HSN.CN.A.1Common CoreMathNumber and QuantityGrades 9-12
HSN.CN.A.1: The Imaginary Unit i and Complex Numbers in a + bi Form
In plain English: HSN.CN.A.1 is the Common Core number standard that asks students to know that there is a number i with i² = -1 and that every complex number can be written as a + bi, where a and b are real numbers. Students use i to rewrite square roots of negative numbers, name the real part a and the imaginary part b, and see each real number as a complex number with b = 0. It is usually taught in Algebra II.
Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.
Common Core State Standards for Mathematics · Domain: The Complex Number System (CN) · Cluster: Perform arithmetic operations with complex numbers. Also written as HSN-CN.A.1 or N-CN.1 · Official standard
Students meet the imaginary unit i, the number whose square is -1, and learn that every complex number can be written in the form a + bi with a and b real. The lesson opens with an equation that has no real solution, x² = -36, and frames i as the next step in a long story: each time an equation had no solution, mathematicians built a larger number system in which it did.
Students then rewrite square roots of negative numbers with i, put numbers into a + bi form, name the real part a and the imaginary part b, and classify numbers as real, pure imaginary or other nonreal. They also use i² = -1 to find powers of i. Arithmetic with complex numbers (HSN.CN.A.2) comes in the next lesson.
Learning Objectives
By the end of this lesson, students will be able to:
State the defining property of the imaginary unit, i² = -1, and use it to check that a number such as 7i squares to a negative number
Rewrite the square root of a negative number as a real multiple of i, for example √-18 = 3i√2
Write a given number in the form a + bi and identify its real part a and its imaginary part b as real numbers
Classify a complex number as real, pure imaginary or other nonreal, and explain why every real number is also a complex number
Use i² = -1 to simplify powers of i such as i⁶ or i¹¹
Prior Knowledge Required
Students should already be comfortable with:
Rational and irrational numbers 8.NS.A.1
Square roots and solutions of x² = p for positive p 8.EE.A.2
Simplifying radicals such as √50 = 5√2 HSN.RN.A.2
Properties of exponents with whole-number exponents 8.EE.A.1
Write three equations on the board and give students two minutes to find every real solution:
Warm-Up Prompt
"Solve x² = 36, x² = 0 and x² = -36. Which one has no real solution, and why can no real number work?"
Collect answers: 6 and -6, then 0, then no real solution, because the square of any real number is zero or positive. Then show a short timeline on the board. The equation x + 7 = 3 has no solution in the whole numbers, so we use the integers. The equation 5x = 2 needs fractions. The equation x² = 2 needs irrational numbers. Ask: "What would we need so that x² = -36 has a solution?" Let students suggest inventing a new number, then name it.
Direct Instruction20 minutes
Part 1: The number i. Define i as a number with i² = -1. It is not a real number, since no real number has a negative square. Show that 6i solves the warm-up equation: (6i)² = 36 · i² = 36(-1) = -36, and (-6i)² = -36 as well. For any positive real number k, define √-k = i√k, the square root with a positive multiple of i. Point out that i is not -1 (because (-1)² = 1) and that i is not "the square root of 1 with a minus sign".
Part 2: The form a + bi. A complex number is any number a + bi where a and b are real. The real number a is the real part, and the real number b (not bi) is the imaginary part. Use Diagram 1: when b = 0 the number is real, so every real number is also a complex number; when b ≠ 0 it is nonreal, and when a = 0 and b ≠ 0 it is pure imaginary.
Find any square root of a negative number and rewrite it: √-k = i√k, then simplify √k.
Separate the terms: collect the real terms into a and the terms with i into bi.
If there is a real denominator, divide it into both terms, not only one.
Write the result as a + bi, with a sign in front of the imaginary term, and state a and b as real numbers.
Square root of a negative number
Rewrite √-49 using i and check the result with i² = -1.
Equation: √-18 = i√18 = i√9 · √2 = 3i√2, so a = 0 and b = 3√2
Writing a number in a + bi form
Write 6 - √-25 in the form a + bi.
Equation: 6 - √-25 = 6 - 5i, so a = 6 and b = -5
A real number as a complex number
Write the real number -4 in the form a + bi.
Equation: -4 = -4 + 0i, so a = -4 and b = 0
Dividing both terms by a real number
Write (10 + √-36)/2 in the form a + bi.
