From the definition of i through De Moivre's Theorem and nth roots of complex numbers — every concept with worked examples.
Stewart Ch. 3 Supplement — the foundational definition that unlocks the complex number system.
Definition
i squared equals negative 1
equivalently: i equals the square root of negative 1
Why i was invented
Real numbers cannot produce a negative result when squared. The imaginary unit i fills this gap, allowing every polynomial equation to have solutions. Before i, equations like x squared plus 1 equals 0 had no solution.
Simplifying square roots of negative numbers
Square root of negative 9 equals square root of 9 times square root of negative 1 equals 3i
Square root of negative 7 equals i times square root of 7
Square root of negative 50 equals 5i times square root of 2
Every complex number can be written in standard form.
Standard Form
a plus bi
a and b are real numbers
Real Part
Re(z) equals a
the non-i component
Imaginary Part
Im(z) equals b
coefficient of i
Special cases
When b equals 0:
a plus 0i equals a (a real number)
Every real number is complex
When a equals 0:
0 plus bi equals bi (pure imaginary)
Lies on the imaginary axis
Equality of complex numbers
Two complex numbers a plus bi and c plus di are equal if and only if a equals c AND b equals d. Both the real parts and the imaginary parts must match separately.
The conjugate flips the sign of the imaginary part and is essential for division.
Definition
conjugate of (a plus bi) equals (a minus bi)
Written with a bar over z: z-bar
Key product property
(a plus bi)(a minus bi) equals a squared plus b squared
Always a non-negative real number
z equals 3 plus 4i
z-bar equals 3 minus 4i
z times z-bar equals 9 plus 16 equals 25
z equals 2 minus 5i
z-bar equals 2 plus 5i
z times z-bar equals 4 plus 25 equals 29
z equals negative 1 plus i
z-bar equals negative 1 minus i
z times z-bar equals 1 plus 1 equals 2
Properties of conjugates
Combine real parts with real parts and imaginary parts with imaginary parts.
Addition rule
(a plus bi) plus (c plus di) equals (a plus c) plus (b plus d)i
Example
(3 plus 2i) plus (1 minus 5i)
equals (3 plus 1) plus (2 minus 5)i
equals 4 minus 3i
Subtraction rule
(a plus bi) minus (c plus di) equals (a minus c) plus (b minus d)i
Example
(5 plus 3i) minus (2 plus 7i)
equals (5 minus 2) plus (3 minus 7)i
equals 3 minus 4i
Additional practice
(negative 4 plus 6i) plus (7 minus 2i)
equals 3 plus 4i
(8 minus i) minus (negative 3 plus 4i)
equals 11 minus 5i
(2 plus square root of 3 i) plus (5 minus square root of 3 i)
equals 7
6i minus (4 plus 2i)
equals negative 4 plus 4i
Treat complex numbers as binomials. Use FOIL, then replace i squared with negative 1.
(a plus bi)(c plus di) equals (ac minus bd) plus (ad plus bc)i
The i squared term becomes negative 1, turning bd positive in the real part and negative in the formula shown as minus bd
Worked Example 1: (2 plus 3i)(1 plus 4i)
First: 2 times 1 equals 2
Outer: 2 times 4i equals 8i
Inner: 3i times 1 equals 3i
Last: 3i times 4i equals 12i squared equals 12 times (negative 1) equals negative 12
Sum: (2 minus 12) plus (8 plus 3)i
equals negative 10 plus 11i
Worked Example 2: (3 minus 2i)(3 plus 2i)
This is (a minus bi)(a plus bi) — conjugate product pattern
equals 3 squared plus 2 squared
equals 9 plus 4
equals 13 (a real number — no i term)
Worked Example 3: (1 plus i) squared
(1 plus i)(1 plus i)
equals 1 plus i plus i plus i squared
equals 1 plus 2i plus (negative 1)
equals 2i
To divide, multiply both numerator and denominator by the conjugate of the denominator. This makes the denominator real.
