Complex numbers

Complex numbers provide an alternative way of representing the location of points on the XY plane. They’re similar to {(x, y)} coordinates, but we can also manipulate them using arithmetic operators, which allows us to express certain geometric concepts very simply and concisely.

This introduction is for students who are familiar with 2-dimensional vectors.

Index

Introducing {i}

If the variable {x} represents a real number, then the following equation doesn’t have a solution:

{x^{\large 2} = -1}

We can provide a solution to that equation if we’re willing to consider numbers that are not real numbers. We can solve the equation using the number {i,} which is a special type of number called an imaginary number. The number {i} is defined as:

{i = \sqrt{-1}}

and it provides a solution to the original equation:

{i^{\large 2} = -1}

It’s important to understand that when {i} is defined to be {\sqrt{-1},} the following are true:

• {i} is not a variable, but is actually a number — specifically, an imaginary number.

• The number {i} is similar to the number {\pi,} in that they are each a special number that’s written using a letter symbol. However, {i} is not a real number like {\pi} is.

• Some of the rules and limitations of the real numbers are no longer true. For example, if we’re limited to using only real numbers, then {\sqrt{-1}} is undefined. But if we allow the imaginary number {i} to be used, then {\sqrt{-1}} is no longer undefined.

Real and imaginary numbers

Every imaginary number can be expressed as a real number multiplied by {i}:

An imaginary number is a number of the form {ri,}
where {r} is any real number, and {i = \sqrt{-1}.}

For example, the numbers {2i} and {-3i} are both imaginary numbers.

The square root of every negative real number is an imaginary number. Here’s an example:

{\sqrt{-9} \ = \ \sqrt{9 \times -1} \ = \ \sqrt{9} \times \sqrt{-1} \ = \ 3i}

When we combine real numbers and imaginary numbers using multiplication and division, we apply the following rules:

1.	real	{\times} real	{=} real
2.	real	{\div} real	{=} real
3.	real	{\times} imaginary	{=} imaginary
4.	real	{\div} imaginary	{=} imaginary
5.	imaginary	{\times} real	{=} imaginary
6.	imaginary	{\div} real	{=} imaginary
7.	imaginary	{\times} imaginary	{=} real
8.	imaginary	{\div} imaginary	{=} real

Here’s an example that shows rule 7:

{3i \times 2i \ \ = \ \ 3 \times i \times 2 \times i \ \ = \ \ 6 \times i^2 \ \ = \ \ 6 \times -1 \ \ = \ \ -6}

Be cautioned that the meaning of the word “real” in mathematics is not the same as the meaning of the word “real” in ordinary English. In mathematics, the word “real” is technical jargon that describes a specific type of number, and is completely unrelated to the concept of existence or reality. This same caution also applies to many other jargon terms that are used to describe numbers, such as “natural”, “irrational”, and “imaginary”. Each of these terms has a specific technical definition, and in mathematics they should never be interpreted in the same way they are in ordinary English.

Complex numbers

If you add a real and an imaginary number together, the result is called a complex number:

A complex number is the sum of a real number and an imaginary number.

Every complex number can be expressed as:

{x + yi}

where:

{x} is a real number, called the real part

{y} is a real number, called the imaginary part

{yi} is an imaginary number, and

{i = \sqrt{-1}}

For example, {3 + 2i} is a complex number. Despite looking like an algebra expression, {3 + 2i} is considered to be a number.

Here are some examples of complex numbers broken into their two separate parts:

{0}	real part {= 0}	imaginary part {= 0}
{1}	real part {= 1}	imaginary part {= 0}
{i}	real part {= 0}	imaginary part {= 1}
{-i}	real part {= 0}	imaginary part {= -1}
{2i}	real part {= 0}	imaginary part {= 2}
{3 + 2i}	real part {= 3}	imaginary part {= 2}
{-3 - 2i}	real part {= -3}	imaginary part {= -2}

A number might have multiple types. For example:

{1 \ }	is both real and complex, because {1 = 1 + 0i}
{i \ }	is both imaginary and complex, because {i = 0 + i}
{0 \ }	is real, imaginary and complex, because {0 = 0i = 0 + 0i}

Complex arithmetic

Arithmetic with complex numbers obeys all the rules of algebra.

