1. Complex Identity: $z = a + bi$.
2. Real Part (Re): The number $a$ (on the horizontal axis).
3. Imaginary Part (Im): The coefficient $b$ (on the vertical axis).
4. Magnitude: The length of the vector $|z| = \sqrt{a^2 + b^2}$.
For each complex number, identify the **Real** and **Imaginary** parts.
$z = 3 + 4i$
$z = -2 + 10i$
$z = 5i$
$z = -7$
On the grid below, plot the four points from Part I.
Label them A, B, C, and D.
Calculate the magnitude $|z|$ for point A ($3+4i$). Show the work using the Pythagorean Theorem. Why is the total length of the vector larger than either individual part? How does this teach us about the **Synergy of Faith and Work**?
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Simplify each expression to its simplest form ($i, -1, -i,$ or $1$).
$i^{10}$
$i^{25}$
$i^{100}$
Find the distance between two complex numbers $z_1 = 1 + 2i$ and $z_2 = 4 + 6i$.
1. Subtract them: $z_2 - z_1 = \langle 4-1, 6-2 \rangle = \langle 3, 4 \rangle$.
2. Calculate the magnitude of the difference.
Objective: Explain the Imaginary Axis to a younger sibling using their hand.
The Activity:
1. Point your finger straight forward. "This is the Real path."
2. Turn your finger 90 degrees to point to the ceiling. "This is the Spirit path."
3. "In math, the 'i' is the turn that lets us see the ceiling."
The Lesson: "Sometimes we get stuck on the ground, and God uses a 'Spirit Turn' to help us look up."
Response: ___________________________________________________________