HSN.CN.B.5Common CoreMathNumber and QuantityGrades 9-12
HSN.CN.B.5: Complex Number Operations on the Complex Plane
In plain English: HSN.CN.B.5 is an advanced (+) Common Core number and quantity standard, usually taught in Precalculus. Students show addition and subtraction of complex numbers as arrows and parallelograms, multiplication as a rotation combined with a stretch (moduli multiply, arguments add), and conjugation as a reflection across the real axis, then use these facts to compute products and powers.
(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.
Common Core State Standards for Mathematics · Domain: The Complex Number System (CN) · Cluster: Represent complex numbers and their operations on the complex plane. Also written as HSN-CN.B.5 or N-CN.5 · Official standard
Students learn to see each operation on complex numbers as a motion of the complex plane. Adding w slides every point by the arrow for w, and the sum z + w is the far corner of the parallelogram built on z and w. Subtracting w slides the other way, and z - w is the arrow that runs from w to z. Taking the conjugate reflects a point across the real axis.
Multiplication is the surprising case: multiplying by w stretches distances from 0 by the factor |w| and turns every point by the angle arg w. In polar form this reads as "multiply the moduli, add the arguments." Students use that rule to compute products and powers quickly, including the official example (-1 + √3 i)³ = 8, and they check each result in rectangular form.
Learning Objectives
By the end of this lesson, students will be able to:
Represent the sum and the difference of two complex numbers with arrows and a parallelogram on the complex plane
Describe multiplication by a complex number w as a rotation by arg w combined with a dilation by |w|
Represent the conjugate of a complex number as its reflection across the real axis and explain why z + z̄ and z·z̄ are real
Use the modulus and argument rules to compute products and powers of complex numbers, and verify results in rectangular form
Prior Knowledge Required
Students should already be comfortable with:
Adding, subtracting and multiplying complex numbers with i² = -1 HSN.CN.A.2
Finding conjugates and moduli of complex numbers HSN.CN.A.3
Writing complex numbers in polar form with a modulus and an argument HSN.CN.B.4
Translations, rotations, reflections and dilations in the coordinate plane HSG.CO.A.2
Students work with a partner on grid paper. The warm-up previews the idea that multiplication moves points in a regular way.
Warm-Up Prompt
"Let z = 3 + 2i. Compute i·z, then multiply your answer by i again, and once more. Plot z and the three results. What motion takes each point to the next one?"
The results are -2 + 3i, -3 - 2i and 2 - 3i. Students usually notice that the points sit at the same distance from 0 and turn a quarter turn counterclockwise each time. Ask a pair to measure the angle between the arrows for z and i·z with a protractor (90°). Tell students that by the end of the lesson they will be able to predict the motion for multiplication by any complex number, not only by i.
Direct Instruction25 minutes
Treat each complex number as a point and as an arrow from 0 to that point. Introduce one operation at a time and, for each, ask: "What does this operation do to the plane?"
Addition is a translation. Place the arrow for w at the tip of the arrow for z. The sum lands where the second arrow ends, which is the fourth vertex of the parallelogram with vertices 0, z and w. Adding the parts matches this: (a + bi) + (c + di) = (a + c) + (b + d)i.
Subtraction is the opposite translation. z - w = z + (-w), and -w is w turned 180° about 0. The difference z - w is also the arrow from w to z, moved so that it starts at 0: the second diagonal of the same parallelogram.
Conjugation is a reflection. The conjugate of z = a + bi is z̄ = a - bi, the mirror image of z across the real axis. It has the same modulus and the opposite argument. So z + z̄ = 2a lies on the real axis, and z·z̄ has argument θ + (-θ) = 0, which makes it the positive real number |z|².
Multiplication is a rotation and a dilation. If z = r(cos α + i sin α) and w = s(cos β + i sin β), then z·w = rs(cos(α + β) + i sin(α + β)). Multiplying by w stretches by the factor s and turns by β. Multiplying by i (modulus 1, argument 90°) is a quarter turn, which explains the warm-up.
Powers repeat the motion. zⁿ has modulus rⁿ and argument nα. This is the rule behind the official example.
