# Complex Number

• ## complex number 4 marks(SAQ)

Note: This section is of mathematical interest and students should be encouraged to read it. In particular, it is helpful for them to understand why the complex numbers are not really any more mathematically abstract than the reals. However, knowledge of this section is not required by the current intermediate  syllabus and is not necessary for an understanding of how to use complex numbers to solve equations.

The set of complex numbers is a number system, just like the set of reals or the set of integers. Like the reals or even the integers, they were developed to close a mathematical ‘gap’ in another number system. So what is this mathematical gap?

Consider the linear equation $Latex formula$. This has one solution in the real numbers: $Latex formula$.
Now consider the equation $Latex formula$. Algebraic manipulation shows that this is equivalent to solving $Latex formula$. Since squares of real numbers are non-negative we would say that this has ‘no real solutions’.
Consider finally a general quadratic equation $Latex formula$. To solve this, we use the quadratic formula, which gives us $Latex formula$ where $Latex formula$ is the discriminant. If the discriminant is negative, we would again say that the equation has ‘no real solutions’.

What has happened here is that squares of real numbers are always non-negative. We cannot square a real number and get a negative number. To close this gap, we extend the reals to a number system where squares can also be negative. The easiest way to achieve this is to introduce some number whose square is $Latex formula$. This number $Latex formula$, is defined by $Latex formula$. Then if we want to solve $Latex formula$ where $Latex formula$ is some positive real, we get $Latex formula$, or $Latex formula$. If we put all numbers of the form $Latex formula$ (where $Latex formula$ is real) in our new number system, we can now solve any quadratic equation with real coefficients. Numbers of the form $Latex formula$ ($Latex formula$ real) are known as ‘imaginary numbers’.

In order to turn our set of numbers into a proper number system, we want to introduce some operations so that we can do things with these numbers. The two most fundamental operations of any set (or field) of numbers are addition ($Latex formula$) and multiplication ($Latex formula$). We will define these operations properly later. For now, all you need to know is that if you take two real numbers and add them using complex addition, the result is the same as if you added them using real addition. Similarly, if you take two real numbers and multiply them using complex multiplication, the result is the same as if you multiplied them using real multiplication.

Since the real numbers are closed under addition ($Latex formula$) and multiplication ($Latex formula$), we want this to hold true for our new number system too. What this means is that if we take any two numbers in the number system (e.g. $Latex formula$ and $Latex formula$), we want their sum $Latex formula$ to be in the number system (closure under addition).We also want their product ($Latex formula$) to be in the number system (closure under multiplication). If we take the closure of the real and imaginary numbers, we get all numbers of the form $Latex formula$ ( $Latex formula$, $Latex formula$ real) must be in the new number system. A little bit of complex number arithmetic shows that this is enough to guarantee closure under addition and multiplication.

So now we have a new set of numbers, the complex numbers $Latex formula$, where each complex number $Latex formula$ can be written in the form $Latex formula$ (where $Latex formula$, $Latex formula$ are real and $Latex formula$). The set of complex numbers is closed under addition and multiplication. Furthermore, each real number $Latex formula$ is in the set of complex numbers, $Latex formula$, so that the real numbers are a subset of the complex numbers (see Figure 1). Finally, any quadratic equation with real coefficients, or even any polynomial with real coefficients, has solutions that can be represented as complex numbers.
Side note: Indeed what makes the complex numbers so powerful is that all degree n polynomials with complex coefficients have n (not necessarily distinct) complex roots. We say that $Latex formula$ is ‘algebraically closed’.

Figure 1:

This figure shows that the naturals are a subset of the integers, which are a subset of the rationals, which are a subset of the reals, which are a subset of the complex numbers. The set of numbers in the reals which are not rationals are known as the irrationals.
Students should note that whilst the real numbers and the imaginary numbers are both subsets of the complex numbers, there are complex numbers (such as $Latex formula$) which are neither real nor imaginary.

## Arithmetic of Complex Numbers

Recall that every complex number is the sum of a real number and imaginary number. We say that $Latex formula$ has a real part, $Latex formula$, and an imaginary part, $Latex formula$. If $Latex formula$, these are given by:

 $Latex formula$ $Latex formula$

In the above notation, notice how much a complex number looks like a surd (e.g. compare $Latex formula$ and $Latex formula$). The only difference is that the number under the square root sign is negative.In fact, when it comes to arithmetic, complex numbers can be treated like surds. This concept is useful when remembering how to add, subtract, multiply or divide complex numbers. Pretend that your complex number is a surd, perform the same operations and everything should work out.

