Argand diagram, representing the complex plane. Despite the historical nomenclature “imaginary”, complex numbers are regarded in the mathematical sciences as just as “real” schaum series complex analysis pdf the real numbers, and are fundamental in many aspects of the scientific description of the natural world. Furthermore, complex numbers can also be divided by nonzero complex numbers. Overall, the complex number system is a field.
Most importantly the complex numbers give rise to the fundamental theorem of algebra: every non-constant polynomial equation with complex coefficients has a complex solution. This property is true of the complex numbers, but not the reals. Complex numbers allow solutions to certain equations that have no solutions in real numbers. Complex numbers provide a solution to this problem. According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers.
An illustration of the complex plane. From this definition, complex numbers can be added or multiplied, using the addition and multiplication for polynomials. William Rowan Hamilton introduced this approach to define the complex number system. A position vector may also be defined in terms of its magnitude and direction relative to the origin. These are emphasized in a complex number’s polar form. Using the polar form of the complex number in calculations may lead to a more intuitive interpretation of mathematical results.
Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Many mathematicians contributed to the full development of complex numbers. The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli. Two complex numbers are equal if and only if both their real and imaginary parts are equal. 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.
Because 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. Moreover, a complex number is real if and only if it equals its own conjugate. Addition of two complex numbers can be done geometrically by constructing a parallelogram. Complex numbers are added by separately adding the real and imaginary parts of the summands. Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram, three of whose vertices are O, A and B. The division of two complex numbers is defined in terms of complex multiplication, which is described above, and real division. This formula can be used to compute the multiplicative inverse of a complex number if it is given in rectangular coordinates.