Multiplying Complex Numbers page 15

Algebraic Methods for Multiplying Complex Numbers

  • Complex numbers can be multiplied together using standard algebraic expansion techniques, specifically the distributive property or the FOIL method.

  • The acronym FOIL defines the sequence of multiplication when expanding two binomial terms:

    • F: First terms

    • O: Outer terms

    • I: Inner terms

    • L: Last terms

General Formula and Derivation

  • To multiply two general complex numbers given in standard form (a+bi)(a + bi) and (c+di)(c + di):

    • Apply the FOIL method or distributive property:     (a+bi)(c+di)=ac+adi+bci+bdi2(a + bi)(c + di) = ac + adi + bci + bdi^2

    • Substitute the fundamental imaginary property i2=−1i^2 = -1 into the product of the last terms:     (a+bi)(c+di)=ac+adi+bci−bd(a + bi)(c + di) = ac + adi + bci - bd

    • Combine real components together and imaginary components together to yield the simplified product in standard form a+bia + bi:     (a+bi)(c+di)=(ac−bd)+(ad+bc)i(a + bi)(c + di) = (ac - bd) + (ad + bc)i

Step-by-Step Procedure for Multiplying Complex Numbers

  • Step 1: Use the distributive property or the FOIL method to expand the two complex terms.

  • Step 2: Simplify the resulting expression by replacing i2i^2 with −1-1 and combining the real parts and imaginary parts.

Worked Examples and Practice Problems

  • Example 1: Multiply (4+3i)(2−5i)(4 + 3i)(2 - 5i)

    • Step 1 (Expand using FOIL):

    • First terms: 4×2=84 \times 2 = 8

    • Outer terms: 4×(−5i)=−20i4 \times (-5i) = -20i

    • Inner terms: 3i×2=6i3i \times 2 = 6i

    • Last terms: 3i×(−5i)=−15i23i \times (-5i) = -15i^2

    • Intermediate expansion: 8−20i+6i−15i28 - 20i + 6i - 15i^2

    • Step 2 (Substitute i2=−1i^2 = -1):

    • 8−20i+6i−15(−1)8 - 20i + 6i - 15(-1)

    • 8−14i+158 - 14i + 15

    • Step 3 (Combine real and imaginary parts):

    • (8+15)−14i=23−14i(8 + 15) - 14i = 23 - 14i

  • Example 2: Multiply (3−4i)(2+3i)(3 - 4i)(2 + 3i)

    • Step 1 (Expand using FOIL):

    • First terms: 3×2=63 \times 2 = 6

    • Outer terms: 3×3i=9i3 \times 3i = 9i

    • Inner terms: −4i×2=−8i-4i \times 2 = -8i

    • Last terms: −4i×3i=−12i2-4i \times 3i = -12i^2

    • Intermediate expansion: 6+9i−8i−12i26 + 9i - 8i - 12i^2

    • Step 2 (Substitute i2=−1i^2 = -1):

    • 6+9i−8i−12(−1)6 + 9i - 8i - 12(-1)

    • 6+i+126 + i + 12

    • Step 3 (Combine real and imaginary parts):

    • (6+12)+i=18+i(6 + 12) + i = 18 + i

  • Example 3: Evaluate the expression (−3−4i)(−1−4i)(-3 - 4i)(-1 - 4i) and write the result in standard form a+bia + bi

    • Step 1 (Expand using FOIL):

    • First terms: (−3)×(−1)=3(-3) \times (-1) = 3

    • Outer terms: (−3)×(−4i)=12i(-3) \times (-4i) = 12i

    • Inner terms: (−4i)×(−1)=4i(-4i) \times (-1) = 4i

    • Last terms: (−4i)×(−4i)=16i2(-4i) \times (-4i) = 16i^2

    • Intermediate expansion: 3+12i+4i+16i23 + 12i + 4i + 16i^2

    • Step 2 (Substitute i2=−1i^2 = -1):

    • 3+12i+4i+16(−1)3 + 12i + 4i + 16(-1)

    • 3+16i−163 + 16i - 16

    • Step 3 (Combine real and imaginary parts):

    • (3−16)+16i=−13+16i(3 - 16) + 16i = -13 + 16i