Question - Complex Number Multiplication
mathdistributive propertyfoil methodcomplex number multiplicationproduct of complex numbersimaginary unit
Solution:
The image shows a handwritten problem which asks to compute the product of the following complex numbers:1) $$ Z_1 = 1 + 3i $$ and $$ Z_2 = 5 + i $$2) $$ Z_1 = 3 - 4i $$ and $$ Z_2 = 3 + 2i $$To compute the product of two complex numbers, you multiply them using the distributive property (also known as the FOIL method for binomials), which stands for "First, Outer, Inner, Last." Also remember that $$ i^2 = -1 $$.Let's solve the first product:$$ Z_1 \cdot Z_2 = (1 + 3i)(5 + i) $$1. First (1 * 5): 52. Outer (1 * i): i3. Inner (3i * 5): 15i4. Last (3i * i): 3i^2Combining these we get:$$ 5 + i + 15i + 3i^2 $$Since $$ i^2 = -1 $$, substitute for $$ i^2 $$:$$ 5 + i + 15i + 3(-1) $$$$ 5 + 16i - 3 $$Combine like terms:$$ (5 - 3) + 16i $$$$ 2 + 16i $$So the product of the first pair is $$ 2 + 16i $$.Now for the second product:$$ Z_1 \cdot Z_2 = (3 - 4i)(3 + 2i) $$1. First (3 * 3): 92. Outer (3 * 2i): 6i3. Inner (-4i * 3): -12i4. Last (-4i * 2i): -8i^2Now combine these:$$ 9 + 6i - 12i - 8i^2 $$Substitute $$ i^2 $$ with -1:$$ 9 + 6i - 12i - 8(-1) $$$$ 9 - 6i - 8 $$Combine like terms:$$ (9 + 8) - 6i $$$$ 17 - 6i $$So the product of the second pair is $$ 17 - 6i $$.
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