{"id":458,"date":"2015-10-26T19:22:59","date_gmt":"2015-10-26T19:22:59","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=458"},"modified":"2017-03-31T18:25:38","modified_gmt":"2017-03-31T18:25:38","slug":"multiply-and-divide-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/multiply-and-divide-complex-numbers\/","title":{"raw":"Multiply and divide complex numbers","rendered":"Multiply and divide complex numbers"},"content":{"raw":"

Multiplying Complex Numbers<\/h2>\r\n
\r\n

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.<\/p>\r\n\r\n

\r\n
\r\n

Example 4: Multiplying a Complex Number by a Real Number<\/h3>\r\n[caption id=\"attachment_2535\" align=\"aligncenter\" width=\"487\"]\"Showing Figure 5<\/b>[\/caption]\r\n\r\nLet\u2019s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,\r\n<\/span>\r\n\r\n<\/div>\r\n
\r\n

How To: Given a complex number and a real number, multiply to find the product.<\/h3>\r\n
    \r\n \t
  1. Use the distributive property.<\/li>\r\n \t
  2. Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n
    \r\n
    \r\n

    Example 5: Multiplying a Complex Number by a Real Number<\/h3>\r\n

    Find the product [latex]4\\left(2+5i\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    Solution<\/h3>\r\n

    Distribute the 4.<\/p>\r\n\r\n

    [latex]\\begin{cases}4\\left(2+5i\\right)=\\left(4\\cdot 2\\right)+\\left(4\\cdot 5i\\right)\\hfill \\\\ =8+20i\\hfill \\end{cases}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
    \r\n

    Try It 4<\/h3>\r\n

    Find the product [latex]-4\\left(2+6i\\right)[\/latex].<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>\r\n<\/section>

    \r\n

    Multiplying Complex Numbers Together<\/h3>\r\n

    Now, let\u2019s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get<\/p>\r\n\r\n

    [latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci+bd{i}^{2}[\/latex]<\/div>\r\n

    Because [latex]{i}^{2}=-1[\/latex], we have<\/p>\r\n\r\n

    [latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci-bd[\/latex]<\/div>\r\n

    To simplify, we combine the real parts, and we combine the imaginary parts.<\/p>\r\n\r\n

    [latex]\\left(a+bi\\right)\\left(c+di\\right)=\\left(ac-bd\\right)+\\left(ad+bc\\right)i[\/latex]<\/div>\r\n
    \r\n

    How To: Given two complex numbers, multiply to find the product.<\/h3>\r\n
      \r\n \t
    1. Use the distributive property or the FOIL method.<\/li>\r\n \t
    2. Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n
      \r\n
      \r\n

      Example 6: Multiplying a Complex Number by a Complex Number<\/h3>\r\n

      Multiply [latex]\\left(4+3i\\right)\\left(2 - 5i\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n

      \r\n

      Solution<\/h3>\r\n

      Use [latex]\\left(a+bi\\right)\\left(c+di\\right)=\\left(ac-bd\\right)+\\left(ad+bc\\right)i[\/latex]<\/p>\r\n\r\n

      [latex]\\begin{cases}\\left(4+3i\\right)\\left(2 - 5i\\right)=\\left(4\\cdot 2 - 3\\cdot \\left(-5\\right)\\right)+\\left(4\\cdot \\left(-5\\right)+3\\cdot 2\\right)i\\hfill \\\\ \\text{ }=\\left(8+15\\right)+\\left(-20+6\\right)i\\hfill \\\\ \\text{ }=23 - 14i\\hfill \\end{cases}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
      \r\n

      Try It 5<\/h3>\r\n

      Multiply [latex]\\left(3 - 4i\\right)\\left(2+3i\\right)[\/latex].<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>\r\n<\/section><\/section>https:\/\/youtu.be\/O9xQaIi0NX0\r\n

      Dividing Complex Numbers<\/h2>\r\n

      Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate<\/strong> of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of [latex]a+bi[\/latex] is [latex]a-bi[\/latex].<\/p>\r\n

      Note that complex conjugates have a reciprocal relationship: The complex conjugate of [latex]a+bi[\/latex] is [latex]a-bi[\/latex], and the complex conjugate of [latex]a-bi[\/latex] is [latex]a+bi[\/latex]. Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.<\/p>\r\n

      Suppose we want to divide [latex]c+di[\/latex] by [latex]a+bi[\/latex], where neither a<\/em>\u00a0nor b<\/em>\u00a0equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.<\/p>\r\n\r\n

      [latex]\\frac{c+di}{a+bi}\\text{ where }a\\ne 0\\text{ and }b\\ne 0[\/latex]<\/div>\r\n

      Multiply the numerator and denominator by the complex conjugate of the denominator.<\/p>\r\n\r\n

      [latex]\\frac{\\left(c+di\\right)}{\\left(a+bi\\right)}\\cdot \\frac{\\left(a-bi\\right)}{\\left(a-bi\\right)}=\\frac{\\left(c+di\\right)\\left(a-bi\\right)}{\\left(a+bi\\right)\\left(a-bi\\right)}[\/latex]<\/div>\r\n

      Apply the distributive property.<\/p>\r\n\r\n

      [latex]=\\frac{ca-cbi+adi-bd{i}^{2}}{{a}^{2}-abi+abi-{b}^{2}{i}^{2}}[\/latex]<\/div>\r\n

      Simplify, remembering that [latex]{i}^{2}=-1[\/latex].<\/p>\r\n\r\n

      [latex]\\begin{cases}=\\frac{ca-cbi+adi-bd\\left(-1\\right)}{{a}^{2}-abi+abi-{b}^{2}\\left(-1\\right)}\\hfill \\\\ =\\frac{\\left(ca+bd\\right)+\\left(ad-cb\\right)i}{{a}^{2}+{b}^{2}}\\hfill \\end{cases}[\/latex]<\/div>\r\n
      \r\n

      A General Note: The Complex Conjugate<\/h3>\r\n

      The complex conjugate<\/strong> of a complex number [latex]a+bi[\/latex] is [latex]a-bi[\/latex]. It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.<\/p>\r\n\r\n