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    Complex Number

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    Abdullah Mughal
    Complex numbers allow solutions to equations that have no real solutions. A complex number is expressed as a + bi, where a and b are real numbers and i is the imaginary unit that satisfies i^2 = -1. Despite the name, complex numbers are considered as fundamental as real numbers in mathematics. Key figures like Cardano, Wallis, de Moivre, Euler, and Cauchy contributed to establishing and developing the foundations and applications of complex numbers over centuries.

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    Complex Number

    Uploaded by

    Abdullah Mughal
    49 views2 pages
    Complex numbers allow solutions to equations that have no real solutions. A complex number is expressed as a + bi, where a and b are real numbers and i is the imaginary unit that satisfies i^2 = -1. Despite the name, complex numbers are considered as fundamental as real numbers in mathematics. Key figures like Cardano, Wallis, de Moivre, Euler, and Cauchy contributed to establishing and developing the foundations and applications of complex numbers over centuries.

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    Complex numbers allow solutions to equations that have no real solutions. A complex number is expressed as a + bi, where a and b are real numbers and i is the imaginary unit that satisfies i^2 = -1. Despite the name, complex numbers are considered as fundamental as real numbers in mathematics. Key figures like Cardano, Wallis, de Moivre, Euler, and Cauchy contributed to establishing and developing the foundations and applications of complex numbers over centuries.

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    49 views2 pages

    Complex Number

    Uploaded by

    Abdullah Mughal
    Complex numbers allow solutions to equations that have no real solutions. A complex number is expressed as a + bi, where a and b are real numbers and i is the imaginary unit that satisfies i^2 = -1. Despite the name, complex numbers are considered as fundamental as real numbers in mathematics. Key figures like Cardano, Wallis, de Moivre, Euler, and Cauchy contributed to establishing and developing the foundations and applications of complex numbers over centuries.

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    © All Rights Reserved
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    Complex number

    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

    A complex number is a number that can be expressed in the form a + bi,


    where a and b are real numbers, and 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, and 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, and are fundamental in many aspects of the
    scientific description of the natural world

    Complex numbers allow solutions to certain equations that have no


    solutions in real numbers. For example, the equation
    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 (sometimes called
    the imaginary unit) that is 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:
    John Wallis (1616-1703) notes in his Algebra that negative numbers, so
    long viewed with suspicion by mathematicians, had a perfectly good
    physical explanation, based on a line with a zero mark, and positive
    numbers being numbers at a distance from the zero point to the right,
    where negative numbers are a distance to the left of zero. Also, he
    made some progress at giving a geometric interpretation to √ −1.

    Abraham de Moivre (1667-1754) left France to seek religious refuge in


    London at eighteen years of age. There he befriended Newton. In 1698
    he mentions that Newton knew, as early as 1676 of an equivalent
    expression to what is today known as de Moivre’s theorem: (cos(θ) +
    isin(θ))n = cos(nθ) + isin(nθ) where n is an integer. Apparently Newton
    used this formula to compute the cubic roots that appear in Cardan
    formulas, in the irreducible case. de Moivre knew and used the formula
    that bears his name, as it is clear from his writings -although he did not
    write it out explicitly.
    L. Euler (1707-1783) introduced the notation i = √ −1 [3], and visualized
    complex numbers as points with rectangular coordinates, but did not
    give a satisfactory foundation for complex numbers. Euler used the
    formula x + iy = r(cos θ + i sin θ), and visualized the roots of z n = 1 as
    vertices of a regular polygon. He defined the complex exponential, and
    proved the identity e iθ = cos θ + i sin θ
    Augustin-Louis Cauchy (1789-1857) initiated complex function theory in
    an 1814 memoir submitted to the French Acad´emie des Sciences. The
    term analytic function was not mentioned in his memoir, but the
    concept is there. The memoir was published in 1825. Contour integrals
    appear in the memoir, but this is not a first, apparently Poisson had a
    1820 paper with a path not on the real line. Cauchy constructed the set
    of complex numbers in 1847 as R[x]/(x 2 + 1)
    “We repudiate the symbol √ −1, abandoning it without regret because
    we do not know what this alleged symbolism signifies nor what
    meaning to give to it.”

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