(a plus bi) divided by (c plus di)
equals (a plus bi)(c minus di) divided by (c squared plus d squared)
Denominator becomes c squared plus d squared — always a positive real number
Worked Example 1: (3 plus 2i) divided by (1 minus i)
Conjugate of denominator (1 minus i) is (1 plus i)
Multiply top and bottom by (1 plus i):
Numerator: (3 plus 2i)(1 plus i) equals 3 plus 3i plus 2i plus 2i squared
equals 3 plus 5i minus 2 equals 1 plus 5i
Denominator: (1 minus i)(1 plus i) equals 1 plus 1 equals 2
Result: (1 plus 5i) divided by 2 equals one-half plus five-halves i
Worked Example 2: (5 plus i) divided by (2 plus 3i)
Conjugate of (2 plus 3i) is (2 minus 3i)
Numerator: (5 plus i)(2 minus 3i) equals 10 minus 15i plus 2i minus 3i squared
equals 10 minus 13i plus 3 equals 13 minus 13i
Denominator: 4 plus 9 equals 13
Result: (13 minus 13i) divided by 13 equals 1 minus i
Worked Example 3: 4 divided by (2i)
Rewrite 2i as 0 plus 2i. Conjugate is 0 minus 2i, or just negative 2i.
Multiply top and bottom by (negative 2i):
4 times (negative 2i) equals negative 8i
2i times (negative 2i) equals negative 4i squared equals 4
Result: negative 8i divided by 4 equals negative 2i
The powers of i repeat in a cycle of length 4. Use the remainder when dividing by 4 to find any power.
i¹
i
i²
negative 1
i³
negative i
i⁴
1
Extended powers table
| Power | Remainder (n mod 4) | Value |
|---|---|---|
| i to the 0 | 0 | 1 |
| i to the 1 | 1 | i |
| i to the 2 | 2 | negative 1 |
| i to the 3 | 3 | negative i |
| i to the 4 | 0 | 1 |
| i to the 5 | 1 | i |
| i to the 6 | 2 | negative 1 |
| i to the 7 | 3 | negative i |
| i to the 8 | 0 | 1 |
How to find i to the n for any integer n
Step 1: Compute n mod 4 (the remainder when n is divided by 4).
Step 2: Remainder 0 gives 1, remainder 1 gives i, remainder 2 gives negative 1, remainder 3 gives negative i.
i to the 27
27 divided by 4 is 6 remainder 3
negative i
i to the 100
100 divided by 4 is 25 remainder 0
1
i to the 53
53 divided by 4 is 13 remainder 1
i
The modulus measures the distance from the origin to the complex number in the complex plane.
absolute value of (a plus bi) equals square root of (a squared plus b squared)
By the Pythagorean theorem — distance from (0,0) to (a,b)
Examples
absolute value of (3 plus 4i)
equals square root of (9 plus 16)
equals 5
absolute value of (1 minus i)
equals square root of (1 plus 1)
equals square root of 2
absolute value of (negative 5 plus 12i)
equals square root of (25 plus 144)
equals 13
Modulus properties
absolute value of (z1 times z2) equals absolute value of z1 times absolute value of z2
absolute value of (z1 divided by z2) equals absolute value of z1 divided by absolute value of z2
absolute value of z-bar equals absolute value of z
z times z-bar equals absolute value of z, quantity squared
absolute value of z squared equals a squared plus b squared
A geometric representation of complex numbers as points in a two-dimensional plane.
Structure of the Argand diagram
Plotting examples
Geometric interpretation of operations
Addition: vector addition — add the coordinate vectors.
Multiplication by i: rotates the point 90 degrees counterclockwise around the origin.
Conjugate: reflection across the real axis (flip the imaginary part sign).
Modulus: length of the vector from origin to the point.
Every complex number can be expressed using its modulus r and argument theta, which is the angle in the Argand diagram.
z equals a plus bi equals r(cosine theta plus i sine theta) equals r cis theta
r equals square root of (a squared plus b squared) — the modulus
theta equals arctan(b divided by a) — the argument (adjust for quadrant)
a equals r times cosine theta — rectangular from polar
b equals r times sine theta — rectangular from polar
Quadrant adjustment for theta
arctan(b divided by a) always returns a value between negative 90 degrees and 90 degrees.
If a is negative (quadrants II and III): add 180 degrees to the arctan result.
If a is positive and b is negative (quadrant IV): add 360 degrees (or leave as negative angle).
Always verify: r times cosine theta should give back a, and r times sine theta should give back b.