To justify step 4, expand (cos α + i sin α)(cos β + i sin β) = (cos α cos β - sin α sin β) + i(sin α cos β + cos α sin β) and name the angle-sum identities for cosine and sine. Use Diagram 1 for steps 1 and 2 and Diagram 2 for steps 3 to 5. Then work the examples below, checking each one in rectangular form.
Addition as a parallelogram
Add z = 3 + i and w = 1 + 2i and draw the parallelogram with vertices 0, z and w.
Equation: z + w = 4 + 3i, the fourth vertex of the parallelogram
Subtraction as the arrow from w to z
For the same z and w, find z - w and compare it with the arrow from w to z.
Equation: z - w = 2 - i; the arrow from 1 + 2i to 3 + i also moves 2 right and 1 down
Multiplication in polar form
Multiply 2(cos 30° + i sin 30°) by 3(cos 60° + i sin 60°), then check with the rectangular forms √3 + i and 3/2 + (3√3/2)i.
Equation: 6(cos 90° + i sin 90°) = 6i, and (√3 + i)(3/2 + (3√3/2)i) = 6i
Conjugation as a reflection
Reflect z = -3 + 2i across the real axis, then find z + z̄ and z·z̄.
Compute (-1 + √3 i)³ using modulus 2 and argument 120°.
Equation: 2³ = 8 and 3 · 120° = 360°, so (-1 + √3 i)³ = 8(cos 360° + i sin 360°) = 8
Verify the official example the long way to build trust in the rule: (-1 + √3 i)² = 1 - 2√3 i + 3i² = -2 - 2√3 i, and (-2 - 2√3 i)(-1 + √3 i) = 2 - 2√3 i + 2√3 i - 6i² = 2 + 6 = 8. Diagram 2 shows the three points: each factor of z doubles the distance and adds 120°.
Guided Practice15-20 minutes
Pairs draw each operation on grid paper before computing it, then confirm the picture with algebra:
Guided practice: draw, then compute
Operation
Geometric description
Result
(-2 + 3i) + (4 - i)
Parallelogram on the two arrows
2 + 2i
(5 + i) - (2 + 4i)
Arrow from 2 + 4i to 5 + i
3 - 3i
i(4 + i)
Quarter turn counterclockwise
-1 + 4i
Conjugate of 2 - 5i
Reflection across the real axis
2 + 5i
2(cos 50° + i sin 50°) · 4(cos 25° + i sin 25°)
Stretch by 4, turn 25°
8(cos 75° + i sin 75°)
Listen for students who multiply the arguments instead of adding them, and for students who add the moduli. Ask them to test their rule on i · i, whose answer they know is -1.
Independent Practice15 minutes
Students complete six items, each with a sketch: (1) (1 + 4i) + (3 - 2i) = 4 + 2i; (2) (-1 + 2i) - (3 + i) = -4 + i; (3) 3(cos 100° + i sin 100°) · 2(cos 140° + i sin 140°) = 6(cos 240° + i sin 240°) = -3 - 3√3 i; (4) -i(2 + 5i) = 5 - 2i, a quarter turn clockwise; (5) the conjugate of -4 - i is -4 + i, and their product is 17; (6) (1 - i)⁶ = 8i, because √2 to the sixth power is 8 and 6 · (-45°) = -270°, which points the same way as 90°.
Closure5 minutes
Exit ticket: (1) Describe what multiplying every point of the plane by 2i does. (Answer: it doubles every distance from 0 and turns every point 90° counterclockwise.) (2) Use modulus and argument to compute (√3 + i)³. (Answer: modulus 8, argument 90°, so 8i.)
Differentiation Strategies
For Struggling Students
Give a four-row reference card: adding slides, subtracting slides back, taking the conjugate flips over the real axis, multiplying turns and stretches
Use patty paper: trace a figure, then physically slide, flip or turn it to predict each result before computing
Start multiplication with factors of modulus 1 (i, -1, -i) so students see pure rotations before stretches
For Advanced Students
Ask students to explain geometrically why |z + w| ≤ |z| + |w|, using the parallelogram
Ask what multiplying by 1/w does to the plane, and use it to describe division in polar form
Challenge: find every complex number z with |z| = 1 whose sixth power is 1, and plot them (this goes beyond the standard)
Assessment Guidance
What to Look For
Look for sketches that match the algebra: sums at the fourth vertex of a parallelogram, differences drawn from w to z, conjugates mirrored across the real axis. For products, check that students add arguments and multiply moduli, and that they can reduce an argument such as 405° to 45°. Ask each student to verify at least one polar product in rectangular form.