### Addition and Subtraction

In order to add two complex numbers, their real and imaginary parts should be added separately. In order to subtract two complex numbers, their real and imaginary parts should be subtracted separately.
Note: Adding $Latex formula$ is the same as subtracting $Latex formula$, and subtracting $Latex formula$\ is the same as adding $Latex formula$.

 Addition:$Latex formula$ Or $Latex formula$ Subtraction: $Latex formula$ Or $Latex formula$

#### Example 1

If the complex numbers $Latex formula$ and $Latex formula$ are given by $Latex formula$ and $Latex formula$, determine:

a) $Latex formula$

b) $Latex formula$

#### Solution 1

a) $Latex formula$

$Latex formula$ $Latex formula$b) $Latex formula$

$Latex formula$$Latex formula$

Notice that this is just like adding or subtracting surds:

$Latex formula$ $Latex formula$

### Multiplication of Complex Numbers

In order to multiply two complex numbers, each number should be treated like a surd and the FOIL method applied as appropriate. Students should note that just as the square of the irrational part of a surd gives a rational number, the square of an imaginary number gives a real number.

 $Latex formula$ $Latex formula$

#### Example 2

If the complex numbers $Latex formula$ and $Latex formula$ are given by $Latex formula$ and $Latex formula$, determine $Latex formula$.

#### Solution 2

$Latex formula$ $Latex formula$ (applying FOIL)
$Latex formula$(collecting ‘like’ terms: real and imaginary parts)

$Latex formula$

• ## What are Complex Numbers and why do we need them?

• ### For a given complex number $Latex formula$, its complex conjugate is denoted by $Latex formula$ and is given by:

 $Latex formula$

Students are required to know how to determine the complex conjugate of a given complex number. The following example illustrates how this is done.

#### Example 3

(HSC 1994 Question 2aii)
Let $Latex formula$, where $Latex formula$ and $Latex formula$ are real.Find $Latex formula$ in the form $Latex formula$, where $Latex formula$ and $Latex formula$ are real.

#### Solution 3

First, express the complex number under the bar as the sum of its real and imaginary components:
$Latex formula$
$Latex formula$
$Latex formula$
To find the conjugate, replace the imaginary component with its negative counterpart
$Latex formula$

Complex conjugates have a number of nice properties:

 $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$

Students will eventually be required to prove all these properties (note that proving the 4th and 6th properties requires reading ahead to the next few section). The next example illustrates how one would go about proving such results.

#### Example 4

For any complex numbers $Latex formula$ and $Latex formula$, prove that $Latex formula$.

#### Solution 4

Let $Latex formula$$Latex formula$

$Latex formula$
$Latex formula$
$Latex formula$
$Latex formula$

### Division of Complex Numbers

One nice thing about complex conjugates is that if a complex number is multiplied with its complex conjugate, the result is a real number: $Latex formula$$Latex formula$$Latex formula$

 $Latex formula$

Notice how this is analogous to multiplying a surd with its conjugate surd, where the result is a rational number.

This result becomes very useful when taking the reciprocals of complex numbers. Just as multiplying the top and bottom of a surd fraction by the conjugate surd of the denominator rationalises the denominator, multiplying top and bottom of a complex number fraction by the complex conjugate of the denominator makes the denominator real.$Latex formula$

Similarly, when dividing by a complex number $Latex formula$, simply multiply top and bottom by the complex conjugate of $Latex formula$ and then apply complex multiplication to the numerator.

#### Example 5

If the complex numbers $Latex formula$ and $Latex formula$ are given by $Latex formula$ and $Latex formula$, determine $Latex formula$.

#### Solution 5

$Latex formula$ $Latex formula$$Latex formula$

### Equality of Complex Numbers

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

 $Latex formula$ if and only if $Latex formula$ AND $Latex formula$

### Finding the Square Roots of Real and Complex Numbers

To determine the square root of a real number $Latex formula$:

• If $Latex formula$$Latex formula$, its square roots are the real numbers $Latex formula$ and $Latex formula$.
• If $Latex formula$,$Latex formula$ and $Latex formula$ are real numbers. Therefore $Latex formula$ and $Latex formula$ are imaginary numbers and $Latex formula$, $Latex formula$. Therefore the square roots of $Latex formula$ are $Latex formula$.