Example: Convert 1 plus i to polar form
r equals square root of (1 plus 1) equals square root of 2
theta equals arctan(1 divided by 1) equals arctan(1) equals 45 degrees (first quadrant — no adjustment)
Result: 1 plus i equals square root of 2 times cis 45 degrees
square root of 2 cis 45 degrees
Example: Convert negative 1 plus i to polar form
r equals square root of (1 plus 1) equals square root of 2
arctan(1 divided by negative 1) equals arctan(negative 1) equals negative 45 degrees
a is negative (second quadrant): add 180 degrees: negative 45 plus 180 equals 135 degrees
Result: negative 1 plus i equals square root of 2 times cis 135 degrees
square root of 2 cis 135 degrees
Example: Convert 2 cis 60 degrees to rectangular form
a equals 2 times cosine 60 degrees equals 2 times one-half equals 1
b equals 2 times sine 60 degrees equals 2 times (square root of 3 divided by 2) equals square root of 3
1 plus i square root of 3
Polar form makes multiplying and dividing complex numbers geometric: multiply the moduli, add the arguments.
Multiplication
r1 cis theta1 times r2 cis theta2
equals r1 r2 cis(theta1 plus theta2)
multiply moduli, add arguments
Division
r1 cis theta1 divided by r2 cis theta2
equals (r1 divided by r2) cis(theta1 minus theta2)
divide moduli, subtract arguments
Multiplication example
2 cis 30 degrees times 3 cis 60 degrees
equals (2 times 3) cis(30 plus 60) degrees
equals 6 cis 90 degrees
cis 90 degrees equals cosine 90 degrees plus i sine 90 degrees equals 0 plus i equals i
equals 6i
Division example
6 cis 120 degrees divided by 2 cis 30 degrees
equals (6 divided by 2) cis(120 minus 30) degrees
equals 3 cis 90 degrees equals 3i
The most powerful tool for computing large powers of complex numbers. It extends the multiplication-in-polar-form rule to integer powers.
(r cis theta) to the power n equals r to the n times cis(n times theta)
for any integer n
Strategy for using De Moivre's Theorem
Worked Example 1: Compute (1 plus i) to the 6th power
Step 1 — Convert to polar: r equals square root of 2, theta equals 45 degrees
1 plus i equals square root of 2 times cis 45 degrees
Step 2 — Apply De Moivre with n equals 6:
(square root of 2) to the 6th equals (2 to the one-half) to the 6th equals 2 to the 3rd equals 8
6 times 45 degrees equals 270 degrees
Step 3 — Convert 8 cis 270 degrees to rectangular:
cosine 270 equals 0, sine 270 equals negative 1
Answer: 8(0 plus i times negative 1) equals negative 8i
Worked Example 2: Compute (negative 1 plus i square root of 3) to the 4th power
Step 1 — Convert: a equals negative 1, b equals square root of 3
r equals square root of (1 plus 3) equals 2
arctan(square root of 3 divided by negative 1) plus 180 equals 120 degrees (second quadrant)
So z equals 2 cis 120 degrees
Step 2 — De Moivre with n equals 4:
2 to the 4th equals 16, angle is 4 times 120 equals 480 degrees
480 degrees minus 360 degrees equals 120 degrees
16 cis 120 degrees equals 16(cosine 120 plus i sine 120)
equals 16(negative one-half plus i times square root of 3 divided by 2)
equals negative 8 plus 8i square root of 3
Worked Example 3: Negative exponent — compute (1 plus i) to the negative 2nd power
1 plus i equals square root of 2 cis 45 degrees
Apply De Moivre with n equals negative 2:
(square root of 2) to the negative 2nd equals one-half
negative 2 times 45 equals negative 90 degrees
one-half cis(negative 90 degrees) equals one-half(0 minus i)
equals negative i divided by 2
Every nonzero complex number has exactly n distinct nth roots, equally spaced on a circle in the complex plane.
nth Root Formula
The nth roots of r cis theta are:
w sub k equals r to the (1/n) times cis((theta plus 360 times k) divided by n)
for k equals 0, 1, 2, up to n minus 1
There are exactly n roots.
All roots have the same modulus: r to the (1/n).