Equation: (10 + 6i)/2 = 5 + 3i, so a = 5 and b = 3
Part 3: Powers of i. Use i² = -1 to build the pattern in Diagram 2: i³ = i² · i = -i and i⁴ = i² · i² = 1, so every fourth power returns to 1. To simplify iⁿ, divide n by 4 and use the remainder. Stress that each step uses only i² = -1 and the rules for exponents.
Guided Practice15 minutes
Pairs work on whiteboards. After each item, one pair shows its work and another pair checks it by squaring or by naming a and b:
Guided practice items and answers
Item
Rewrite
a
b
√-64
8i
0
8
√-28
2i√7
0
2√7
-3 + √-100
-3 + 10i
-3
10
12
12 + 0i
12
0
(8 - √-4)/4
2 - (1/2)i
2
-1/2
Listen for three errors: writing √-64 as -8 (moving the negative sign outside the root), giving the imaginary part as 8i instead of 8, and dividing only the real term by 4 in the last item.
Independent Practice10-15 minutes
Students work alone on a half sheet. (1) Write each number in the form a + bi and classify it as real, pure imaginary or other nonreal: √-121 (0 + 11i, pure imaginary), -√121 (-11 + 0i, real), 5 - 2i (other nonreal) and i² (-1 + 0i, real). (2) Find real numbers x and y with x + 4i = 9 + yi (x = 9, y = 4), and explain why two complex numbers are equal only when their real parts match and their imaginary parts match. (3) Simplify i⁸ and i¹⁰ and explain the method. Students who finish early write their own example of each type of number from Diagram 1.
Closure5 minutes
Exit ticket: (1) Write √-45 in the form a + bi. (Answer: 0 + 3√5 i.) (2) Write one complex number that is real and one that is pure imaginary, each in the form a + bi. (Sample answers: 7 + 0i and 0 + 2i.) (3) A classmate writes "i = -1". Explain in one sentence what is wrong. (Squaring -1 gives 1, but i² must equal -1.)
Differentiation Strategies
For Struggling Students
Give a reference card with i² = -1 and √-k = i√k, plus one worked example of each rewrite type
Have students write every number in a two-column table labeled "real part a" and "imaginary part b" before writing a + bi
Start with perfect-square radicands such as √-4 and √-100 before radicands that need simplifying
For Advanced Students
Ask students to explain why √-4 · √-9 is not √36, and to find the correct value using i
Ask students to show that i and -i are the only two numbers of the form bi (b real) that square to -1
Ask students to find a rule for i raised to a negative whole number, starting from i · (-i) = 1
Assessment Guidance
What to Look For
Check that students write the imaginary part as a real number (b = -5, not -5i) and that they treat real numbers as complex numbers with b = 0. When students rewrite √-k, look for i placed in front of or after the radical, such as 3i√2, so it is not read as being under the root sign. Ask students to check a rewrite by squaring: if (bi)² does not return the original negative radicand, the rewrite is wrong.
02
Classroom Activities
3 Activities
1
Number System Timeline
15 minGroups of 3-4
Groups build a timeline of number systems, each one created to solve an equation the previous system could not. The complex numbers become the last entry, with i as the new number.
Equation Cards
x + 7 = 3: no whole-number solution; the integers give x = -4
5x = 2: no integer solution; the rational numbers give x = 2/5
x² = 7: no rational solution; the real numbers give x = ±√7
x² = -10: no real solution; the complex numbers give x = ±i√10
Procedure
Groups place the four cards in order on poster paper and label each system: whole numbers, integers, rational numbers, real numbers, complex numbers
For each card, the group writes one sentence explaining why the earlier system has no solution
For the last card, the group checks both solutions by squaring and using i² = -1
Discussion Questions
Each new system contains the old one. How does the form a + bi show that the real numbers are inside the complex numbers?
Why is i called "imaginary" if it solves a real equation?
2
Real or Imaginary Sort
20 minPairs
Pairs receive 12 number cards. For each card they write the number in the form a + bi, name a and b, and sort it into one of three piles: real (b = 0), pure imaginary (a = 0 and b ≠ 0) or other nonreal (a ≠ 0 and b ≠ 0).
The 12 Cards and Their Piles
Real: 8, -√2, 3i² (which is -3), √25 (which is 5)
Pure imaginary: √-1 (which is i), √-3 (which is i√3), -i, -√-25 (which is -5i)
Other nonreal: -2 + √-4 (which is -2 + 2i), 1/2 - 4i, 5 + √-64 (which is 5 + 8i), π + i
Procedure
Partner A rewrites a card in a + bi form; Partner B names a and b and places the card; they swap roles on each card
When all 12 cards are placed, pairs compare piles with a neighboring pair and settle disagreements by rewriting the card together
Each pair writes one sentence explaining why 3i² belongs in the real pile
Modification for Distance Learning
Put the 12 cards on a shared slide with three labeled boxes. Pairs drag each card into a box and type its a + bi form next to it, then share the slide with the class.
