Section C — Imaginary Numbers and the Complex Plane
Learning Objectives
Understand that the symbol i means a particular square root of −1, and that an imaginary number is a real number times i.
Represent solutions to equations using i and the complex plane.
Activity 1 — Activity Narrative
Math Talk: Squaring Products with Square Roots
This Math Talk focuses on squaring numbers written as the product of an integer and a square root. It encourages students to think about how a square is applied to a product and to rely on what they know about exponents to mentally solve the problems.
The key idea to surface: (a · b)² = a² · b² — the square distributes across a product. Students use this to square each expression mentally.
The understanding built here — especially with the last two expressions involving √(−1) — will be critical when students are formally introduced to i in Activity 2.
MP8: Look for repeated reasoning. What pattern do students notice across all four expressions?
Warm-Up — Math Talk
Evaluate each expression mentally.
(2√(3))²
(12√(3))²
(2√(−1))²
(12√(−1))²
Split the square across the product: (a · b)² = a² · b²
Warm-Up — Math Talk Results
(2√(3))²
=
2² · (√(3))²
=
4 · 3
=
12
(12√(3))²
=
(12)² · (√(3))²
=
14 · 3
=
34
(2√(−1))²
=
4 · (√(−1))²
=
4 · (−1)
=
−4
(12√(−1))²
=
14 · (−1)
=
−14
The last two give negative results — what is √(−1)?
Activity 1 — Synthesis
Discussion: Squaring with Square Roots
Goal: Review students' strategies for squaring expressions involving square roots.
"Who can restate [a classmate]'s reasoning in a different way?"
"Did anyone solve the problem in a different way?"
"What pattern do you see across all four expressions?"
"What does (√(−1))² equal? How do you know?"
"Does it bother you that √(−1) doesn't seem to be a real number?"
MLR8 — Display sentence frames: "First, I ___ because ___." and "I noticed ___, so I ___."
Activity 2 — Activity Narrative
Solving x² = k and the Complex Plane
Students are formally introduced to the symbol i used to represent the imaginary unit — the number we had been calling √(−1). Students solve quadratic equations of the form x² = k for various integers k, and plot their solutions in the complex plane.
Key idea: Just as positive numbers have two real square roots, negative numbers have two imaginary square roots. Solutions lie on the imaginary axis, not the real axis.
The complex plane is introduced as the coordinate system for visualizing complex numbers. Each point represents a single number, not a pair.
Concept — Introducing i
No real number satisfies x² = −1.
We define a new number:
i
i² = −1
Both i and −i are square roots of −1:
(i)² = −1 ✓
(−i)² = (−1)² · i² = 1 · (−1) = −1 ✓
Just like √(4) = ±2, we get x = ±i as the two square roots of −1.
Vocabulary — Math As A Second Language
Symbol
Spoken Phrase
imaginary unit
i
"the imaginary unit"
imaginary number
bi
"b i" · "b times i"
An imaginary number is any real number multiplied by i. Examples: 3i −7i √(5) · i23i
Concept — Solving x² = k
k > 0 — positive
Two real solutions (on real axis)
x² = 4
→
x = ±2
x² = 25
→
x = ±5
x² = 35
→
x = ±√(35)
k < 0 — negative
Two imaginary solutions (on imaginary axis)
x² = −1
→
x = ±i
x² = −4
→
x = ±2i
x² = −25
→
x = ±5i
Positive k → ±√(k) (real). Negative k → ±√(|k|) · i (imaginary).
Practice — Solving Equations
1
Solve: a² = 16
2
Solve: b² = −9
3
Solve: c² = −5
Write both solutions. Which are real? Which are imaginary?
Activity 2 — Synthesis
Two Square Roots of Every Number
Goal: Highlight that negative real numbers have two imaginary square roots — one on the positive imaginary axis and one on the negative imaginary axis.
"What do you notice about solutions to x² = k when k is positive?"
"What if k is negative? Where do the solutions live?"
"How is the complex plane the same as a coordinate plane? How is it different?"
"Can a number be both real and imaginary? What about 0?"
Display: If k > 0, solutions are ±√(k) on real axis. If k < 0, solutions are ±√(|k|) · i on imaginary axis.
Activity 3 — Activity Narrative
Convention: √(−a) = √a · i
In this activity, students learn and apply the convention that for any positive real number a:
√(−a) = √a · i
This will be essential later in the unit when students use the quadratic formula and encounter terms like √(−36).
Writing convention to emphasize: Write i after a rational coefficient (3i, not i3). When i is multiplied by a radical, write √a · i, not √(a · i) — do not extend the radical symbol over the i.
Convention — Simplifying √(−a)
For any positive real a: √(−a) = √(a) · i
√(−4)
= √(4) · i = 2i
√(−10)
= √(10) · i
−√(−100)
= −10i
√(−25) = 5i,
check: (5i)·(5i) = 25i² = −25 ✓
√(−17)
= √(17) · i
Write iafter a number: 3i ✓ i3 ✗ | Write ioutside the radical: √(10) · i ✓ √(10i) ✗
Practice — Imaginary Arithmetic
6
Write −√(−49) as an imaginary number using i.
7
Write (5i)² as either an integer or an integer multiple of i.
For #7: use the same squaring strategy from the Math Talk.
Activity 3 — Synthesis
Writing Imaginary Numbers Correctly
Select students to share responses. Ask them to show their answers make sense by squaring.
"Show that your answer to #7 is correct by squaring it."
"Why do we write 3i and not i3?"
"Why does √10 · i look different from √(10i)? Which is our answer?"
"When would you need this convention? (Hint: quadratic formula, upcoming unit.)"
Activity 4 — Activity Narrative
Complex Numbers and the Complex Plane
Students combine real and imaginary numbers with addition to form complex numbers of the form a + bi, then plot them in the complex plane.
Crucial distinction: In the coordinate planes students have used before, each point represents a pair of real numbers like (−8, 7). In the complex plane, each point represents a single complex number like −8 + 7i.
The real axis (horizontal) and imaginary axis (vertical) together form the complex plane. Real numbers like 5 sit on the real axis (5 + 0i). Imaginary numbers like −3i sit on the imaginary axis.