C. That is to say, the properties of nearness and continuity , which matter in areas such as analysis and topology , are not dealt with. The following description of C as a topological field that is, a field that is equipped with a topology , which allows the notion of convergence does take into account the topological properties.

C contains a subset P namely the set of positive real numbers of nonzero elements satisfying the following three conditions:.

What is it to multiply something with complex numbers?

With this topology F is isomorphic as a topological field to C. The only connected locally compact topological fields are R and C.

Complex Numbers (part 1)

This gives another characterization of C as a topological field, since C can be distinguished from R because the nonzero complex numbers are connected , while the nonzero real numbers are not. Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of C more immediately.

This characterization relies on the notion of fields and polynomials. A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law. The set R of real numbers does form a field. A polynomial p X with real coefficients is an expression of the form. The usual addition and multiplication of polynomials endows the set R [ X ] of all such polynomials with a ring structure. This ring is called the polynomial ring over the real numbers. So the two definitions of the field C are isomorphic as fields.

Accepting that C is algebraically closed, since it is an algebraic extension of R in this approach, C is therefore the algebraic closure of R. Here the entries a and b are real numbers. The sum and product of two such matrices is again of this form, and the sum and product of complex numbers corresponds to the sum and product of such matrices, the product being:. The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices.

Moreover, the square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix:. The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis see prime number theorem for an example. Unlike real functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

The notions of convergent series and continuous functions in real analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. From a more abstract point of view, C , endowed with the metric. Like in real analysis, this notion of convergence is used to construct a number of elementary functions : the exponential function exp z , also written e z , is defined as the infinite series. The series defining the real trigonometric functions sine and cosine , as well as the hyperbolic functions sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as tangent , things are slightly more complicated, as the defining series do not converge for all complex values.

Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of analytic continuation. Unlike in the situation of real numbers, there is an infinitude of complex solutions z of the equation. It can be shown that any such solution z — called complex logarithm of w — satisfies. For example, they do not satisfy.