02
Classroom Activities
3 Activities
1
Arrow Walk: Sums and Differences
15 minPairs
Pairs draw each sum or difference tip-to-tail on grid paper, read the answer from the picture, and then confirm it by adding or subtracting the parts.
The 6 Cards
(2 + 3i) + (4 - i)
(-3 + i) + (2 + 4i)
(5 - 2i) + (-5 + 2i)
(6 + 2i) - (2 + 5i)
(-1 - 3i) - (2 - i)
4i - 3
Procedure
For a sum, draw the first arrow from 0, then draw the second arrow starting at its tip. Mark where you end
For a difference, draw both arrows from 0 and then the arrow from the second number to the first. Slide that arrow so it starts at 0 and read its tip
Check with the parts. The answers are 6 + 2i, -1 + 5i, 0, 4 - 3i, -3 - 2i and -3 + 4i
Discussion Questions
Why does the third card land exactly at 0? What do the two arrows look like?
For the fourth card, how does the arrow from 2 + 5i to 6 + 2i compare with the arrow for 4 - 3i?
2
Rotate and Stretch a Triangle
20 minGroups of 3
Groups multiply every vertex of a triangle by a complex number and describe the motion, then take conjugates of the vertices. Patty paper and a protractor let them measure the turn and the stretch.
Procedure
Plot triangle T with vertices 1, 2 + i and 1 + 2i
Multiply each vertex by 1 + i to get 1 + i, 1 + 3i and -1 + 3i. Measure: the new triangle is turned 45° about 0 and each side is √2 times as long
Multiply each vertex of T by 2i to get 2i, -2 + 4i and -4 + 2i. Measure the turn (90°) and the stretch (factor 2)
Take the conjugate of each vertex of T to get 1, 2 - i and 1 - 2i, and describe the result as a reflection
Discussion Questions
How could you predict the turn and the stretch from the modulus and argument of 1 + i and of 2i?
Trace the vertices of T in order. Do the images after multiplication go around the same way? After conjugation?
Modification for Distance Learning
Use a free graphing tool with sliders for a multiplier w = s(cos β + i sin β). Students move the sliders and watch the triangle turn and grow, then record three settings and the vertices they produce.
3
Power Spiral
15-20 minSmall groups
Groups use the modulus and argument rules to compute a sequence of powers without expanding, plot them, and see the spiral that repeated multiplication traces.
Procedure
Let z = 1 + i, with modulus √2 and argument 45°. For n = 1 to 8, write zⁿ as (√2)ⁿ(cos 45n° + i sin 45n°)
Convert each power to rectangular form: 1 + i, 2i, -2 + 2i, -4, -4 - 4i, -8i, 8 - 8i, 16
Plot the eight points and connect them in order to see the spiral
Check z² = 2i and z⁴ = -4 by multiplying out
Extension Variation
Return to the official example with z = -1 + √3 i. Continue past z³ = 8 to z⁶ = 64. Then use the conjugate: since conjugation flips the argument, the conjugate of zⁿ equals (z̄)ⁿ, so (-1 - √3 i)³ is also 8. Groups explain why.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Adding and Subtracting on the Complex Plane
The sum z + w = 4 + 3i is the fourth vertex of the parallelogram with vertices 0, z and w. The difference z - w = 2 - i is the green arrow from 0; it is parallel to and as long as the dashed arrow from w to z. Drawn to scale, one grid square is one unit.
Diagram 2: Powers of -1 + √3 i and Its Conjugate
z = -1 + √3 i has modulus 2 and argument 120°. Squaring gives modulus 4 and argument 240°, and cubing gives modulus 8 and argument 360°, which is the real number 8. The conjugate -1 - √3 i is the reflection of z across the real axis. It lies on the same ray as z², since -120° and 240° are the same direction. Axis labels every 2 units; drawn to scale.
