| Term Paper Title | Complex Number |
| # of Words | 923 |
| # of Pages (250 words per page double spaced) | 3.69 |
Complex Number
Complex Number, in mathematics, the sum of a real number and an imaginary number. An imaginary number is a multiple of i, where i is the square root of -1. Complex numbers can be expressed in the form a + bi, where a and b are real numbers. They have the algebraic structure of a field in mathematics. In engineering and physics, complex numbers are used extensively to describe electric circuits and electromagnetic waves (see Electromagnetic Radiation). The number i appears explicitly in the Schrödinger wave equation (see Schrödinger, Erwin), which is fundamental to the quantum theory of the atom. Complex analysis, which combines complex numbers with ideas from calculus, has been widely applied to subjects as different as the theory of numbers and the design of airplane wings.
History
Historically, complex numbers arose in the search for solutions to equations such as x2 = -1. Because there is no real number x for which the square is -1, early mathematicians believed this equation had no solution. However, by the middle of the 16th century, Italian mathematician Gerolamo Cardano and his contemporaries were experimenting with solutions to equations that involved the square roots of negative numbers. Cardano suggested that the real number 40 could be expressed as
Swiss mathematician Leonhard Euler introduced the modern symbol i for
in 1777 and expressed the famous relationship epi = -1 which connects four of the fundamental numbers of mathematics. For his doctoral dissertation in 1799, German mathematician Carl Friedrich Gauss proved the fundamental theorem of algebra, which states that every polynomial with complex coefficients has a complex root. The study of complex functions was continued by French mathematician Augustin Louis Cauchy, who in 1825 generalized the real definite integral of calculus to functions of a complex variable.
Properties
For a complex number a + bi, a is called the real part and b is called the imaginary part. Thus, the complex number -2 + 3i has the real part -2 and the imaginary part 3. Addition of complex numbers is performed by adding the real and imaginary parts separately. To add 1 + 4i and 2 - 2i, for example, add the real parts 1 and 2 and then the imaginary parts 4 and -2 to obtain the complex number 3 + 2i. The general rule for addition is
(a + bi) + (c + di) = (a + c) + (b + d)i
Multiplication of complex numbers is based on the premise that i × i = -1 and the assumption that multiplication distributes over addition. This gives ...Read entire document
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