Determining the square root of a complex number $Latex formula$ is a slightly more complicated process. One way to think about it is that it is equivalent to determining which complex numbers can be squared to give $Latex formula$. That is, find all $Latex formula$ real such that $Latex formula$. The general method is as follows:

• Use complex multiplication to obtain the real and imaginary parts of the LHS.
• $Latex formula$
• Since two complex numbers are equal if and only if their real and imaginary parts are equal, equate the real and imaginary parts of the LHS and RHS to obtain two simultaneous equations:
• Comparing real parts gives $Latex formula$
• Comparing imaginary parts gives $Latex formula$
• Solve for $Latex formula$ and $Latex formula$, either by inspection or by substitution.

The following example demonstrates how both variants of the above method can be used to determine the square roots of a complex number.

#### Example 6

Find all complex solutions to the equation $Latex formula$

#### Solution 6

Let the solutions be of the form $Latex formula$.

Then $Latex formula$

so that $Latex formula$ and $Latex formula$.

Variant 1:

By inspection, $Latex formula$, $Latex formula$ satisfies these simultaneous equations

Checking:

$Latex formula$ and $Latex formula$ so that $Latex formula$ is a solution

$Latex formula$ and $Latex formula$ so that $Latex formula$ is a solution

Variant 2:

From the second equation, $Latex formula$ so that $Latex formula$.

Substituting into the first equation yields $Latex formula$

so that $Latex formula$$Latex formula$ and $Latex formula$ or $Latex formula$.

Since the second equation has no real solutions, $Latex formula$ and $Latex formula$.

Corresponding y can be determined using $Latex formula$ so that $Latex formula$.

Therefore $Latex formula$ and $Latex formula$ are the solutions to the given equation.

Note that the first method, solve by inspection, is a lot shorter and tidier than the second method. Thus if students are able to determine the solutions to the two simultaneous equations by inspection, they are encouraged to use this method. Just to safe, however, they should always include a few lines of checking.

The second method has the advantage of providing a series of steps which is guaranteed to yield an answer. If a student cannot easily determine the answer by inspection they should proceed to solve the equations algebraically.

Students should note that since two simultaneous equations in two variables ($Latex formula$ and $Latex formula$) yield exactly two (not necessarily distinct) solutions, each complex number has two, not necessarily distinct,complex square roots. This can be seen in Figure 2. Furthermore, as the above example illustrates, if $Latex formula$ is a square root then $Latex formula$ is also one. Therefore if the two complex square roots are not distinct, $Latex formula$ and $Latex formula$ so that $Latex formula$.

The conclusion is that each non-zero complex number has exactly two distinct complex square roots, and $Latex formula$ has exactly one square root.

## Using Complex Numbers to Solve Equations

### Solving Quadratic Equations with Real Coefficients

Consider a general quadratic equation $Latex formula$.This can be solved by factorising the equation, completing the square or using the quadratic formula. The first method is usually done by inspection and so extending it to encompass complex roots would requires a degree of intuition about the behaviour of complex numbers that is beyond that which is expected from students. The second and third, however, involve algebraic manipulations that can easily be extended by using complex numbers and performing standard arithmetic calculations on complex numbers. The following example demonstrates how the roots of a quadratic equation with real coefficients can be determined using both the extended versions both of completing the square and using the quadratic formula.

#### Example 7

Find all complex solutions to the equation $Latex formula$.

#### Solution 7

Method 1- Completing the Square:

Starting with the equation $Latex formula$, completing the square yields

$Latex formula$$Latex formula$

$Latex formula$ $Latex formula$Method 2- Using the Quadratic Formula:

$Latex formula$ $Latex formula$$Latex formula$

### Solving Quadratic Equations with Complex Coefficients

Consider again a general quadratic equation $Latex formula$. This time, $Latex formula$, $Latex formula$ and $Latex formula$ are complex numbers. However, this doesn’t change the method that we use to determine the possible value of $Latex formula$. Solving by inspection may be a little harder, however completing the square and using the quadratic formula are still equally viable methods.

The following example illustrates how to find the roots of a quadratic equation with complex coefficients.

#### Example 8

Find all complex solutions to the equation $Latex formula$.

#### Solution 8

Method 1- Completing the Square:

Completing the square on the left hand side of the equation yields

$Latex formula$
$Latex formula$
$Latex formula$
$Latex formula$

so that $Latex formula$ and $Latex formula$

Let $Latex formula$, $Latex formula$, $Latex formula$ real.