04
Homework Assignment
~30 min
HSN.CN.B.5 Homework: Operations on the Complex Plane
Directions: Draw a complex plane sketch for every problem and label the points. Give exact answers. For each product or power, describe the motion (turn and stretch) and check your answer in rectangular form.
Part 1: Addition and Subtraction (Problems 1-2)
Let z = -2 + i and w = 5 + 3i. Draw the parallelogram with vertices 0, z and w, and find its fourth vertex. Explain why that vertex is z + w, and compute z + w.
For the same z and w, draw the arrow from w to z and the arrow from z to w. Use them to find z - w and w - z, and explain how the two answers are related on the plane.
Part 2: Multiplication and Conjugation (Problems 3-4)
Multiply z = 3(cos 20° + i sin 20°) by w = 2(cos 115° + i sin 115°). Write the product in polar and rectangular form, and describe how multiplying by w moves the point z.
Let z = 5 - 2i. Plot z and its conjugate. Find z + z̄ and z·z̄, and explain with the picture why both results are real numbers.
Part 3: Computing with the Properties (Problems 5-6)
Use the modulus and the argument of √3 - i to compute (√3 - i)⁶ without expanding. Show the modulus and argument at each step.
Compute (-2 + 2i)³ with modulus and argument, and reduce the argument to an angle between 0° and 360°. Then check your answer by squaring -2 + 2i and multiplying the result by -2 + 2i.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Sketches
Each operation drawn correctly and labeled
Most sketches correct
Sketches missing
Geometric Description
Names the translation, reflection, or turn and stretch correctly
Description incomplete
No description
Computation
All results exact and correct in both forms
One arithmetic or argument error
Most results incorrect
Checking
Polar results confirmed in rectangular form
Some results checked
No checks
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which complex number is the fourth vertex of the parallelogram with vertices 0, 3 - 4i and -1 + 7i?
Answer: B
The fourth vertex is the sum (3 - 4i) + (-1 + 7i) = 2 + 3i. Choice A is the difference (3 - 4i) - (-1 + 7i), which is the other diagonal. Choice D adds the imaginary parts with the wrong sign on -4i.
Question 2 of 20 · Multiple Choice
Every point of a figure on the complex plane is replaced by the point plus 2 + 5i. What happens to the figure?
Answer: A
Adding a fixed number adds 2 to every real part and 5 to every imaginary part, which is a translation. Choice C describes multiplication by a number with modulus √29, not addition. Choice D describes conjugation.
Question 3 of 20 · Multiple Choice
Points z and w are plotted on the complex plane. Which arrow, moved to start at 0, represents z - w?
Answer: C
Since w + (z - w) = z, the difference is the arrow that takes w to z. Choice B points the other way and represents w - z. Choice A is the sum, the other diagonal of the parallelogram.
Question 4 of 20 · Multiple Choice
Compute (6 + i) - (2 + 3i).
Answer: D
Subtract the parts: (6 - 2) + (1 - 3)i = 4 - 2i. Choice A adds the numbers. Choice B computes (2 + 3i) - (6 + i), the arrow in the opposite direction. Choice C subtracts only the real parts and adds the imaginary parts.
Question 5 of 20 · Multiple Choice
Multiplying by i turns a point 90° counterclockwise about 0. What is i(5 + 2i)?
Answer: A
i(5 + 2i) = 5i + 2i² = -2 + 5i. The point (5, 2) turned a quarter turn counterclockwise lands at (-2, 5). Choice B is a quarter turn clockwise, which is multiplication by -i. Choice C forgets that i² = -1.
Question 6 of 20 · Multiple Choice
Find the product of 5(cos 80° + i sin 80°) and 2(cos 130° + i sin 130°).
Answer: C
Multiply the moduli and add the arguments: 5 · 2 = 10 and 80° + 130° = 210°. Choice A adds the moduli. Choice B subtracts the arguments, which describes division. Choice D multiplies the arguments.
Question 7 of 20 · Multiple Choice
What does multiplying every point of the plane by 2(cos 45° + i sin 45°) do?
Answer: D
Multiplying by a number with modulus 2 and argument 45° multiplies every modulus by 2 and adds 45° to every argument. Choice A describes adding the number, not multiplying by it. Choice B doubles the angle as well as the distance.
