Chapter Notes: Complex Numbers

Mathematics has always been about solving problems, and sometimes those problems lead us to equations that seem impossible to solve using only the numbers we know. For example, what number multiplied by itself equals -1? In the world of real numbers, no such number exists because any real number squared is always positive or zero. To solve such equations, mathematicians expanded the number system to include complex numbers-a powerful extension that combines real numbers with a new type of number called imaginary numbers. Complex numbers are not just theoretical curiosities; they are essential in fields like electrical engineering, signal processing, quantum mechanics, and fluid dynamics. In this chapter, you will learn what complex numbers are, how to perform operations with them, and how to represent them in multiple ways.

The Imaginary Unit and Imaginary Numbers

The foundation of complex numbers rests on defining a new number that solves the equation \( x^2 = -1 \). We call this number the imaginary unit and represent it with the symbol \( i \).

Definition: The imaginary unit \( i \) is defined by the property:

\[ i^2 = -1 \]

This means that \( i = \sqrt{-1} \), though we must be careful when working with square roots of negative numbers. From this definition, we can find powers of \( i \):

• \( i^1 = i \)
• \( i^2 = -1 \)
• \( i^3 = i^2 \cdot i = -1 \cdot i = -i \)
• \( i^4 = i^2 \cdot i^2 = (-1)(-1) = 1 \)
• \( i^5 = i^4 \cdot i = 1 \cdot i = i \)

Notice that the powers of \( i \) repeat in a cycle of four: \( i, -1, -i, 1, i, -1, -i, 1, \ldots \) This pattern is extremely useful when simplifying higher powers of \( i \).

Example:  Simplify \( i^{27} \).

Solution:

To simplify \( i^{27} \), we use the fact that powers of \( i \) repeat every 4 powers.

Divide the exponent by 4: 27 ÷ 4 = 6 remainder 3

This means \( i^{27} = i^{4 \cdot 6 + 3} = (i^4)^6 \cdot i^3 = 1^6 \cdot i^3 = i^3 \)

Since \( i^3 = -i \), we have \( i^{27} = -i \)

The value of \( i^{27} \) is -i.

An imaginary number is any number that can be written as a real number multiplied by \( i \). For example, \( 3i \), \( -7i \), and \( \frac{5}{2}i \) are all imaginary numbers. We can also write square roots of negative numbers as imaginary numbers:

\[ \sqrt{-9} = \sqrt{9 \cdot (-1)} = \sqrt{9} \cdot \sqrt{-1} = 3i \]

Example:  Write \( \sqrt{-50} \) in terms of \( i \).

Solution:

First, factor out -1 from under the square root: \( \sqrt{-50} = \sqrt{50 \cdot (-1)} = \sqrt{50} \cdot \sqrt{-1} \)

Since \( \sqrt{-1} = i \), we have \( \sqrt{-50} = \sqrt{50} \cdot i \)

Simplify \( \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \)

Therefore, \( \sqrt{-50} = 5\sqrt{2} \, i \) (often written as \( 5i\sqrt{2} \))

The simplified form is 5√2 i.

Definition and Standard Form of Complex Numbers

A complex number is a number that combines a real part and an imaginary part. Every complex number can be written in standard form (also called rectangular form or Cartesian form):

\[ z = a + bi \]

In this form, \( a \) and \( b \) are real numbers, \( a \) is called the real part of the complex number, and \( b \) is called the imaginary part of the complex number. Note that the imaginary part is the coefficient \( b \), not \( bi \).

For example, in the complex number \( 4 + 7i \), the real part is 4 and the imaginary part is 7.

Special cases:

• If \( b = 0 \), then \( z = a + 0i = a \), which is a real number. This shows that all real numbers are also complex numbers.
• If \( a = 0 \), then \( z = 0 + bi = bi \), which is a pure imaginary number.
• The number 0 can be written as \( 0 + 0i \), so it is both real and imaginary.

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, \( a + bi = c + di \) if and only if \( a = c \) and \( b = d \).

Operations with Complex Numbers

Addition and Subtraction

To add or subtract complex numbers, we combine the real parts separately and the imaginary parts separately. This is similar to combining like terms in algebra.

Addition: If \( z_1 = a + bi \) and \( z_2 = c + di \), then:

\[ z_1 + z_2 = (a + c) + (b + d)i \]

Subtraction: If \( z_1 = a + bi \) and \( z_2 = c + di \), then:

\[ z_1 - z_2 = (a - c) + (b - d)i \]

Example:  Find the sum and difference of \( (5 + 3i) \) and \( (2 - 7i) \).

Solution:

For addition: \( (5 + 3i) + (2 - 7i) \)

Combine real parts: 5 + 2 = 7

Combine imaginary parts: 3 + (-7) = -4

Sum = 7 - 4i

For subtraction: \( (5 + 3i) - (2 - 7i) \)

Combine real parts: 5 - 2 = 3

Combine imaginary parts: 3 - (-7) = 3 + 7 = 10

Difference = 3 + 10i

The sum is 7 - 4i and the difference is 3 + 10i.