Notice how in this step, we don’t automatically jump to ‘let $Latex formula$ ’. This is because we know that the quantities that we’re most familiar with dealing with the real and imaginary parts of the number being squared. Of course, it is also fine to let $Latex formula$, but the working will probably be a little more tedious and a little less familiar.

Then $Latex formula$ so that $Latex formula$ and $Latex formula$.

The first equation gives $Latex formula$

The second equation gives $Latex formula$ so that $Latex formula$ and $Latex formula$.

$Latex formula$ or $Latex formula$

$Latex formula$ or $Latex formula$

Method 2- Using the Quadratic Formula:

$Latex formula$
$Latex formula$
$Latex formula$
$Latex formula$

Here the square roots of $Latex formula$ need to be obtained. This can be done using the same method as the one used above, where all pairs of $Latex formula$ and $Latex formula$ satisfying $Latex formula$ are determined. This leads to solving the simultaneous equations $Latex formula$ and $Latex formula$.

The solutions to the simultaneous equations can be attained by inspection or through algebraic manipulation and yield $Latex formula$.

$Latex formula$ or $Latex formula$

$Latex formula$ or $Latex formula$

## The Argand Diagram

For every real $Latex formula$ and $Latex formula$ there exists a complex number given by $Latex formula$. From before, if the real parts and the imaginary parts of two complex numbers are equal, then they are the same number. This means that for every real $Latex formula$ and $Latex formula$ there exists a unique complex number given by $Latex formula$. In other words, there is a 1-1 correspondence between ordered pairs of reals $Latex formula$ and the complex numbers. Extending the idea a little further, there is a 1-1 correspondence between the complex numbers and the points in the x-y plane.

 $Latex formula$

This correspondence immediately suggests a way to diagrammatically represent a complex number $Latex formula$: on the x-y plane, with x and y coordinates corresponding to $Latex formula$ and $Latex formula$ respectively. Such a diagram is known as an ‘Argand diagram’ and the plane on which the points are plotted is sometimes referred to as the ‘Argand plane’ or the ‘complex plane’. The x-axis, along which the value of $Latex formula$ can be determined, is referred to as the ‘real axis’ and the y-axis is referred to as the ‘imaginary axis’.

The following example demonstrates how to represent complex numbers of the form $Latex formula$ on an Argand diagram.

### Example 9

$Latex formula$ and $Latex formula$

If $Latex formula$ is restricted by $Latex formula$, the value of $Latex formula$ is unique. If not, there is an infinite family of solutions given by adding $Latex formula$ ($Latex formula$ an integer) to the value of $Latex formula$ between $Latex formula$ and $Latex formula$.
Note that if $Latex formula$ (i.e. $Latex formula$), $Latex formula$ and all values of $Latex formula$ are possible.

For a compvvWhat are Complex Numbers and why do we need them?

Note: This section is of mathematical interest and students should be encouraged to read it. In particular, it is helpful for them to understand why the complex numbers are not really any more mathematically abstract than the reals. However, knowledge of this section is not required by the current HSC syllabus and is not necessary for an understanding of how to use complex numbers to solve equations.

The set of complex numbers is a number system, just like the set of reals or the set of integers. Like the reals or even the integers, they were developed to close a mathematical ‘gap’ in another number system. So what is this mathematical gap?

Consider the linear equation $Latex formula$. This has one solution in the real numbers: $Latex formula$.
Now consider the equation $Latex formula$. Algebraic manipulation shows that this is equivalent to solving $Latex formula$. Since squares of real numbers are non-negative we would say that this has ‘no real solutions’.
Consider finally a general quadratic equation $Latex formula$. To solve this, we use the quadratic formula, which gives us $Latex formula$ where $Latex formula$ is the discriminant. If the discriminant is negative, we would again say that the equation has ‘no real solutions’.

What has happened here is that squares of real numbers are always non-negative. We cannot square a real number and get a negative number. To close this gap, we extend the reals to a number system where squares can also be negative. The easiest way to achieve this is to introduce some number whose square is $Latex formula$. This number $Latex formula$, is defined by $Latex formula$. Then if we want to solve $Latex formula$ where $Latex formula$ is some positive real, we get $Latex formula$, or $Latex formula$. If we put all numbers of the form $Latex formula$ (where $Latex formula$ is real) in our new number system, we can now solve any quadratic equation with real coefficients. Numbers of the form $Latex formula$ ($Latex formula$ real) are known as ‘imaginary numbers’.

