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Algebra 2 · Unit 4 · Lesson 12
Arithmetic with Complex Numbers
Section C — Imaginary Numbers and the Complex Plane
Learning Objectives
- Determine powers of imaginary numbers and represent them on the complex plane.
- Add and subtract complex numbers and write the result in the form a + bi.
- Use the complex plane to visualize why the sum of two complex numbers is another complex number.
Warm-Up — Co-Craft Questions
What questions do you have?
You will see a diagram of a complex plane with a path of arrows.
Your task: Write at least two questions about what you see. Your questions will be discussed as a class.
Think about: What do the arrows represent? Where does the path start? Where does it end?
Math Language Routine: Co-Craft Questions — listen for language like complex plane, sum, and complex number.
I notice ___ . I wonder ___ .
Visual — Complex Plane
A path on the complex plane
The arrows represent steps from one complex number to the next. What complex number does point A represent?
The path starts at 0. Each arrow represents adding one complex number.
Visual / MASL
The standard form of a complex number
Meaning
●
The standard form for any complex number. a and b are real numbers. i equals √(−1).
Examples: 2 + 3i · −8 − 8i · 0 + 5i · 7 + 0i
Visual / MASL
Inside a complex number: two parts
a
+
bi
A real number — plotted on the horizontal axis
A real number b times i — plotted on the vertical axis
In 2 + 3i: the real part is 2 and the imaginary part is 3i.
Concept
Adding complex numbers
Add the real parts. Add the imaginary parts.
Rule
(
a +
bi)
+
(
c +
di)
=
(
a + c) + (
b + d)
i
Think of i like a variable — combine like terms.
a and c are both real, so they combine. bi and di both have an i, so they combine.
(2 + 3i) + (4 + 5i) = (2+4) + (3+5)i = 6 + 8i
Worked Example
(2 + 3i) + (−8 − 8i)
1
Write out the expression
(2 + 3i) + (−8 − 8i)
2
Group the real parts
2 + (−8) + 3i + (−8i)
3
Combine each group
−6 + (−5)i
✓
Answer in a + bi
−6 − 5i
Blue = real parts combine · Teal = imaginary parts combine
Concept
Subtracting complex numbers
Subtract the real parts. Subtract the imaginary parts.
Rule
(
a +
bi)
−
(
c +
di)
=
(
a − c) + (
b − d)
i
(2 + 3i) − (4 + 5i) = (2−4) + (3−5)i = −2 − 2i
Distribute the subtraction sign across both parts of the second complex number.
Concept
Powers of 2i — write each in the form a + bi
Use i² = −1
(2i)²
= 4i² = 4(−1)
−4 + 0i
(2i)³
= 8i³ = 8(−i)
0 − 8i
(2i)⁴
= 16i⁴ = 16(1)
16 + 0i
On the complex plane, each power rotates 90° counterclockwise and doubles the distance from 0.
Visual — Complex Plane
Powers of 2i rotate around the origin
Each power of 2i:
↺ rotates 90° counterclockwise
× 2 the distance from 0
The points land on the real and imaginary axes, alternating with each power.
This is why i² = −1: multiplying by i twice = 180° rotation = landing on the negative real axis.
Practice
Write each sum or difference in the form a + bi
B
(−37 − 45i) + (11 + 81i)
D
(−37 − 45i) − (11 + 81i)
For problem A: draw a diagram on the complex plane to check your answer.
Answers
Check your work
A
(−3 + 2i) + (4 − 5i) =
(−3+4) + (2−5)i = 1 − 3i
B
(−37 − 45i) + (11 + 81i) =
−26 + 36i
C
(−3 + 2i) − (4 − 5i) = (−3−4) + (2−(−5))i =
−7 + 7i
D
(−37 − 45i) − (11 + 81i) =
−48 − 126i
Synthesis — Class Discussion
Is addition of complex numbers commutative?
(2 + 3i) + (−8 − 8i)
vs.
(−8 − 8i) + (2 + 3i)
- What does each expression equal? Are they the same?
- On the complex plane: going right 2 then left 8 — is that the same as going left 8 then right 2?
- Use the phrases real part and imaginary part in your explanation.
Adding ___ and ___ gives ___ because the real parts ___ and the imaginary parts ___ .
Concept — Summary
The general rules
(a + bi) + (c + di)
= (a + c) + (b + d)i
(a + bi) − (c + di)
= (a − c) + (b − d)i
In both cases: real parts combine with real parts, imaginary parts combine with imaginary parts.
Cool-Down
Write in the form a + bi
−(3i)³
Answer:
Hint: Start by computing (3i)³. Then apply the negative sign.
Answer key: −(3i)³ = −(27i³) = −(27 · (−i)) = 27i → 0 + 27i