3
Powers of i Pattern Hunt
15 minPairs
Pairs compute powers of i one at a time, using only i² = -1 and the rule iⁿ⁺¹ = iⁿ · i, and discover that the values repeat every four powers.
Procedure
Give each pair a table with the rows i¹ through i¹² and a blank value column
Pairs fill in each row by multiplying the row above by i and replacing i² with -1 whenever it appears
Pairs circle the rows where the value is 1 and describe the pattern in words
Pairs write a rule that uses the remainder when n is divided by 4, then test it on i¹⁵ and i²²
Discussion Questions
Why does the pattern restart after i⁴?
Two students simplified i¹⁸ in different ways: one wrote (i²)⁹ and one wrote i¹⁶ · i². Do both methods work?
Challenge Variation
Ask pairs to add the first four powers of i, i + i² + i³ + i⁴, and explain the result. Then ask what the sum of the first 40 powers must be, without adding them one by one.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: How the Complex Numbers Are Organized
Every complex number has the form a + bi with a and b real. When b = 0 the number is real, so each real number is also a complex number. When b ≠ 0 the number is nonreal, and when in addition a = 0 it is pure imaginary.
Diagram 2: The Powers of i Repeat Every Four Steps
Each arrow multiplies by i. Starting from i and using i² = -1, the powers cycle through i, -1, -i and 1, so iⁿ depends only on the remainder when n is divided by 4. The four points sit where these numbers appear on the complex plane, which students meet in the (+) standard HSN.CN.B.4.
04
Homework Assignment
~30 min
HSN.CN.A.1 Homework: The Number i and the Form a + bi
Directions: Show your work. Write every complex number in the form a + bi and state a and b as real numbers. Check each square root of a negative number by squaring your answer and using i² = -1.
Part 1: The Number i (Problems 1-2)
Rewrite each number using i and simplify: (a) √-144 (b) √-7 (c) √-50 (d) -√-196
Show that both 18i and -18i are solutions of x² = -324 by squaring each one and using i² = -1. Then explain why no real number is a solution.
Part 2: Writing Numbers in a + bi Form (Problems 3-4)
Write each number in the form a + bi and state a and b: (a) 9 - √-400 (b) √-27 + 2 (c) -15 (d) (12 - √-72)/3
Find the real numbers x and y that make each equation true: (a) 2x + (y - 3)i = 10 + 5i (b) x - 6i = -7 + 2yi
Part 3: Classifying Numbers and Powers of i (Problems 5-6)
Classify each number as real, pure imaginary or other nonreal, and write it in the form a + bi: (a) -√36 (b) -3i (c) √-1 + √1 (d) 2i² + 3 (e) π - 2i. Then answer Sam, who says "Every complex number is either real or pure imaginary."
Simplify i⁶, i¹¹, i²⁰ and i³³. For each one, show how you used i² = -1 or the remainder when the exponent is divided by 4.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Using i
Every √-k rewritten as i√k, simplified and checked by squaring
Rewrites correct but not simplified or not checked
Negative moved outside the root or i missing
Form a + bi
Every number in a + bi form with a and b named as real numbers
Correct form, but b written as bi or a sign lost
Form missing or incorrect
Classifying and Equality
Correct piles and correct x and y, with reasons
Most correct, reasons incomplete
Mostly incorrect
Powers of i
All four powers correct with the method shown
Two or three correct
One or none correct
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 statement defines the imaginary unit i?
Answer: B
By definition, i is a number whose square is -1, so i² = -1. Choice A confuses i with its square: -1 cannot be i, because (-1)² = 1. Choice D is the same mistake, since -√1 = -1. Choice C describes 1 and -1, not i.
Question 2 of 20 · Multiple Choice
Which number is equal to √-81?
Answer: C
√-81 = i√81 = 9i. Check: (9i)² = 81i² = -81. Choice A moves the negative sign outside the root, but (-9)² = 81, not -81. Choice B forgets to take the square root of 81. Choice D lists real numbers, and neither one squares to -81.
Question 3 of 20 · Multiple Choice
Which is √-32 written in simplest form?