Question 8 of 20 · Multiple Choice
Which transformation takes every complex number z to its conjugate z̄?
Answer: B
a + bi becomes a - bi: the real part stays and the imaginary part changes sign, which is a reflection across the real axis. Choice A would give -a + bi. Choice C would give -a - bi, which is -z.
Question 9 of 20 · Multiple Choice
Where is the conjugate of -6 + 5i plotted?
Answer: C
The conjugate is -6 - 5i, so it is plotted at (-6, -5), the mirror image of (-6, 5) across the real axis. Choice A reflects across the imaginary axis instead. Choice B is -z, a half turn about 0.
Question 10 of 20 · Multiple Choice
For z = 2 + 7i, what is z·z̄?
Answer: D
(2 + 7i)(2 - 7i) = 4 - 49i² = 4 + 49 = 53, which is |z|². The arguments θ and -θ add to 0, so the product lies on the positive real axis. Choice A uses i² = +1. Choice C adds the parts, and choice B puts the product on the imaginary axis.
Question 11 of 20 · Multiple Choice
Use modulus and argument to find (1 + √3 i)³.
Answer: B
1 + √3 i has modulus 2 and argument 60°. Cubing gives modulus 8 and argument 180°, so the result is 8(cos 180° + i sin 180°) = -8. Choice A uses the argument 120° from the official example, which belongs to -1 + √3 i, not 1 + √3 i.
Question 12 of 20 · Multiple Choice
A complex number z has modulus 3 and argument 40°. Which is z³?
Answer: C
z³ has modulus 3³ = 27 and argument 3 · 40° = 120°. Choice A multiplies the modulus by 3 instead of cubing it. Choice B adds 3 to the argument. Choice D leaves the modulus unchanged.
Question 13 of 20 · Multiple Choice
The argument of z is 25°. What is an argument of z̄?
Answer: A
Reflecting across the real axis turns the angle 25° above the axis into 25° below it, so arg z̄ = -25°, or 335°. Choice B reflects across the imaginary axis. Choice C is the argument of -z.
Question 14 of 20 · Multiple Choice
Using |zw| = |z|·|w|, what is |(3 + 4i)(5 - 12i)|?
Answer: B
|3 + 4i| = 5 and |5 - 12i| = 13, so the product has modulus 5 · 13 = 65. Multiplying out gives 63 - 16i, and √(63² + 16²) = √4225 = 65. Choice C adds the moduli. Choice A adds the two numbers instead of multiplying them.
Question 15 of 20 · Short Answer
Let z = 1 + 5i and w = 4 - 2i. Find z + w and z - w, and describe where each one appears in the parallelogram with vertices 0, z and w.
z + w = 5 + 3i is the fourth vertex, at the end of the diagonal from 0. z - w = -3 + 7i is the other diagonal, the arrow from w to z, moved so that it starts at 0.
Question 16 of 20 · Short Answer
Write 1 + i and √3 + i in polar form and use the forms to find the modulus and argument of (1 + i)(√3 + i). Then multiply in rectangular form to check.
1 + i = √2(cos 45° + i sin 45°) and √3 + i = 2(cos 30° + i sin 30°). The product has modulus 2√2 and argument 75°. Rectangular: (1 + i)(√3 + i) = √3 + i + √3 i + i² = (√3 - 1) + (√3 + 1)i. Check: (√3 - 1)² + (√3 + 1)² = 8, so the modulus is √8 = 2√2, and both parts are positive, as an argument of 75° requires.
Question 17 of 20 · Short Answer
Let z = -3 + 4i. Plot z and z̄, then compute z + z̄ and z - z̄. On which axis does each result lie, and why?
z̄ = -3 - 4i, the reflection of z across the real axis. z + z̄ = -6 lies on the real axis because the imaginary parts cancel. z - z̄ = 8i lies on the imaginary axis because the real parts cancel. In the picture, the arrows for z and z̄ are mirror images, so their sum has no vertical part and their difference has no horizontal part.
Question 18 of 20 · Short Answer
Use modulus and argument to compute (√2 - √2 i)⁵. Give the answer in rectangular form.