In order to turn our set of numbers into a proper number system, we want to introduce some operations so that we can do things with these numbers. The two most fundamental operations of any set (or field) of numbers are addition ($Latex formula$) and multiplication ($Latex formula$). We will define these operations properly later. For now, all you need to know is that if you take two real numbers and add them using complex addition, the result is the same as if you added them using real addition. Similarly, if you take two real numbers and multiply them using complex multiplication, the result is the same as if you multiplied them using real multiplication.

Since the real numbers are closed under addition ($Latex formula$) and multiplication ($Latex formula$), we want this to hold true for our new number system too. What this means is that if we take any two numbers in the number system (e.g. $Latex formula$ and $Latex formula$), we want their sum $Latex formula$ to be in the number system (closure under addition).We also want their product ($Latex formula$) to be in the number system (closure under multiplication). If we take the closure of the real and imaginary numbers, we get all numbers of the form $Latex formula$ ( $Latex formula$, $Latex formula$ real) must be in the new number system. A little bit of complex number arithmetic shows that this is enough to guarantee closure under addition and multiplication.

So now we have a new set of numbers, the complex numbers $Latex formula$, where each complex number $Latex formula$ can be written in the form $Latex formula$ (where $Latex formula$, $Latex formula$ are real and $Latex formula$). The set of complex numbers is closed under addition and multiplication. Furthermore, each real number $Latex formula$ is in the set of complex numbers, $Latex formula$, so that the real numbers are a subset of the complex numbers (see Figure 1). Finally, any quadratic equation with real coefficients, or even any polynomial with real coefficients, has solutions that can be represented as complex numbers.
Side note: Indeed what makes the complex numbers so powerful is that all degree n polynomials with complex coefficients have n (not necessarily distinct) complex roots. We say that $Latex formula$ is ‘algebraically closed’.

Figure 1:

This figure shows that the naturals are a subset of the integers, which are a subset of the rationals, which are a subset of the reals, which are a subset of the complex numbers. The set of numbers in the reals which are not rationals are known as the irrationals.
Students should note that whilst the real numbers and the imaginary numbers are both subsets of the complex numbers, there are complex numbers (such as $Latex formula$) which are neither real nor imaginary.

## Arithmetic of Complex Numbers

Recall that every complex number is the sum of a real number and imaginary number. We say that $Latex formula$ has a real part, $Latex formula$, and an imaginary part, $Latex formula$. If $Latex formula$, these are given by:

 $Latex formula$ $Latex formula$

In the above notation, notice how much a complex number looks like a surd (e.g. compare $Latex formula$ and $Latex formula$). The only difference is that the number under the square root sign is negative.In fact, when it comes to arithmetic, complex numbers can be treated like surds. This concept is useful when remembering how to add, subtract, multiply or divide complex numbers. Pretend that your complex number is a surd, perform the same operations and everything should work out.

### Addition and Subtraction

In order to add two complex numbers, their real and imaginary parts should be added separately. In order to subtract two complex numbers, their real and imaginary parts should be subtracted separately.
Note: Adding $Latex formula$ is the same as subtracting $Latex formula$, and subtracting $Latex formula$\ is the same as adding $Latex formula$.

 Addition:$Latex formula$ Or $Latex formula$ Subtraction: $Latex formula$ Or $Latex formula$

#### Example 1

If the complex numbers $Latex formula$ and $Latex formula$ are given by $Latex formula$ and $Latex formula$, determine:

a) $Latex formula$

b) $Latex formula$

#### Solution 1

a) $Latex formula$

$Latex formula$ $Latex formula$b) $Latex formula$

$Latex formula$$Latex formula$

Notice that this is just like adding or subtracting surds:

$Latex formula$ $Latex formula$

### Multiplication of Complex Numbers

In order to multiply two complex numbers, each number should be treated like a surd and the FOIL method applied as appropriate. Students should note that just as the square of the irrational part of a surd gives a rational number, the square of an imaginary number gives a real number.

 $Latex formula$ $Latex formula$

#### Example 2

If the complex numbers $Latex formula$ and $Latex formula$ are given by $Latex formula$ and $Latex formula$, determine $Latex formula$.