Multiplication

To multiply complex numbers, we use the distributive property (also known as the FOIL method for binomials) and remember that \( i^2 = -1 \).

If \( z_1 = a + bi \) and \( z_2 = c + di \), then:

\[ z_1 \cdot z_2 = (a + bi)(c + di) = ac + adi + bci + bdi^2 \]

Since \( i^2 = -1 \), we substitute:

\[ z_1 \cdot z_2 = ac + adi + bci + bd(-1) = (ac - bd) + (ad + bc)i \]

Example:  Multiply \( (3 + 2i)(4 - 5i) \).

Solution:

Use the distributive property: \( (3 + 2i)(4 - 5i) = 3 \cdot 4 + 3 \cdot (-5i) + 2i \cdot 4 + 2i \cdot (-5i) \)

Simplify each term: = 12 - 15i + 8i - 10i²

Replace \( i^2 \) with -1: = 12 - 15i + 8i - 10(-1) = 12 - 15i + 8i + 10

Combine real parts and imaginary parts: = (12 + 10) + (-15 + 8)i = 22 - 7i

The product is 22 - 7i.

Complex Conjugates

The complex conjugate of a complex number \( z = a + bi \) is denoted \( \overline{z} \) (read "z bar") and is defined as:

\[ \overline{z} = a - bi \]

The complex conjugate has the same real part but the opposite imaginary part. For example, the conjugate of \( 3 + 4i \) is \( 3 - 4i \), and the conjugate of \( -2 - 5i \) is \( -2 + 5i \).

Important properties of complex conjugates:

• The product of a complex number and its conjugate is always a real number: \( z \cdot \overline{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2i^2 = a^2 + b^2 \)
• The sum of a complex number and its conjugate is always real: \( z + \overline{z} = 2a \)
• The conjugate of a conjugate gives back the original number: \( \overline{\overline{z}} = z \)

Division

To divide complex numbers, we multiply both the numerator and denominator by the complex conjugate of the denominator. This process eliminates the imaginary part from the denominator, making it a real number.

If we want to compute \( \frac{a + bi}{c + di} \), we multiply by \( \frac{c - di}{c - di} \):

\[ \frac{a + bi}{c + di} = \frac{a + bi}{c + di} \cdot \frac{c - di}{c - di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} \]

This simplifies to:

\[ \frac{a + bi}{c + di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i \]

Example:  Divide \( \frac{6 + 8i}{1 + 2i} \).

Solution:

Identify the conjugate of the denominator: The conjugate of \( 1 + 2i \) is \( 1 - 2i \)

Multiply numerator and denominator by the conjugate:

\( \frac{6 + 8i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{(6 + 8i)(1 - 2i)}{(1 + 2i)(1 - 2i)} \)

Expand the numerator: \( (6 + 8i)(1 - 2i) = 6 - 12i + 8i - 16i^2 = 6 - 4i - 16(-1) = 6 - 4i + 16 = 22 - 4i \)

Expand the denominator: \( (1 + 2i)(1 - 2i) = 1 - 4i^2 = 1 - 4(-1) = 1 + 4 = 5 \)

Divide: \( \frac{22 - 4i}{5} = \frac{22}{5} - \frac{4}{5}i = \frac{22}{5} - \frac{4}{5}i \)

The quotient is 22/5 - 4/5 i.

The Complex Plane

Just as real numbers can be represented on a number line, complex numbers can be represented graphically on a two-dimensional plane called the complex plane or Argand diagram. The horizontal axis is called the real axis, and the vertical axis is called the imaginary axis.

A complex number \( z = a + bi \) is plotted as the point \( (a, b) \) in the complex plane, where \( a \) is the horizontal coordinate and \( b \) is the vertical coordinate. Think of the complex plane as a map where every complex number has a unique location defined by its real and imaginary parts.

For example:

• The complex number \( 3 + 2i \) is plotted at the point (3, 2)
• The complex number \( -4 + i \) is plotted at the point (-4, 1)
• The complex number \( 5 \) (which is \( 5 + 0i \)) is plotted at (5, 0) on the real axis
• The complex number \( 3i \) (which is \( 0 + 3i \)) is plotted at (0, 3) on the imaginary axis

Absolute Value (Modulus) of a Complex Number

The absolute value or modulus of a complex number \( z = a + bi \), denoted \( |z| \), represents the distance from the origin to the point \( (a, b) \) in the complex plane. Using the distance formula from geometry:

\[ |z| = |a + bi| = \sqrt{a^2 + b^2} \]

Notice that \( |z|^2 = a^2 + b^2 = z \cdot \overline{z} \), which connects the modulus with the complex conjugate.

Example:  Find the absolute value of \( z = -3 + 4i \).