When performing algebra with complex numbers, note that the expression {i^{\large 2}} can be simplified to {-1.}

Here are some examples of complex arithmetic:

Conjugate pair

In the division example from the previous section, notice that multiplying {(3 + 2i)} and {(3 - 2i)} together produced the real number {13.} This simplified the denominator, which allowed us to split the result into its real and imaginary components.

In general, the numbers {\, x + yi \,} and {\, x - yi \,} are called a conjugate pair:

The conjugate of {\, x + yi \,} is {\, x - yi, \,} and vice versa.

If we define the complex variable {z} to be:

{z = x + yi}

then it’s common to use the symbol {\skew1\overline{z}} or {z^*} to specify its conjugate:

Exercises (part 1)

Evaluate each of the following to get a complex number result:

1. [[ (2 - 3i) \times i ]] Answer

   {(2 \times i) - (3i \times i)}	(distribute)
   {2i - 3i^{\large 2}}	(simplify)
   {2i + 3}	(substitute {i^{\large 2} = -1})
   {3 + 2i}	(place the terms in standard order)

2. [[ (3 - i)^{\large 2} ]] Answer

   {(3 - i) \times (3 - i)}	(expand)
   {3^{\large 2} - 3i - 3i + i^{\large 2}}	(distribute using FOIL)
   {9 - 6i - 1}	(simplify)
   {8 - 6i}	(simplify)

3. [[ {10} \over {2 + i} ]] Answer

   [[ {10 \times (2 - i)} \over {(2 + i) \times (2 - i)} ]]	(multiply by [[ {(2 - i)} \over {(2 - i) } ]])
   [[ {20 - 10i} \over {4 - 2i + 2i - i^{\large 2}} ]]	(distribute)
   [[ {20 - 10i} \over {4 - 2i + 2i + 1} ]]	(substitute {i^{\large 2} = -1})
   [[ {20 - 10i} \over {5} ]]	(simplify)
   [[ 4 - 2i ]]	(simplify)

Complex plane

Every real number has a corresponding point on the real number line that represents it.

Likewise, we can construct an imaginary number line on which every point represents an imaginary number.

These two number lines are placed at right angles to each other to define the complex plane, in the same way that the {x} and {y} axes define the XY plane:

Imaginary axis 🡇
	🡄 Real axis

Every point on this plane represents a complex number.

There’s a natural one-to-one correspondence between every 2D vector on the XY plane and every complex number on the complex plane:

The 2D vector {(x,y)} corresponds to the complex number {x + yi.}

When switching from the XY plane to the complex plane, we make the following changes in notation and terminology:

• The horizontal {x} axis is called the real axis.
• The vertical {y} axis is called the imaginary axis.
• The point at coordinates {(x,0)} is represented by the real number {x.}
• The point at coordinates {(0,y)} is represented by the imaginary number {yi.}
• The point at coordinates {(x,y)} is represented by the complex number {x + yi.}

Comparison with 2D vectors

There are several 2D vector operators that translate directly to complex numbers, with only a change in notation:

To show a 2D vector on the XY plane, we draw an arrow that illustrates its length and direction. The arrow suggests the idea of displacement, which is the movement from one point to another on the plane.

We can likewise draw an arrow for any complex number on the complex plane. For example, here’s the arrow for the number {3 + 2i}:

The arrow helps explain the geometric meaning of the operators that were listed above:

• Addition is used to “chain” displacements together, end to end, combining them together into a single displacement.

• Subtraction is used to find the displacement from one point to another. The order of the subtraction is always end {-} start.

• Negation is used to reverse the direction of the arrow, without changing its length.

• Multiplication by a real number {(r(x,y)} or {r(x + yi))} results in an arrow that’s {r} times longer, and points in the same direction, or points in the opposite direction if {r < 0.}

Notice that the arrow is drawn with its tail at {0.} The tail of an arrow is always assumed to be at {0,} unless otherwise specified.