#### Solution 2

$Latex formula$ $Latex formula$ (applying FOIL)
$Latex formula$(collecting ‘like’ terms: real and imaginary parts)

$Latex formula$

### Complex Conjugates and their Properties

For a given complex number $Latex formula$, its complex conjugate is denoted by $Latex formula$ and is given by:

 $Latex formula$

Students are required to know how to determine the complex conjugate of a given complex number. The following example illustrates how this is done.

#### Example 3

(HSC 1994 Question 2aii)
Let $Latex formula$, where $Latex formula$ and $Latex formula$ are real.Find $Latex formula$ in the form $Latex formula$, where $Latex formula$ and $Latex formula$ are real.

#### Solution 3

First, express the complex number under the bar as the sum of its real and imaginary components:
$Latex formula$
$Latex formula$
$Latex formula$
To find the conjugate, replace the imaginary component with its negative counterpart
$Latex formula$

Complex conjugates have a number of nice properties:

 $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$

Students will eventually be required to prove all these properties (note that proving the 4th and 6th properties requires reading ahead to the next few section). The next example illustrates how one would go about proving such results.

#### Example 4

For any complex numbers $Latex formula$ and $Latex formula$, prove that $Latex formula$.

#### Solution 4

Let $Latex formula$$Latex formula$

$Latex formula$
$Latex formula$
$Latex formula$
$Latex formula$

### Division of Complex Numbers

One nice thing about complex conjugates is that if a complex number is multiplied with its complex conjugate, the result is a real number: $Latex formula$$Latex formula$$Latex formula$

 $Latex formula$

Notice how this is analogous to multiplying a surd with its conjugate surd, where the result is a rational number.

This result becomes very useful when taking the reciprocals of complex numbers. Just as multiplying the top and bottom of a surd fraction by the conjugate surd of the denominator rationalises the denominator, multiplying top and bottom of a complex number fraction by the complex conjugate of the denominator makes the denominator real.$Latex formula$

Similarly, when dividing by a complex number $Latex formula$, simply multiply top and bottom by the complex conjugate of $Latex formula$ and then apply complex multiplication to the numerator.

#### Example 5

If the complex numbers $Latex formula$ and $Latex formula$ are given by $Latex formula$ and $Latex formula$, determine $Latex formula$.

#### Solution 5

$Latex formula$ $Latex formula$$Latex formula$

### Equality of Complex Numbers

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

 $Latex formula$ if and only if $Latex formula$ AND $Latex formula$

### Finding the Square Roots of Real and Complex Numbers

To determine the square root of a real number $Latex formula$:

• If $Latex formula$$Latex formula$, its square roots are the real numbers $Latex formula$ and $Latex formula$.
• If $Latex formula$,$Latex formula$ and $Latex formula$ are real numbers. Therefore $Latex formula$ and $Latex formula$ are imaginary numbers and $Latex formula$, $Latex formula$. Therefore the square roots of $Latex formula$ are $Latex formula$.

Determining the square root of a complex number $Latex formula$ is a slightly more complicated process. One way to think about it is that it is equivalent to determining which complex numbers can be squared to give $Latex formula$. That is, find all $Latex formula$ real such that $Latex formula$. The general method is as follows:

• Use complex multiplication to obtain the real and imaginary parts of the LHS.
• $Latex formula$
• Since two complex numbers are equal if and only if their real and imaginary parts are equal, equate the real and imaginary parts of the LHS and RHS to obtain two simultaneous equations:
• Comparing real parts gives $Latex formula$
• Comparing imaginary parts gives $Latex formula$
• Solve for $Latex formula$ and $Latex formula$, either by inspection or by substitution.

The following example demonstrates how both variants of the above method can be used to determine the square roots of a complex number.

#### Example 6

Find all complex solutions to the equation $Latex formula$

#### Solution 6

Let the solutions be of the form $Latex formula$.

Then $Latex formula$

so that $Latex formula$ and $Latex formula$.

Variant 1:

By inspection, $Latex formula$, $Latex formula$ satisfies these simultaneous equations

Checking:

$Latex formula$ and $Latex formula$ so that $Latex formula$ is a solution

$Latex formula$ and $Latex formula$ so that $Latex formula$ is a solution

Variant 2:

From the second equation, $Latex formula$ so that $Latex formula$.

Substituting into the first equation yields $Latex formula$

so that $Latex formula$$Latex formula$ and $Latex formula$ or $Latex formula$.

Since the second equation has no real solutions, $Latex formula$ and $Latex formula$.