Answer: A
√-32 = i√32 = i√16 · √2 = 4i√2. Choice B moves the negative sign outside the root, which gives a real number whose square is +32. Choice C divides 32 by 2 instead of taking a square root. Choice D drops the i, so its square is 32, not -32.
Question 4 of 20 · Multiple Choice
What is the real part of -7 + 3i?
Answer: D
In a + bi, the real part is a. Here a = -7. Choices A and B give the imaginary part (and B wrongly includes i). Choice C is the whole number, not one of its parts.
Question 5 of 20 · Multiple Choice
What is the imaginary part of 5 - 8i?
Answer: C
Write 5 - 8i as 5 + (-8)i. The imaginary part is the real number b, so it is -8. Choice B includes the i, a frequent slip: the imaginary part is b, not bi. Choice D loses the sign. Choice A is the real part.
Question 6 of 20 · Multiple Choice
Which shows 4 - √-289 written in the form a + bi?
Answer: B
√-289 = i√289 = 17i, so 4 - √-289 = 4 - 17i, with a = 4 and b = -17. Choice A drops the minus sign in front of the root. Choice C treats √-289 as the real number 17. Choice D forgets to take the square root.
Question 7 of 20 · Multiple Choice
Which of these numbers is not a real number?
Answer: A
√-16 = 4i, which has imaginary part 4, so it is not real. Choices B and C are 4 and -4. Choice D may look imaginary, but i² = -1 is a real number.
Question 8 of 20 · Multiple Choice
What is i⁷?
Answer: D
i⁷ = i⁴ · i³ = 1 · i³, and i³ = i² · i = -i. So i⁷ = -i. Equivalently, 7 leaves remainder 3 when divided by 4. Choice B stops at i² = -1 and forgets the extra factor of i. Choice C uses the remainder 1 instead of 3.
Question 9 of 20 · Multiple Choice
What is i⁴²?
Answer: B
42 = 4 · 10 + 2, so i⁴² = (i⁴)¹⁰ · i² = 1 · (-1) = -1. Choice D assumes that every even power of i equals 1, but only multiples of 4 give 1. Choices A and C use the wrong remainder.
Question 10 of 20 · Multiple Choice
What is the value of √-9 · √-16?
Answer: C
Rewrite each root first: √-9 · √-16 = 3i · 4i = 12i² = -12. Choice A uses the rule √a · √b = √(ab), which is true for nonnegative a and b but fails when both are negative. Choices B and D forget that i · i = i² = -1.
Question 11 of 20 · Multiple Choice
Which statement about the number 0 is true?
Answer: A
0 = 0 + 0i, with a = 0 and b = 0, both real. Since b = 0, it is real, and like every real number it is also complex. Choice B makes the error of thinking a complex number must show an i; a real number simply has b = 0. Choices C and D contradict the form 0 + 0i.
Question 12 of 20 · Multiple Choice
Find the real numbers x and y such that x + 2yi = -3 + 10i.
Answer: D
Two complex numbers are equal when their real parts are equal and their imaginary parts are equal: x = -3 and 2y = 10, so x = -3, y = 5. Choice A sets y equal to 10 and forgets the factor 2. Choice B swaps the real and imaginary parts. Choice C loses the sign of x.
Question 13 of 20 · Multiple Choice
Which number is pure imaginary (real part 0 and imaginary part not 0)?
Answer: B
-6i = 0 + (-6)i, so a = 0 and b = -6 ≠ 0. Choice C has real part 6, so it is nonreal but not pure imaginary. Choice D equals -1, which is real. Choice A is real.
Question 14 of 20 · Multiple Choice
Which shows (6 + √-20)/2 written in the form a + bi?
Answer: A
√-20 = i√4 · √5 = 2i√5. Then (6 + 2i√5)/2 = 3 + i√5, dividing both terms by 2. Choice B divides only the real term. Choice C divides only the imaginary term. Choice D halves 20 instead of taking its square root.
Question 15 of 20 · Short Answer
Explain why no real number solves x² = -169. Then name two numbers that do, and check one of them.
The square of any real number is zero or positive, so it can never equal -169. The solutions are 13i and -13i. Check: (13i)² = 169 · i² = 169(-1) = -169, and (-13i)² = 169 · i² = -169.
Question 16 of 20 · Short Answer
Write -√-75 in the form a + bi, and state a and b.
√-75 = i√75 = i√25 · √3 = 5i√3, so -√-75 = 0 - 5√3 i. The real part is a = 0 and the imaginary part is b = -5√3 (a real number).
Question 17 of 20 · Short Answer
Write √-12 - 8 in the form a + bi, and state a and b.