Solution:

Identify the real part \( a = -3 \) and imaginary part \( b = 4 \)

Use the modulus formula: \( |z| = \sqrt{(-3)^2 + 4^2} \)

Calculate: \( |z| = \sqrt{9 + 16} = \sqrt{25} = 5 \)

The absolute value of \( -3 + 4i \) is 5.

Polar Form of Complex Numbers

While the standard form \( a + bi \) uses rectangular coordinates, complex numbers can also be expressed using polar coordinates. In polar form, a complex number is described by its distance from the origin (the modulus \( r \)) and the angle it makes with the positive real axis (called the argument, denoted \( \theta \)).

The polar form of a complex number is:

\[ z = r(\cos\theta + i\sin\theta) \]

or more compactly using cis notation:

\[ z = r \, \text{cis} \, \theta \]

where \( r = |z| = \sqrt{a^2 + b^2} \) is the modulus and \( \theta \) is the argument.

Finding the Argument

The argument \( \theta \) is the angle measured counterclockwise from the positive real axis to the line segment connecting the origin to the point \( (a, b) \). We can find \( \theta \) using:

\[ \tan\theta = \frac{b}{a} \]

However, we must be careful about which quadrant the complex number lies in, since the arctangent function only gives values between -90° and 90°. The principal argument is typically chosen to be in the range \( -\pi < \theta="" \leq="" \pi="" \)="" (or="" \(="" 0="" \leq="" \theta="">< 2\pi="">

Converting Between Forms

From rectangular to polar:

• Calculate \( r = \sqrt{a^2 + b^2} \)
• Calculate \( \theta = \arctan\left(\frac{b}{a}\right) \), adjusting for the correct quadrant

From polar to rectangular:

• Calculate \( a = r\cos\theta \)
• Calculate \( b = r\sin\theta \)

Example:  Convert \( z = 1 + \sqrt{3}i \) to polar form.

Solution:

Find the modulus: \( r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \)

Find the argument: \( \tan\theta = \frac{\sqrt{3}}{1} = \sqrt{3} \)

Since both \( a = 1 \) and \( b = \sqrt{3} \) are positive, the point is in the first quadrant.

The angle whose tangent is \( \sqrt{3} \) is \( \theta = \frac{\pi}{3} \) radians (or 60°)

Therefore, \( z = 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right) \) or \( z = 2 \, \text{cis} \, \frac{\pi}{3} \)

The polar form is 2(cos π/3 + i sin π/3).

Multiplying and Dividing in Polar Form

One of the great advantages of polar form is that multiplication and division become simpler. When multiplying complex numbers in polar form, we multiply their moduli and add their arguments. When dividing, we divide their moduli and subtract their arguments.

Multiplication: If \( z_1 = r_1(\cos\theta_1 + i\sin\theta_1) \) and \( z_2 = r_2(\cos\theta_2 + i\sin\theta_2) \), then:

\[ z_1 \cdot z_2 = r_1r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)] \]

Division: If \( z_1 = r_1(\cos\theta_1 + i\sin\theta_1) \) and \( z_2 = r_2(\cos\theta_2 + i\sin\theta_2) \), then:

\[ \frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)] \]

Example:  Multiply \( z_1 = 2(\cos 30° + i\sin 30°) \) and \( z_2 = 3(\cos 45° + i\sin 45°) \).

Solution:

Multiply the moduli: \( r = 2 \times 3 = 6 \)

Add the arguments: \( \theta = 30° + 45° = 75° \)

Write the result in polar form: \( z_1 \cdot z_2 = 6(\cos 75° + i\sin 75°) \)

The product is 6(cos 75° + i sin 75°).

De Moivre's Theorem

De Moivre's Theorem provides a powerful method for raising complex numbers to integer powers. If \( z = r(\cos\theta + i\sin\theta) \) and \( n \) is an integer, then:

\[ z^n = r^n(\cos n\theta + i\sin n\theta) \]

This theorem tells us that to raise a complex number in polar form to the \( n \)th power, we raise the modulus to the \( n \)th power and multiply the argument by \( n \).

Example:  Use De Moivre's Theorem to find \( (1 + i)^6 \).

Solution:

First, convert \( 1 + i \) to polar form.

Modulus: \( r = \sqrt{1^2 + 1^2} = \sqrt{2} \)

Argument: \( \tan\theta = \frac{1}{1} = 1 \), so \( \theta = \frac{\pi}{4} \) (first quadrant)

Polar form: \( 1 + i = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right) \)

Apply De Moivre's Theorem with \( n = 6 \):

\( (1 + i)^6 = (\sqrt{2})^6\left(\cos\frac{6\pi}{4} + i\sin\frac{6\pi}{4}\right) \)

Simplify: \( (\sqrt{2})^6 = 2^3 = 8 \) and \( \frac{6\pi}{4} = \frac{3\pi}{2} \)

\( (1 + i)^6 = 8\left(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2}\right) = 8(0 + i(-1)) = -8i \)

The value of \( (1 + i)^6 \) is -8i.