Corresponding y can be determined using $Latex formula$ so that $Latex formula$.

Therefore $Latex formula$ and $Latex formula$ are the solutions to the given equation.

Note that the first method, solve by inspection, is a lot shorter and tidier than the second method. Thus if students are able to determine the solutions to the two simultaneous equations by inspection, they are encouraged to use this method. Just to safe, however, they should always include a few lines of checking.

The second method has the advantage of providing a series of steps which is guaranteed to yield an answer. If a student cannot easily determine the answer by inspection they should proceed to solve the equations algebraically.

Students should note that since two simultaneous equations in two variables ($Latex formula$ and $Latex formula$) yield exactly two (not necessarily distinct) solutions, each complex number has two, not necessarily distinct,complex square roots. This can be seen in Figure 2. Furthermore, as the above example illustrates, if $Latex formula$ is a square root then $Latex formula$ is also one. Therefore if the two complex square roots are not distinct, $Latex formula$ and $Latex formula$ so that $Latex formula$.

The conclusion is that each non-zero complex number has exactly two distinct complex square roots, and $Latex formula$ has exactly one square root.

## Using Complex Numbers to Solve Equations

### Solving Quadratic Equations with Real Coefficients

Consider a general quadratic equation $Latex formula$.This can be solved by factorising the equation, completing the square or using the quadratic formula. The first method is usually done by inspection and so extending it to encompass complex roots would requires a degree of intuition about the behaviour of complex numbers that is beyond that which is expected from students. The second and third, however, involve algebraic manipulations that can easily be extended by using complex numbers and performing standard arithmetic calculations on complex numbers. The following example demonstrates how the roots of a quadratic equation with real coefficients can be determined using both the extended versions both of completing the square and using the quadratic formula.

#### Example 7

Find all complex solutions to the equation $Latex formula$.

#### Solution 7

Method 1- Completing the Square:

Starting with the equation $Latex formula$, completing the square yields

$Latex formula$$Latex formula$

$Latex formula$ $Latex formula$Method 2- Using the Quadratic Formula:

$Latex formula$ $Latex formula$$Latex formula$

### Solving Quadratic Equations with Complex Coefficients

Consider again a general quadratic equation $Latex formula$. This time, $Latex formula$, $Latex formula$ and $Latex formula$ are complex numbers. However, this doesn’t change the method that we use to determine the possible value of $Latex formula$. Solving by inspection may be a little harder, however completing the square and using the quadratic formula are still equally viable methods.

The following example illustrates how to find the roots of a quadratic equation with complex coefficients.

#### Example 8

Find all complex solutions to the equation $Latex formula$.

#### Solution 8

Method 1- Completing the Square:

Completing the square on the left hand side of the equation yields

$Latex formula$
$Latex formula$
$Latex formula$
$Latex formula$

so that $Latex formula$ and $Latex formula$

Let $Latex formula$, $Latex formula$, $Latex formula$ real.

Notice how in this step, we don’t automatically jump to ‘let $Latex formula$ ’. This is because we know that the quantities that we’re most familiar with dealing with the real and imaginary parts of the number being squared. Of course, it is also fine to let $Latex formula$, but the working will probably be a little more tedious and a little less familiar.

Then $Latex formula$ so that $Latex formula$ and $Latex formula$.

The first equation gives $Latex formula$

The second equation gives $Latex formula$ so that $Latex formula$ and $Latex formula$.

$Latex formula$ or $Latex formula$

$Latex formula$ or $Latex formula$

Method 2- Using the Quadratic Formula:

$Latex formula$
$Latex formula$
$Latex formula$
$Latex formula$

Here the square roots of $Latex formula$ need to be obtained. This can be done using the same method as the one used above, where all pairs of $Latex formula$ and $Latex formula$ satisfying $Latex formula$ are determined. This leads to solving the simultaneous equations $Latex formula$ and $Latex formula$.

The solutions to the simultaneous equations can be attained by inspection or through algebraic manipulation and yield $Latex formula$.

$Latex formula$ or $Latex formula$

$Latex formula$ or $Latex formula$

## The Argand Diagram

For every real $Latex formula$ and $Latex formula$ there exists a complex number given by $Latex formula$. From before, if the real parts and the imaginary parts of two complex numbers are equal, then they are the same number. This means that for every real $Latex formula$ and $Latex formula$ there exists a unique complex number given by $Latex formula$. In other words, there is a 1-1 correspondence between ordered pairs of reals $Latex formula$ and the complex numbers. Extending the idea a little further, there is a 1-1 correspondence between the complex numbers and the points in the x-y plane.