√-12 = i√4 · √3 = 2i√3. Put the real term first: -8 + 2√3 i, so a = -8 and b = 2√3.
Question 18 of 20 · Short Answer
Kai says that the imaginary part of 2 - 9i is -9i. Is Kai right? Explain.
No. In the form a + bi, the imaginary part is the real number b. Since 2 - 9i = 2 + (-9)i, the imaginary part is -9. The term -9i is the imaginary term, but the imaginary part itself does not include i.
Question 19 of 20 · Short Answer
Simplify i²⁵ and explain your method.
25 = 4 · 6 + 1, so i²⁵ = (i⁴)⁶ · i = 1 · i = i. Every group of four factors of i multiplies to i⁴ = (i²)² = (-1)² = 1, so only the remainder 1 matters.
Question 20 of 20 · Short Answer
Show that (-15i)² = -225, using i² = -1. What does this tell you about the equation x² = -225?
(-15i)² = (-15)² · i² = 225(-1) = -225. So -15i is a solution of x² = -225. The same work shows (15i)² = -225, so the equation has the two solutions 15i and -15i, and √-225 = 15i names the one with the positive multiple of i.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSN.CN.A.1 mean?
HSN.CN.A.1 means students know that there is a number i with i² = -1, and that every complex number can be written as a + bi with a and b real. In practice, students rewrite square roots of negative numbers with i, write numbers in a + bi form, and name the real part a and the imaginary part b. It is the first standard in the Common Core cluster on arithmetic with complex numbers.
Is HSN.CN.A.1 taught in Algebra 1 or Algebra 2?
It is usually taught in Algebra II. Algebra I courses often say that an equation such as x² = -4 has "no real solution" and stop there; Algebra II introduces i so that those equations do have solutions. Integrated Math III courses cover the same content.
If imaginary numbers are "imaginary," why do we need them?
They make every quadratic equation solvable. With i, an equation like x² + 1 = 0 has the solutions i and -i, and the quadratic formula always gives an answer. The name comes from a time when mathematicians doubted these numbers; today they are as well defined as negative numbers or irrational numbers, and engineers use them to describe alternating current and waves.
Is every real number a complex number?
Yes. Any real number r can be written as r + 0i, which has the form a + bi with a = r and b = 0. The complex numbers contain the real numbers the same way the real numbers contain the integers.
What is the difference between an imaginary number and a complex number?
A complex number is any number a + bi with a and b real. A pure imaginary number is the special case with a = 0 and b ≠ 0, such as 4i or -i√3. So 2 + 4i is complex but not pure imaginary, 4i is both, and 2 is complex and real. Textbooks differ on the exact wording, so tell students which definition your course uses.
Why is √-4 · √-9 not equal to √36?
Because the rule √a · √b = √(ab) holds only when a and b are not both negative. Rewrite each root with i first: √-4 · √-9 = 2i · 3i = 6i² = -6, not 6. This is why students should always rewrite √-k as i√k before doing anything else.
How do you simplify a large power of i, like i¹⁰⁰?
Divide the exponent by 4 and use the remainder. Since i⁴ = 1, every block of four factors of i equals 1. For i¹⁰⁰, 100 is a multiple of 4, so i¹⁰⁰ = (i⁴)²⁵ = 1. A remainder of 1 gives i, 2 gives -1 and 3 gives -i.
Is the imaginary part of a complex number b or bi?
It is b, a real number. For 3 - 4i, the real part is 3 and the imaginary part is -4. The phrase "the imaginary part" is defined this way so that both parts of a complex number are real numbers, which makes it easy to compare and combine complex numbers.
How does HSN.CN.A.1 connect to quadratic equations?
When the discriminant b² - 4ac of a quadratic equation is negative, the quadratic formula produces the square root of a negative number. HSN.CN.A.1 gives students the tool to rewrite that root with i and to write the solutions in the form a + bi. Solving those equations is the focus of HSN.CN.C.7 and part of HSA.REI.B.4.
What mistakes do students often make with i?
Common ones are: writing i = -1 instead of i² = -1; moving the negative sign outside the root, so √-9 becomes -3; giving the imaginary part as bi instead of b; and dividing only one term of an expression such as (8 + √-16)/4 by the denominator. A quick check catches most of them: square your answer and see whether you get the original negative number back.
07
Related Standards
6 standards
These standards connect to HSN.CN.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.NS.A.1Prerequisite
Know that numbers that are not rational are irrational; every number has a decimal expansion
