A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
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. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th-century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i. Formally, the complex numbers are defined as the quotient ring of the p
Rodman Wilson Paul was an American historian who taught at the California Institute of Technology. He was known as a foremost authority on California mining and agricultural history. Paul was raised near Boston, he received his AB, AM and PhD from Harvard. His PhD adviser at Harvard was Frederick Merk. From 1943 to 1946 he served in the Navy Reserve. In 1947 he went to Caltech, his choice for going there was based on an interest he developed in the history of the far West after a trip he took to Arizona to recuperate from an illness. He became the Edward S. Harkness Professor of History. After retirement in 1972, he continued work as a researcher at the Huntington Library. Paul wrote many books and articles, was recognized with several awards including the 1984 Henry R. Wagner Memorial Award, he was a fellow of the California Historical Society, served on the board of the Pasadena and Santa Barbara historical societies, was a member of the NASA Historical Advisory Committee. The Mining History Association's Rodman Paul Award recognizes individuals who have contributed to the understanding of American mining history.
The following is a selected list of the works of Rodman Paul. Not listed are his many book reviews of works by other authors. Paul, Rodman W.. California Gold: The Beginning of Mining in the Far West. Lincoln: Univ. of Nebraska Press. ISBN 0803251491. Retrieved November 10, 2013. Paul, Rodman W.. The Frontier and the American West. Arlington Heights, IL: AHM Pub. Corp. Retrieved November 10, 2013. Paul, Rodman W.. When Culture Came to Boise: Mary Hallock Foote in Idaho. Boise, Idaho: Idaho State Historical Society. Retrieved November 10, 2013. Paul, Rodman W.. The Far West and the Great Plains In Transition, 1859-1900. New York: Harper & Row. ISBN 0060158360. Retrieved November 10, 2013. Paul, Rodman W.. Mining Frontiers of the Far West, 1848-1880. Albuquerque: University of New Mexico Press. Retrieved November 10, 2013. Paul, Rodman W.. "The Origin of the Chinese Issue in California". The Mississippi Valley Historical Review. Organization of American Historians. 25: 181–196. Doi:10.2307/1896498. JSTOR 1896498. Paul, Rodman Wilson.
"The Great California Grain War: The Grangers Challenge the Wheat King". Pacific Historical Review. University of California Press. 27: 331–349. Doi:10.2307/3636810. JSTOR 3636810. Paul, Rodman Wilson. "Colorado as a Pioneer of Science in the Mining West". The Mississippi Valley Historical Review. Organization of American Historians. 47: 34–50. Doi:10.2307/1891277. JSTOR 1891277. Paul, Rodman Wilson. "Mining Frontiers as a Measure of Western Historical Writing". Pacific Historical Review. University of California Press. 33: 25–34. Doi:10.2307/3636376. JSTOR 3636376. Paul, Rodman W.. "The Mormons As a Theme in Western Historical Writing". The Journal of American History. Organization of American Historians. 54: 511–523. Doi:10.2307/2937404. JSTOR 2937404. Paul, Rodman W.. "The New Western History: A Review of Two Recent Examples". Agricultural History. Agricultural History Society. 43: 297–300. JSTOR 4617668. Paul, Rodman W.. "Historical Advisory Committees: NASA and the National Archives". Pacific Historical Review. University of California Press.
44: 385–394. Doi:10.2307/3638034. JSTOR 3638034. Paul, Rodman W.. "Frederick Merk and Scholar: A Tribute". The Western Historical Quarterly. Utah State University. 9: 141–148. JSTOR 966823. Paul, Rodman W.. "A Tenderfoot Discovers There Once Was a Mining West". The Western Historical Quarterly. Utah State University. 10: 4–20. JSTOR 967129. Paul, Rodman W.. "After the Gold Rush: San Francisco and Portland". Pacific Historical Review. University of California Press. 51: 1–21. Doi:10.2307/3639818. JSTOR 3639818. Paul, Rodman W.. "Tradition and Challenge in Western Historiography". The Western Historical Quarterly. Utah State University. 16: 27–53. JSTOR 968156. "Rodman W. Paul, 74. Los Angeles Times. May 21, 1987. Retrieved November 10, 2013. "Finding Aid for the Rodman W. Paul Papers 1929-1986". Online Archive of California. Retrieved November 10, 2013. Paul, Rodman. "Rodman W. Paul". Interviewed by Carol Bugé. Pasadena, CA. Retrieved November 10, 2013. "Mining History Association Awards". Mining History Association. Retrieved May 3, 2014
The 2014 Newcastle City Council election took place on 22 May 2014 to elect members of Newcastle City Council in England. This was on the same day as other local elections; the election saw the governing Labour party gain a net extra seat in the council chambers, gaining the wards of Denton and South Jesmond. An Independent councillor was elected in Westerhope, with Bill Corbett winning with a 451 majority over the Labour Party; the Conservatives improved their performance from previous years, coming second in wards where they were third. The election saw UKIP field candidates around the city; the party came second in wards including Byker and Lemington