√2 - √2 i has modulus 2 and argument -45°. The fifth power has modulus 2⁵ = 32 and argument -225°, which is the same direction as 135°. (√2 - √2 i)⁵ = 32(cos 135° + i sin 135°) = -16√2 + 16√2 i.
Question 19 of 20 · Short Answer
A complex number z has modulus 2 and argument 150°. Find z² and z⁴ in polar and rectangular form, and describe the motion from z to z².
z² = 4(cos 300° + i sin 300°) = 2 - 2√3 i. z⁴ = 16(cos 600° + i sin 600°) = 16(cos 240° + i sin 240°) = -8 - 8√3 i. Going from z to z² multiplies by z itself, so the point turns another 150° and its distance from 0 doubles, from 2 to 4.
Question 20 of 20 · Short Answer
Point A is 2 + 6i and point B is 7 + 3i. What complex number, added to A, moves it to B? Explain how you found it and show it on the plane.
We need A + d = B, so d = B - A = (7 + 3i) - (2 + 6i) = 5 - 3i. On the plane, d is the arrow from A to B: 5 units right and 3 units down. Drawn from 0, it ends at the point (5, -3).
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSN.CN.B.5 mean?
It means students can show each basic operation on complex numbers as a picture on the complex plane and then use the picture to compute. Addition and subtraction are translations, conjugation is a reflection across the real axis, and multiplication is a rotation combined with a stretch. The official example uses these facts to show that (-1 + √3 i)³ = 8.
Which course covers HSN.CN.B.5?
It is usually part of Precalculus. HSN.CN.B.5 is a (+) standard, the Common Core label for additional mathematics meant for students who go on to advanced courses. Students need polar form (HSN.CN.B.4) and unit circle trigonometry before the multiplication part makes sense.
Why does multiplying complex numbers rotate them?
Because the arguments add. If w has argument β, every product z·w has the argument of z plus β, so every point turns by β about 0. At the same time the moduli multiply, so every distance from 0 is scaled by |w|. The angle-sum identities for sine and cosine are what make the arguments add.
What does multiplying by i do on the complex plane?
It turns every point a quarter turn (90°) counterclockwise about 0 without changing its distance from 0. That is because i has modulus 1 and argument 90°. Multiplying by i twice turns 180°, which is why i² = -1: the point 1 ends up at -1.
How is adding complex numbers like adding vectors?
It is the same operation. Each complex number a + bi acts like the vector with components (a, b), and the sum is found tip-to-tail or with the parallelogram rule. Students who learn one can reuse the picture for the other when they study vectors (HSN.VM.B.4).
What does the complex conjugate look like on the graph?
The conjugate a - bi is the mirror image of a + bi across the real axis. It has the same modulus and the opposite argument. This picture explains two facts from algebra: z + z̄ is always real, and z·z̄ is always the nonnegative real number |z|².
Why is (-1 + √3 i)³ equal to 8?
The number -1 + √3 i is 2 units from 0 at an angle of 120°. Cubing multiplies the distance 2 · 2 · 2 = 8 and adds the angle three times, 3 · 120° = 360°, which points along the positive real axis. So the cube is the real number 8. Multiplying out (-1 + √3 i)(-1 + √3 i)(-1 + √3 i) gives the same answer.
Should students add in rectangular form and multiply in polar form?
That is usually the efficient choice. Adding and subtracting only need the real and imaginary parts, so rectangular form is quickest. Products and especially powers are much faster in polar form, because the moduli multiply and the arguments add. Strong students convert between the forms as the task requires.
Is De Moivre's theorem part of HSN.CN.B.5?
The power rule is: De Moivre's theorem says [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ), and it follows from applying the multiplication rule n times. The official example is a direct use of it. Finding nth roots of complex numbers is not required by this standard and is usually treated as enrichment.
What mistakes do students make with these operations?
A frequent one is multiplying the arguments or adding the moduli when multiplying in polar form. Others include drawing z - w from z to w instead of from w to z, reflecting across the wrong axis for the conjugate, and forgetting to reduce an argument such as 405° to 45°. Asking students to sketch before computing catches many of these.
07
Related Standards
6 standards
These standards connect to HSN.CN.B.5: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSN.CN.A.2Prerequisite
Use i² = -1 and the properties of operations to add, subtract and multiply complex numbers