Complex absolute value

In the previous section, we introduced the idea of drawing an arrow for any complex number on the complex plane. For example, here’s the arrow for {z,} where {z} represents the number {3 + 2i}:

We can calculate the length of the arrow using the Pythagorean:

{|z| = |x + yi| = \sqrt{x^{\large 2} + y^{\large 2}} }

The result {|z|} is called the absolute value of {z}. It’s the distance between {0} and {z,} and is defined the same way as the length of the corresponding 2D vector:

{|(x, y)| = \sqrt{x^{\large 2} + y^{\large 2}} }

The absolute value is also sometimes called the magnitude or the modulus.

Complex angle

Let’s continue looking at the arrow from {0} to {z} on the complex plane.

We can define the direction of this arrow by measuring the angle from the positive real axis to the arrow line, going counterclockwise. This angle is shown as {\theta} in the following diagram:

The angle of {z} is also frequently called the argument of {z.}

A special function called {\mathbf{\text{arg}}} is used to convert a complex number to its direction angle. The {\mathbf{\text{arg}}} function is defined using trigonometry:

[[ \arg(z) = \begin{cases} \phantom{-} \arccos \left( \displaystyle {{{\text {Re}}(z)} \over {|z|}} \right) \space\space \text{ if } \, {{\text {Im}}(z)} \ge 0 \\[6pt] - \arccos \left( \displaystyle {{{\text {Re}}(z)} \over {|z|}} \right) \space\space \text{ if } \, {{\text {Im}}(z)} < 0 \\ \end{cases} ]]

The {\mathbf{\text{arg}}} function produces an angle {\theta} that’s in the range {-180° < \theta \le 180°, } and {\theta} will always have the same sign as {{\text {Im}}(z).} This means that all arrows having a downward vertical displacement also have a negative angle. This convention was chosen because it’s natural to associate downward with negative.

The direction of an angle is unchanged if {360°} is added to it. As a result, every direction has an infinite number of angles that represent it. The {\mathbf{\text{arg}}} function produces the principal direction angle, which is the value of {\theta} that’s in the range {-180° < \theta \le 180°.}

Exercise (part 2)

[[ |z| \ = \ \sqrt{(-2)^{\large 2} + (-2)^{\large 2}} \ = \ \sqrt{4+4} \ = \ \sqrt{8} ]]

[[ \arg(z) \ = \ \mathord{-}135° ]]

Multiplication by {i}

Let’s multiply the number {x + yi} by {i}:

{(x + yi) \times i}	(given)
{(x \times i) + (yi \times i)}	(distribute)
{xi + yi^{\large 2}}	(simplify)
{xi - y}	(substitute {i^{\large 2} = -1})
{- y + xi}	(place the terms in standard order)

If you compare the first and last lines, you can see that multiplication by {i} swaps the real and imaginary parts, and then negates the new real part.

Geometrically, this causes the number’s arrow to rotate counterclockwise by {90°} around its tail, without changing its length.

If you click the mouse in the following diagram, you can change the position of the complex number {z,} and see where {zi} is located.

The integer powers of {i}

It’s interesting to look at the integer powers of {i}:

{ \begin{array}{lllr} i^{\large 0} &\hspace-5pt &\hspace-5pt &\hspace-5pt = &\hspace-5pt 1 \\[-2pt] i^{\large 1} &\hspace-5pt &\hspace-5pt &\hspace-5pt = &\hspace-5pt i \\[-2pt] i^{\large 2} &\hspace-5pt &\hspace-5pt &\hspace-5pt = &\hspace-5pt -1 \\[-2pt] i^{\large 3} &\hspace-5pt = &\hspace-5pt i^{\large 2} \times i &\hspace-5pt = &\hspace-5pt -i \\[-2pt] i^{\large 4} &\hspace-5pt = &\hspace-5pt i^{\large 2} \times i^{\large 2} &\hspace-5pt = &\hspace-5pt 1 \\[-2pt] i^{\large 5} &\hspace-5pt = &\hspace-5pt i^{\large 4} \times i &\hspace-5pt = &\hspace-5pt i \\[-2pt] i^{\large 6} &\hspace-5pt = &\hspace-5pt i^{\large 4} \times i^{\large 2} &\hspace-5pt = &\hspace-5pt -1 \\[-2pt] i^{\large 7} &\hspace-5pt = &\hspace-5pt i^{\large 6} \times i &\hspace-5pt = &\hspace-5pt -i \\[-2pt] i^{\large 8} &\hspace-5pt = &\hspace-5pt i^{\large 6} \times i^{\large 2} &\hspace-5pt = &\hspace-5pt 1 \\[-2pt] \cdots \end{array} }