The roots are equally spaced at angles of 360 divided by n degrees apart.
Worked Example 1: Cube roots of 8
Write 8 in polar form: r equals 8, theta equals 0 degrees
8 equals 8 cis 0 degrees
Cube root modulus: 8 to the (1/3) equals 2
k equals 0: 2 cis(0 divided by 3) equals 2 cis 0 equals 2
k equals 1: 2 cis(360 divided by 3) equals 2 cis 120 equals negative 1 plus i square root of 3
k equals 2: 2 cis(720 divided by 3) equals 2 cis 240 equals negative 1 minus i square root of 3
Three cube roots: 2, negative 1 plus i square root of 3, negative 1 minus i square root of 3
Worked Example 2: Square roots of i
Write i in polar form: r equals 1, theta equals 90 degrees
i equals 1 cis 90 degrees
Square root modulus: 1 to the (1/2) equals 1
k equals 0: 1 cis(90 divided by 2) equals cis 45 degrees equals (square root of 2 divided by 2) plus i(square root of 2 divided by 2)
k equals 1: 1 cis((90 plus 360) divided by 2) equals cis 225 degrees equals negative (square root of 2 divided by 2) minus i(square root of 2 divided by 2)
Two square roots of i, each with modulus 1
Worked Example 3: Fourth roots of 16 cis 60 degrees
r equals 16, theta equals 60 degrees, n equals 4
Modulus of each root: 16 to the (1/4) equals 2
Angles: (60 plus 360k) divided by 4 for k equals 0, 1, 2, 3
k equals 0: 60 divided by 4 equals 15 degrees — root is 2 cis 15 degrees
k equals 1: 420 divided by 4 equals 105 degrees — root is 2 cis 105 degrees
k equals 2: 780 divided by 4 equals 195 degrees — root is 2 cis 195 degrees
k equals 3: 1140 divided by 4 equals 285 degrees — root is 2 cis 285 degrees
Four roots equally spaced at 90-degree intervals on a circle of radius 2
The nth roots of unity are the solutions to z to the n equals 1. They are equally spaced on the unit circle.
General formula for nth roots of unity
w sub k equals cis(360k divided by n) for k equals 0, 1, ..., n minus 1
All have modulus 1. Equally spaced at 360/n degrees apart. Start at (1, 0) when k equals 0.
Square roots of unity (n equals 2)
k equals 0: cis 0 equals 1
k equals 1: cis 180 equals negative 1
Spaced 180 degrees apart
Cube roots of unity (n equals 3)
k equals 0: 1
k equals 1: cis 120
k equals 2: cis 240
Spaced 120 degrees apart
4th roots of unity (n equals 4)
k equals 0: 1
k equals 1: i
k equals 2: negative 1
k equals 3: negative i
Spaced 90 degrees apart
Primitive roots of unity
A primitive nth root of unity is one that generates all others by taking powers. For example, i is a primitive 4th root of unity because i to the 1 equals i, i squared equals negative 1, i cubed equals negative i, and i to the 4 equals 1 gives all four 4th roots of unity.
Complex numbers allow every polynomial equation to have solutions. Real polynomials have complex roots in conjugate pairs.
Conjugate Root Theorem
If a polynomial has real coefficients and (a plus bi) is a root with b not equal to zero, then the conjugate (a minus bi) is also a root. Complex roots of real polynomials always come in conjugate pairs.
Example 1: Solve x squared plus 4 equals 0
x squared equals negative 4
x equals plus or minus square root of negative 4
x equals plus or minus 2i
Solutions: x equals 2i and x equals negative 2i (conjugate pair)
Example 2: Solve x squared plus 2x plus 5 equals 0
Discriminant: b squared minus 4ac equals 4 minus 20 equals negative 16
x equals (negative 2 plus or minus square root of negative 16) divided by 2
x equals (negative 2 plus or minus 4i) divided by 2
Solutions: x equals negative 1 plus 2i and x equals negative 1 minus 2i
Example 3: Factor x to the 4th minus 1 completely over the complex numbers
x to the 4th minus 1 equals (x squared minus 1)(x squared plus 1)
equals (x minus 1)(x plus 1)(x minus i)(x plus i)
Four roots: x equals 1, negative 1, i, negative i (the 4th roots of unity)
Example 4: Find a polynomial with roots 2, 3 plus i, and 3 minus i
Factors: (x minus 2), (x minus (3 plus i)), (x minus (3 minus i))
Multiply the complex pair: (x minus 3 minus i)(x minus 3 plus i)
equals ((x minus 3) minus i)((x minus 3) plus i)
equals (x minus 3) squared plus 1
equals x squared minus 6x plus 9 plus 1 equals x squared minus 6x plus 10
Final polynomial: (x minus 2)(x squared minus 6x plus 10)
equals x cubed minus 8x squared plus 22x minus 20
The complex number system is algebraically closed — every polynomial has a complete set of roots.
