1. Addition/Subtraction: Like-with-like. Real + Real, Imaginary + Imaginary.
2. Multiplication: Use FOIL ($F \cdot O \cdot I \cdot L$).
3. The Spirit Swap: Always replace $i^2$ with $-1$.
4. Rotation Rule: Multiplying by $i$ turns the vector 90 degrees.
Perform the indicated operation. Simplify your result to $a + bi$.
$(5 + 2i) + (3 + 10i)$
$(10 - 5i) - (4 + 2i)$
$(0 + 7i) + (3 - 7i)$
If you add a purely Real number (5) to a purely Imaginary number ($5i$), what is the result? Why is it NOT $10i$? Draw the point $(5, 5i)$ on a small sketch. Is it on the line or in the plane?
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Multiply and simplify. Remember $i^2 = -1$.
$i \cdot (3 + 2i)$
$(1 + i)(1 - i)$
$(2 + 3i)(4 + 5i)$
The Pruning: Multiply $(1 + i)$ by itself 4 times.
1. $(1+i)^2 = 1 + 2i + i^2 = ...$
2. Take that result and square it again.
What is the final "Scalar" value of your life after 4 cycles of this spirit-growth?
A person has Work $a=1$ and Faith $b=1$ ($z_1 = 1+i$).
Another person has Work $c=1$ and Faith $d=1$ ($z_2 = 1+i$).
Task: Find the product $z_1 \cdot z_2$.
Explain why the "Real" part of the result is **Zero**.
Why did the physical work disappear in the interaction?
Objective: Explain Complex Multiplication to a younger student using a paper spinner.
The Activity:
1. Draw a Cross on a piece of paper. Label the directions (1, i, -1, -i).
2. Spin the spinner.
3. "Every time we multiply by the Spirit, the spinner turns one quarter turn."
The Lesson: "In math, the Spirit is the 'Energy' that makes our life spin so we don't get stuck in one spot."
Response: ___________________________________________________________