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3A. Given Z1 = 2+4i and Z2 = 1+i.

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3A. Given Z1=2+4iZ_1 = 2+4i and Z2=1+iZ_2 = 1+i.

i) Product Z1Z2Z_1 Z_2

  1. Multiply the complex numbers.

    Z1Z2=(2+4i)(1+i)Z_1 Z_2 = (2+4i)(1+i)

  2. Expand the product.

    (2+4i)(1+i)=2(1)+2(i)+4i(1)+4i(i)(2+4i)(1+i) = 2(1) + 2(i) + 4i(1) + 4i(i) =2+2i+4i+4i2= 2 + 2i + 4i + 4i^2

  3. Simplify using i2=−1i^2 = -1.

    2+6i+4(−1)=2+6i−4=−2+6i2 + 6i + 4(-1) = 2 + 6i - 4 = -2 + 6i

    The product is −2+6i\boxed{-2 + 6i}.

    ii) Division Z1/Z2Z_1 / Z_2

  4. Write the division as a fraction.

    Z1Z2=2+4i1+i\frac{Z_1}{Z_2} = \frac{2+4i}{1+i}

  5. Multiply the numerator and denominator by the conjugate of the denominator, which is 1−i1-i.

    2+4i1+i=(2+4i)(1−i)(1+i)(1−i)\frac{2+4i}{1+i} = \frac{(2+4i)(1-i)}{(1+i)(1-i)}

  6. Expand the numerator and denominator.

    Numerator: (2+4i)(1−i)=2(1)+2(−i)+4i(1)+4i(−i)=2−2i+4i−4i2=2+2i−4(−1)=2+2i+4=6+2i(2+4i)(1-i) = 2(1) + 2(-i) + 4i(1) + 4i(-i) = 2 - 2i + 4i - 4i^2 = 2 + 2i - 4(-1) = 2 + 2i + 4 = 6 + 2i
    Denominator: (1+i)(1−i)=12−i2=1−(−1)=1+1=2(1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2

  7. Combine the results and simplify.

    6+2i2=62+2i2=3+i\frac{6+2i}{2} = \frac{6}{2} + \frac{2i}{2} = 3 + i

    The division is 3+i\boxed{3 + i}.

    3B. Given Z1=1−2iZ_1 = 1-2i and Z2=1+2iZ_2 = 1+2i.

    i) Product Z1Z2Z_1 Z_2

  8. Multiply the complex numbers.

    Z1Z2=(1−2i)(1+2i)Z_1 Z_2 = (1-2i)(1+2i)

  9. Use the difference of squares formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2.

    12−(2i)21^2 - (2i)^2

  10. Simplify using i2=−1i^2 = -1.

    1−4i2=1−4(−1)=1+4=51 - 4i^2 = 1 - 4(-1) = 1 + 4 = 5

    The product is 5\boxed{5}.

    ii) Division Z1/Z2Z_1 / Z_2

  11. Write the division as a fraction.

    Z1Z2=1−2i1+2i\frac{Z_1}{Z_2} = \frac{1-2i}{1+2i}

  12. Multiply the numerator and denominator by the conjugate of the denominator, which is 1−2i1-2i.

    1−2i1+2i=(1−2i)(1−2i)(1+2i)(1−2i)\frac{1-2i}{1+2i} = \frac{(1-2i)(1-2i)}{(1+2i)(1-2i)}

  13. Expand the numerator and denominator.

    Numerator: (1−2i)(1−2i)=1(1)+1(−2i)−2i(1)−2i(−2i)=1−2i−2i+4i2=1−4i+4(−1)=1−4i−4=−3−4i(1-2i)(1-2i) = 1(1) + 1(-2i) - 2i(1) - 2i(-2i) = 1 - 2i - 2i + 4i^2 = 1 - 4i + 4(-1) = 1 - 4i - 4 = -3 - 4i
    Denominator: (1+2i)(1−2i)=12−(2i)2=1−4i2=1−4(−1)=1+4=5(1+2i)(1-2i) = 1^2 - (2i)^2 = 1 - 4i^2 = 1 - 4(-1) = 1 + 4 = 5

  14. Combine the results and simplify.

    −3−4i5=−35−45i\frac{-3-4i}{5} = -\frac{3}{5} - \frac{4}{5}i

    The division is −35−45i\boxed{-\frac{3}{5} - \frac{4}{5}i}.

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