Fundamental Theorem of Algebra
Every polynomial of degree n with complex coefficients has exactly n roots in the complex number system, counting multiplicity.
What this means for real polynomials
Connection to factoring
Every degree-n polynomial can be factored into exactly n linear factors over the complex numbers:
p(x) equals a(x minus r1)(x minus r2) times ... times (x minus rn)
where r1 through rn are the complex roots (some may be repeated).
Fundamentals
i squared equals negative 1
i to the 1 equals i, i squared equals negative 1, i cubed equals negative i, i to the 4 equals 1
i to the n: use remainder of n mod 4
Standard form: a plus bi
Conjugate of (a plus bi): a minus bi
(a plus bi)(a minus bi) equals a squared plus b squared
Modulus: square root of (a squared plus b squared)
Polar Form and De Moivre
z equals r cis theta, r equals modulus, theta equals argument
r equals square root of (a squared plus b squared)
theta equals arctan(b over a), adjusted for quadrant
z1 z2 equals r1 r2 cis(theta1 plus theta2)
z to the n equals r to the n cis(n theta)
De Moivre's Theorem
nth roots: r to the (1/n) cis((theta plus 360k) over n)
k equals 0, 1, ..., n minus 1
The imaginary unit i is defined so that i squared equals negative 1. This extends the real number system to allow square roots of negative numbers. Every complex number has standard form a plus bi, where a is the real part and b is the imaginary part. When b equals 0 the number is real; when a equals 0 it is called pure imaginary.
Multiply complex numbers exactly like binomials using FOIL: First, Outer, Inner, Last. For example, (2 plus 3i)(1 plus 4i) gives 2 plus 8i plus 3i plus 12i squared. Replace i squared with negative 1 to get 2 plus 11i minus 12, which simplifies to negative 10 plus 11i. The formula is (a plus bi)(c plus di) equals (ac minus bd) plus (ad plus bc)i.
De Moivre's Theorem states that (r cis theta) to the power n equals r to the n times cis(n times theta), where cis theta means cosine theta plus i sine theta. To use it: convert to polar form to find r and theta, raise r to the nth power and multiply theta by n, then convert back to rectangular form. Example: (1 plus i) to the 6th power. Convert to polar: r equals square root of 2, theta equals 45 degrees. Apply De Moivre: (square root of 2) to the 6th equals 8, angle is 6 times 45 equals 270 degrees. Result is 8 cis 270 degrees equals negative 8i.
The nth roots of r cis theta are given by r to the power (1/n) times cis((theta plus 360k)/n) for k equals 0, 1, 2, up to n minus 1. There are always exactly n distinct nth roots, equally spaced on a circle of radius r to the (1/n) in the complex plane. For the nth roots of unity (cube roots, 4th roots, etc.), set r equals 1 and theta equals 0, giving cis(360k/n) for each k.
The Fundamental Theorem of Algebra states that every polynomial of degree n with complex coefficients has exactly n roots in the complex number system (counting multiplicity). For polynomials with real coefficients, complex roots always come in conjugate pairs: if a plus bi is a root, then a minus bi is also a root. This means every odd-degree real polynomial has at least one real root.
The complex plane (Argand diagram) is a coordinate plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part of complex numbers. The complex number a plus bi corresponds to the point (a, b). The distance from the origin to this point is the modulus (absolute value) equal to the square root of (a squared plus b squared). The angle from the positive real axis is called the argument of the complex number.
Interactive problems on De Moivre's Theorem, nth roots, polar form, and every topic on this page — free to start.
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