Observe the sequence numbers on the right side of the equations. Notice that they repeat in an endless cycle around four points on the complex plane:

And here are those four numbers written as powers of {i}:

and also

In the previous section, we saw how can we multiply a complex number {z} by {i} to obtain a new number ({zi}) that has an arrow rotated {90°} counterclockwise from {z.}

Likewise, if we multiply {z} by an integer power of {i,} the arrow of {z} will undergo multiple {90°} rotations:

{z \times i^{\large 0}}	{\ = z}	(rotates the arrow of {z} by {0°})
{z \times i^{\large 1}}	{\ = zi}	(rotates the arrow of {z} by {90°})
{z \times i^{\large 2}}	{\ = -z}	(rotates the arrow of {z} by {180°})
{z \times i^{\large 3}}	{\ = -zi}	(rotates the arrow of {z} by {270°})
{z \times i^{\large 4}}	{\ = z}	(rotates the arrow of {z} by {360°})
{\cdots}

Exercise (part 3)

The square root of {i}

In the previous section, we saw how the integer powers of {i} are arranged in a circle around the origin, creating an endless cycle of four numbers: {1, i, -1,} and {-i.}

We can also see that those four numbers are evenly spaced along the circumference of the circle, so, for example, {i^{\large 1}} is exactly half-way between {i^{\large 0}} and {i^{\large 2}} along the circumference.

This evenly-spaced circular arrangement raises an interesting question:

Is {i^{\large{\left({1 \over 2}\right)}}} exactly half-way between {i^{\large 0}} and {i^{\large 1}} along the circumference of the circle?

The answer is yes, as you can see in this diagram:

Since { {i^{\large{\left({1 \over 2}\right)}}} } is half-way between {i^{\large 0}} and {i^{\large 1},} its arrow must form a {45°} angle with the positive real axis, as shown in the following enlargement:

The shaded area is a {45°} right triangle with a hypotenuse of {1.} Using the Pythagorean theorem, we can easily find that the lengths of the other two sides are both {\vphantom{\large \sqrt{{1 \over 2}}} \sqrt{{1 \over 2}}.} Those side lengths give us the real and imaginary parts of the complex number { {i^{\large{\left({1 \over 2}\right)}}} }:

{ {i^{\large{\left({1 \over 2}\right)}}} \ = \ \sqrt{{1 \over 2}} + \sqrt{{1 \over 2}} \, i }

The number {i^{\large{\left({1 \over 2}\right)}}} is the same as {\sqrt{i},} so if we calculate its square, we should obtain {i.} The following algebra steps show that this is indeed the case:

{ {i^{\large{\left({1 \over 2}\right)}}} \ = \ \sqrt{{1 \over 2}} + \sqrt{{1 \over 2}} \, i }	(given)
{ i \ = \ \left( \sqrt{{1 \over 2}} + \sqrt{{1 \over 2}} \, i \right)^{\large 2} }	(square both sides)
{ \phantom{i} \ = \ \left( \sqrt{{1 \over 2}} + \sqrt{{1 \over 2}} \, i \right) \times \left( \sqrt{{1 \over 2}} + \sqrt{{1 \over 2}} \, i \right) }	(expand)
{ \phantom{i} \ = \ {1 \over 2} + {1 \over 2}i + {1 \over 2}i + {1 \over 2}i^{\large 2} }	(distribute using FOIL)
{ \phantom{i} \ = \ {1 \over 2} + {1 \over 2}i + {1 \over 2}i - {1 \over 2} }	(substitute {i^{\large 2} = -1})
{ \phantom{i} \ = \ 0 + i }	(simplify)
{ \phantom{i} \ = \ i }	(simplify)

The real powers of {i}

In the past few sections, we looked at the geometric relationships among some of the powers of {i}:

The exponents of {i} can serve as a way of identifying the various points along the circumference of the circle, ranging from {0} to {4,} going counterclockwise, in an evenly-spaced arrangement around the circumference.

