Complex Plane
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Introduction to the Complex Plane
Definition:
The complex plane (also called the Argand plane or Argand diagram) is a geometric representation of complex numbers as points in a two-dimensional coordinate system
Components:
• Horizontal Axis (x-axis): Real axis - represents real part
• Vertical Axis (y-axis): Imaginary axis - represents imaginary part
• Point (a, b): Represents complex number \( z = a + bi \)
2. Graph Complex Numbers
How to Plot:
For \( z = a + bi \):
1. Move \( a \) units along the real axis (horizontal)
2. Move \( b \) units along the imaginary axis (vertical)
3. Plot the point at (a, b)
Examples:
| Complex Number | Point on Plane | Location |
|---|---|---|
| \( 3 + 4i \) | (3, 4) | Quadrant I |
| \( -2 + 5i \) | (-2, 5) | Quadrant II |
| \( -3 - 2i \) | (-3, -2) | Quadrant III |
| \( 4 - 3i \) | (4, -3) | Quadrant IV |
| \( 5 \) (pure real) | (5, 0) | On real axis |
| \( 3i \) (pure imaginary) | (0, 3) | On imaginary axis |
3. Addition in the Complex Plane
Geometric Interpretation:
Adding complex numbers can be visualized using the parallelogram rule (similar to vector addition)
For \( z_1 = a + bi \) and \( z_2 = c + di \):
\[ z_1 + z_2 = (a + c) + (b + d)i \]
Plotted at point \( (a + c, b + d) \)
Parallelogram Method:
1. Plot both complex numbers as vectors from the origin
2. Form a parallelogram with these vectors as adjacent sides
3. The diagonal from the origin represents the sum
Example:
Add: \( (2 + 3i) + (4 + i) \)
Real parts: \( 2 + 4 = 6 \)
Imaginary parts: \( 3 + 1 = 4 \)
Result: \( 6 + 4i \) plotted at (6, 4)
4. Subtraction in the Complex Plane
Formula:
Subtraction is equivalent to adding the additive inverse (opposite)
For \( z_1 = a + bi \) and \( z_2 = c + di \):
\[ z_1 - z_2 = (a - c) + (b - d)i \]
Geometric Interpretation:
1. Rewrite as addition: \( z_1 - z_2 = z_1 + (-z_2) \)
2. Plot \( -z_2 \) (opposite of \( z_2 \))
3. Use parallelogram method to add \( z_1 + (-z_2) \)
Example:
Subtract: \( (5 + 6i) - (2 + 3i) \)
Real parts: \( 5 - 2 = 3 \)
Imaginary parts: \( 6 - 3 = 3 \)
Result: \( 3 + 3i \) plotted at (3, 3)
5. Graph Complex Conjugates
Definition:
The conjugate of \( z = a + bi \) is \( \overline{z} = a - bi \)
Geometric Property:
• Complex conjugates are reflections of each other across the real axis
• They have the same real part
• Their imaginary parts are opposites
Examples:
| Complex Number z | Conjugate \( \overline{z} \) | Relationship |
|---|---|---|
| \( 3 + 4i \) at (3, 4) | \( 3 - 4i \) at (3, -4) | Reflected over x-axis |
| \( -2 + 5i \) at (-2, 5) | \( -2 - 5i \) at (-2, -5) | Reflected over x-axis |
| \( 5 \) at (5, 0) | \( 5 \) at (5, 0) | On real axis (same point) |
6. Absolute Value in the Complex Plane
Geometric Meaning:
The absolute value (modulus) of a complex number is its distance from the origin
\[ |z| = |a + bi| = \sqrt{a^2 + b^2} \]
This is the length of the vector from the origin (0, 0) to point (a, b)
Example:
Find \( |3 + 4i| \) on the complex plane
Point: (3, 4)
Distance from origin: \( \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
The point is 5 units from the origin
7. Distance in the Complex Plane
Distance Formula:
The distance between two complex numbers is the distance between their corresponding points
For \( z_1 = a + bi \) and \( z_2 = c + di \):
\[ d = |z_1 - z_2| = \sqrt{(a - c)^2 + (b - d)^2} \]
This is identical to the distance formula in coordinate geometry
Alternative Formula:
\[ d = |z_1 - z_2| \]
The distance is the absolute value of the difference
Example:
Find distance between \( z_1 = 3 + 2i \) and \( z_2 = 7 + 5i \)
Points: (3, 2) and (7, 5)
Difference: \( z_1 - z_2 = (3 - 7) + (2 - 5)i = -4 - 3i \)
Distance: \( d = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} \)
Distance = 5 units
8. Midpoints in the Complex Plane
Midpoint Formula:
The midpoint between two complex numbers is found by averaging their real and imaginary parts
For \( z_1 = a + bi \) and \( z_2 = c + di \):
\[ M = \frac{z_1 + z_2}{2} = \frac{a + c}{2} + \frac{b + d}{2}i \]
Plotted at point \( \left(\frac{a+c}{2}, \frac{b+d}{2}\right) \)
Steps:
1. Average the real parts: \( \frac{a + c}{2} \)
2. Average the imaginary parts: \( \frac{b + d}{2} \)
3. Combine: \( M = \frac{a+c}{2} + \frac{b+d}{2}i \)
Examples:
Find midpoint between \( z_1 = 2 + 4i \) and \( z_2 = 6 + 10i \)
Real part: \( \frac{2 + 6}{2} = \frac{8}{2} = 4 \)
Imaginary part: \( \frac{4 + 10}{2} = \frac{14}{2} = 7 \)
Midpoint: \( M = 4 + 7i \) at (4, 7)
Find midpoint between \( z_1 = -3 + 5i \) and \( z_2 = 7 - 3i \)
Real part: \( \frac{-3 + 7}{2} = \frac{4}{2} = 2 \)
Imaginary part: \( \frac{5 + (-3)}{2} = \frac{2}{2} = 1 \)
Midpoint: \( M = 2 + i \) at (2, 1)
9. Quick Reference Summary
Essential Formulas:
Plot \( z = a + bi \) at point (a, b)
Conjugate: \( \overline{z} = a - bi \) (reflect over real axis)
Absolute Value: \( |z| = \sqrt{a^2 + b^2} \)
Distance: \( d = |z_1 - z_2| = \sqrt{(a-c)^2 + (b-d)^2} \)
Midpoint: \( M = \frac{z_1 + z_2}{2} = \frac{a+c}{2} + \frac{b+d}{2}i \)
Key Points:
• Real axis = horizontal (x-axis)
• Imaginary axis = vertical (y-axis)
• Operations work like coordinate geometry
📚 Study Tips
✓ Complex plane is like coordinate plane: (real, imaginary) instead of (x, y)
✓ Conjugates are mirror images across the real (horizontal) axis
✓ Absolute value = distance from origin (use Pythagorean theorem)
✓ Distance formula same as coordinate geometry
✓ Midpoint = average of real parts + average of imaginary parts