Multiplication of Complex Numbers

Key Questions

  • What is the geometric interpretation of multiplying two complex numbers?

    Let z_1 and z_2 be two complex numbers.

    By rewriting in exponential form,

    {(z_1=r_1e^{i theta_1}),(z_2=r_2 e^{i theta_2}):}

    So,

    z_1 cdot z_2 =r_1e^{i theta_1}cdot r_2 e^{i theta_2} =(r_1 cdot r_2)e^{i(theta_1+theta_2)}

    Hence, the product of two complex numbers can be geometrically interpreted as the combination of the product of their absolute values (r_1 cdot r_2) and the sum of their angles (theta_1+theta_2) as shown below.

    enter image source here

    I hope that this was clear.

    Wataru · 1 · Dec 12 2014
  • What is the formula for multiplying complex numbers in trigonometric form?

    In trigonometric form, a complex number looks like this:
    a + bi = c*cis(theta)
    where a, b and c are scalars.

    Let two complex numbers:
    -> k_(1) = c_(1)*cis(alpha)
    -> k_(2) = c_(2)*cis(beta)
    k_(1)*k_(2) = c_(1) * c_(2) * cis(alpha) * cis(beta) =
    = c_(1) * c_(2) * (cos(alpha) + i*sin(alpha)) * (cos(beta) + i*sin(beta))

    This product will end up leading to the expression
    k_(1)*k_(2) =
    = c_(1)*c_(2)*(cos(alpha + beta) + i*sin(alpha + beta)) =
    = c_(1)*c_(2)*cis(alpha+beta)

    By analyzing the steps above, we can infer that, for having used generic terms c_(1), c_(2), alpha and beta, the formula of the product of two complex numbers in trigonometric form is:
    (c_(1) * cis(alpha)) * (c_(2) * cis(beta)) = c_(1)*c_(2)*cis(alpha+beta)

    Hope it helps.

    Bernardo Russo G. Ferreira · 1 · Oct 2 2014
  • How do I multiply complex numbers in polar form?

    To explain this, I will name two generic complex.
    c_1 = a*cis(alpha) and c_2 = b*cis(beta)

    The product between c_1 and c_2 is:
    ab*cis(alpha)cis(beta) =
    ab*(cos(alpha)+isin(alpha)) (cos(beta)+isin(beta)) =
    ab*({cos(alpha)cos(beta)-sin(alpha)sin(beta)} +
    {i(sin(alpha)sin(beta)+cos(alpha)sin(beta)}) =
    ab*{cos(a+b)+isin(a+b)}//

    Therefore, we can assume that the product of the two complex numbers c_1 and c_2 can be generaly given by the form above.

    Ex.:
    (2*cis(pi)) * (3*cis(2pi)) = 6*cis(3pi) = 6*cis(pi)

    Hope it helps.

    Bernardo Russo G. Ferreira · 1 · Mar 5 2015
  • How do I multiply complex numbers?

    Answer:

    Hope this helps.

    Explanation:

    Step 1: Distribute (or FOIL) to remove the parenthesis.
    Step 2 : Simplify the powers of i, specifically remember that i^2 = –1.
    Step 3 : Combine like terms, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers.

    Example 1 – Multiply: (4 – 3i)(2 + 5i)

    Step 1:
    "Distribute to remove the parenthesis. " color(green)(8 + 20 i - 6 i - 15 i^2

    Step 2:
    "Simplify the powers of i, specifically remember that " i^2 = –1
    #color(blue)(8 + 20 i - 6 i + 15.

    Step 3:
    "Combine like terms, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers."
    color(crimson)(23 + 14 i

    Example 2 – Multiply: (7 – 9i)(4 - 6i)

    Step 1:
    "Distribute to remove the parenthesis. " color(green)(28 - 42 i - 36 i + 54 i^2

    Step 2:
    "Simplify the powers of i, specifically remember that " i^2 = –1
    #color(blue)(28 - 42 i - 36 i - 54.

    Step 3:
    "Combine like terms, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers."
    color(crimson)(-26 - 78 i

    Example 3 – Multiply: (7 - 9i)(4 - 6i)(2 + 5i)

    "Step 1: Distribute (or FOIL) using only the first two complex numbers " color(green)((28 - 42 i - 36 i + 54 i^2) * (2 + 5 i)

    "Step 2: Simplify the powers of i, specifically remember that i2 = –1" color(blue)(28 - 42 i - 36 i - 54)(2 + 5i)

    "Step 3: Combine like terms, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers."
    color(crimson)((-26 - 78 i) * (2 + 5 i)

    "Step 4 : Distribute to remove the parenthesis. " color(chocolate)(-52 - 130 i - 156 i - 390 i^2

    "Step 5: Simplify the powers of i, specifically remember that " i^2 = –1
    color(magenta)(-52 - 130 i - 156 i + 390.

    "Step 6: Combine like terms, that is, combine real numbers with real numbers and imaginary numbers with imaginary numbers."
    color(purple)(328 - 286 i

    sankarankalyanam · 1 · Jul 21 2018
  • How do I find the product of two imaginary numbers?

    First, complex numbers can come in a variety of forms!

    Ex: multiply 3i*-4i =

    Remember, with multiplication you can rearrange the order (called the Commutative Property):

    3*-4*i*i =-12i^2

    ... and then always substitute -1 for i^2:

    -12*-1 = 12

    Ex: the numbers might come in a radical form:

    sqrt(-3)*4sqrt(-12) =

    You should always "factor" out the imaginary part from the square roots like this:

    sqrt(-1)sqrt(3)*4*sqrt(-1)sqrt(4)sqrt(3) =

    and simplify again:

    =i*4*sqrt(3)*sqrt(3)*sqrt(4)
    =i*4*3*2 = 24i

    Ex: what about the Distributive Property? 3i(4i - 6) =

    =12i^2- 18i
    =12(-1) - 18i
    = -12 - 18i

    And last but not least, a pair of binomials in a + bi form:

    Ex: (3 - 2i)(4 + i) =

    =12 + 3i - 8i - 2i^2
    = 12 - 2(-1) + 3i - 8i
    = 12 + 2 - 5i
    = 14 - 5i

    MrsRudolph221 · · Sep 4 2014

Questions

Complex Numbers in Trigonometric Form