This observation leads us to the following identity:

[[ \arg(i^{\Large q}) \ = \ q \times {90°} ]]

The letter {q} is used here because it means “quadrant”.

This identity means that if you draw an arrow from {0} to {1} and rotate it {q \times {90°}} counterclockwise around its tail, the tip of the arrow will arrive at {i^{\Large q}.\ } If {q < 0,} the rotation is clockwise.

It’s notable that {q} can be any real number. For example, if {q = {\large {1 \over 2}},} then the rotation is “one-half of a quadrant”, or {45 °.}

All the real powers of {i} lie along the circle with radius {1} that’s centered on {0,} as shown in the diagram above. We can express this fact with the following identity:

[[ |i^{\Large q}| \ = \ 1 \ ]] (for all real {q})

Complex multiplication

In this section, we explore the geometric meaning of complex multiplication.

Given two complex numbers {a} and {b,} their product {ab} has the following two properties:

{|ab| = |a| |b|}
{\arg(ab) = \arg(a) + \arg(b)}

If you click the mouse in the following diagram, you can change the positions of {a} and {b.} As you move {a} and {b} around, observe the two properties listed above.

Once you click on {a} or {b} to position it, you can then use the keyboard’s arrow keys to make fine adjustments to the point’s position.

A good way to understand the geometric relationship of {a,} {b,} and {ab} is to compare the triangles Δ {0,1,a} and Δ {0,b,ab} and observe that the two triangles are similar. You can click this button: Triangles to show or hide the triangles in the above diagram. This allows you to better see how multiplication by {b} simultaneously scales and rotates.

Complex multiplication: summary of special cases

Here is a summary of the special cases of complex multiplication:

1. Multiplication of a complex number and a real number

{r(x + yi) = (rx + ryi)}

This results in an arrow that’s {r} times longer, and points in the same direction, or points in the opposite direction if {r < 0.}

2. Multiplication of a complex number and {i}

{i(x + yi) = (-y + xi)}

This results in an arrow of the same length that’s rotated {90°} counterclockwise

3. Multiplication of a complex number and an imaginary number

{ri(x + yi) = (-ry + rxi)}

This results in an arrow that’s {r} times longer, and is rotated {90°.} The rotation is counterclockwise if {r > 0,} or clockwise if {r < 0.}

4. Multiplication of a complex number and {i^{\Large q}}

This results in an arrow of the same length that’s rotated {q \times 90°.} The rotation is counterclockwise if {q > 0,} or clockwise if {q < 0.}

5. Multiplication of a complex number and {ri^{\Large q}}:

This results in an arrow that’s {r} times longer, and is rotated {q \times 90°.} The rotation is counterclockwise if {q > 0,} or clockwise if {q < 0.} The direction of the arrow is reversed if {r < 0.}

Rotation forms

In the previous section, we saw that this expression:

{z \times i^{\Large q}}

produces a number whose arrow is the same length as {z,} but is rotated {q \times 90°.} Recall that {q} can be any real number, so this expression is capable of rotation by any angle.

In this section, we look at several alternative forms for this expression.

Here’s a form that allows you to specify the angle of rotation in degrees:

{z \times i^{\raise 5pt \LARGE{\left({\theta °} \over {90 °}\right)}} \ \ } rotates {z} by {\theta °} degrees

and this form allows you to specify the angle in radians:

{ z \times i^{\raise 5pt \LARGE{\left({2 \theta} \over {\pi}\right)}} \ \ } rotates {z} by {\theta} radians

Here’s a form that multiplies {z} by a number that’s in the more familiar “real {+} imaginary” format:

{ z \times (\cos \theta + i \sin \theta) \ \ } rotates {z} by {\theta} radians

and, finally, here’s a form that’s especially concise:

{ z \times e^{\Large{i\theta}} \ \ } rotates {z} by {\theta} radians, where {e} is Euler’s number {(= 2.71828\text{...})}

It’s beyond the scope of this chapter to explain why the form {(z \times e^{\Large{i\theta}})} rotates {z} by {\theta} radians, but it’s introduced here because it’s very commonly used.