{"id":4489,"date":"2013-12-26T06:05:59","date_gmt":"2013-12-25T21:05:59","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=4489"},"modified":"2014-08-01T19:04:26","modified_gmt":"2014-08-01T10:04:26","slug":"complex-numbers","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4489","title":{"rendered":"Complex numbers"},"content":{"rendered":"
\n

Complex numbers arose in order to solve polynomial equations such as x^2 + 1 = 0 or x^2 + x + 1 = 0 which are not satisfied by real numbers. It’s assumed that a complex number has the form a + bi<\/span> where a, b<\/span> are real numbers and i<\/span>, called imaginary unit<\/span>, has the property that i2<\/sup><\/span> = -1. Complex numbers are defined as follows.<\/p>\n

    \n
  1. Addition.
    \n\\displaystyle (a + bi) + (c + di) = (a + c) + (b + d)i<\/li>\n
  2. Subtraction.
    \n\\displaystyle (a + bi) - (c + di) = (a - c) + (b - d)i<\/li>\n
  3. Multiplication.
    \n\\displaystyle (a + bi)\\times(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i<\/li>\n
  4. Division.
    \n\\displaystyle \\frac{a + bi}{c + di} = \\frac{a + bi}{c + di}\\times\\frac{c - di}{c - di} = \\frac{ac + bd}{c^2 + d^2} + \\frac{bc - ad}{c^2 + d^2}i<\/li>\n<\/ol>\n

    The ordinary rules of algebra has been used except that replaces i2<\/sup><\/span> by -1 wherever it occurs. The commutative, associative and distributive laws also apply to complex numbers. It’s called a<\/span> and b<\/span> of a + bi<\/span> the real<\/span> and imaginary parts<\/span>, respectively. Two complex numbers are equal<\/span> if and only if their real and imaginary parts are respectively equal. <\/p>\n

    A complex number z = x + iy<\/span> can be considered as a point P with coordinates (x, y<\/span>) on a rectangular xy<\/span> plane called in this case the complex plane<\/span> or Argand diagram<\/span>. If the line would be constructed from origin O<\/span> to P<\/span> and let ρ be the distance OP<\/span> and φ<\/span> the angle made by OP<\/span> with the positive x<\/span> axis, you could have from Figure<\/p>\n

    \"Graph\"<\/a><\/p>\n

    \\displaystyle x = \\rho \\cos\\phi,\\ y = \\rho\\sin\\phi,\\ \\rho = \\sqrt{x^2 + y^2}<\/p>\n

    and could write the complex number in so-called polar form<\/span> as <\/p>\n

    \\displaystyle z = x + iy = \\rho(\\cos\\phi + i\\sin\\phi) = \\rho cis\\phi<\/p>\n

    It’s often called that ρ<\/span> the modulus<\/span> or absolute value<\/span> of of z<\/span> and denote it by |z<\/span>|. The angle φ<\/span> is called the amplitude<\/span> or argument<\/span> of z<\/span> abbreviated arg z<\/span>. It could be also written \\rho = \\sqrt{z\\bar{z}} where \\bar{z} = x - iy is called the conjugate<\/span> of z = x + iy. <\/p>\n

    If you write two complex numbers in polar form as<\/p>\n

    \\displaystyle z_1 = \\rho_1(\\cos\\phi_1 + i\\sin\\phi_1),\\ z_2 = \\rho_2(\\cos\\phi_2 + i\\sin\\phi_2)<\/p>\n

    then<\/p>\n

    \\displaystyle z_1z_2 = \\rho_1\\rho_2[\\cos(\\phi_1 + \\phi_2) + i\\sin(\\phi_1 + \\phi_2)]\\\\\\vspace{0.2 in} \\frac{z_1}{z_2} = \\frac{\\rho_1}{\\rho_2}[\\cos(\\phi_1 - \\phi_2) + i\\sin(\\phi_1 - \\phi_2)]<\/p>\n

    Also if n<\/span> is any real number, you have<\/p>\n

    \\displaystyle z^n = [\\rho(\\cos\\phi + i\\sin\\phi)]^n = \\rho^n(\\cos n\\phi + i\\sin n\\phi)<\/p>\n

    which is often called De Moivre’s theorem<\/span>. You can use this to determine roots of complex numbers. For example if n<\/span> is a positive integer, <\/p>\n

    \\displaystyle z^{\\frac{1}{n}} = [\\rho(\\cos\\phi + i\\sin\\phi)]^\\frac{1}{n} = \\rho^{\\frac{1}{n}}\\left\\{\\cos\\left(\\frac{\\phi + 2k\\pi}{n}\\right) + i\\sin\\left(\\frac{\\phi + 2k\\pi}{n}\\right)\\right\\}\\\\\\vspace{0.2 in} \\ \\ k = 0,\\ 1,\\ 2,\\ \\cdots,\\ n-1<\/p>\n

    Using the series for ex<\/sup><\/span>, sin x<\/span>, cos x<\/span>, you are led to define<\/p>\n

    \\displaystyle e^{i\\phi} = \\cos\\phi + i\\sin\\phi,\\ e^{-i\\phi} = \\cos\\phi - i\\sin\\phi<\/p>\n

    which are called Euler’s formulas<\/span> and which enable you to rewrite equations in terms of exponentials. <\/p>\n<\/blockquote>\n