 $Latex formula$

This correspondence immediately suggests a way to diagrammatically represent a complex number $Latex formula$: on the x-y plane, with x and y coordinates corresponding to $Latex formula$ and $Latex formula$ respectively. Such a diagram is known as an ‘Argand diagram’ and the plane on which the points are plotted is sometimes referred to as the ‘Argand plane’ or the ‘complex plane’. The x-axis, along which the value of $Latex formula$ can be determined, is referred to as the ‘real axis’ and the y-axis is referred to as the ‘imaginary axis’.

The following example demonstrates how to represent complex numbers of the form $Latex formula$ on an Argand diagram.

### Example 9

lex number $Latex formula$, the corresponding values of $Latex formula$ and $Latex formula$ are given special names.

• The quantity $Latex formula$ is called the modulus of $Latex formula$ and is denoted by |z|.
• All possible values of $Latex formula$ are called arguments of $Latex formula$ and are denoted by $Latex formula$. The unique value of $Latex formula$ that is between $Latex formula$ and $Latex formula$ is known as the principal argument of $Latex formula$ and is denoted by $Latex formula$.
• Note: The capitalised A is very important: it differentiates the principal argument from other arguments.
• If $Latex formula$ is not the principal argument then it is incorrect to write $Latex formula$.
• If $Latex formula$ it is still correct to write $Latex formula$, but some information is lost in doing so.
• Note also that the complex number 0 does not have a defined principal argument. By convention, 0 does not have an argument.
 Modulus:$Latex formula$ Arguments: $Latex formula$ where $Latex formula$ and $Latex formula$ Principal Argument: $Latex formula$ where $Latex formula$ and $Latex formula$

Just as two complex numbers are equal if and only if their real and imaginary parts are both equal, two complex numbers are equal if and only if their moduli and principal arguments are equal. (The obvious exception is the complex number 0, which does not have a defined principal argument.)

The modulus of a complex number of the form $Latex formula$ is easily determined. Students tend to struggle more with determining a correct value for the argument. One method is to find the principal argument using a diagram and some trigonometry. The following example illustrates how this can be done.

### Example 10

For the complex number $Latex formula$, find $Latex formula$ and $Latex formula$.

### Solution 10

$Latex formula$

Let $Latex formula$. From the diagram, $Latex formula$ and $Latex formula$.

Most students would solve the trigonometric equation by plugging $Latex formula$ into their calculator. This gives $Latex formula$, not necessarily $Latex formula$. In this case, we see from checking possible values of $Latex formula$ that $Latex formula$ is indeed the value of the principal argument.

$Latex formula$

From the relationships between $Latex formula$, $Latex formula$, $Latex formula$ and $Latex formula$, it can be determined that a complex number $Latex formula$ can also be written in the form $Latex formula$ or $Latex formula$, where $Latex formula$ is standard shorthand notation for $Latex formula$. This is known as the ‘modulus-argument form’, or ‘mod-arg form’ of a complex number.

 Mod – Arg Form:$Latex formula$

The next example demonstrates the process in a typical conversion between Cartesian form and mod-arg form, and illustrates how to decide on a form to use.

### Example 11

Let $Latex formula$, where $Latex formula$ and $Latex formula$ are real numbers and $Latex formula$

i. Express $Latex formula$ and $Latex formula$ in terms of $Latex formula$ and $Latex formula$.

ii. Express $Latex formula$ in the form $Latex formula$, where $Latex formula$ and $Latex formula$ are real.

### Solution 11

i. $Latex formula$
Let $Latex formula$. Then $Latex formula$, $Latex formula$ so that dividing the second equation by the first gives $Latex formula$

$Latex formula$ii. This part involves division of complex numbers, which is usually simpler to do using mod-arg form. However, it presents everything in Cartesian form, so it makes sense to do the question in that form.
$Latex formula$(multiplying top and bottom by the complex conjugate of the denominator)
$Latex formula$(using multiplication of complex numbers)

$Latex formula$

$Latex formula$

## Multiplication and Division of Complex Numbers and Properties of the Modulus and Argument

Students should ensure that they are familiar with how to transform between the Cartesian form and the mod-arg form of a complex number. This is because questions involving complex numbers are often much simpler to solve using one form than the other form. In particular, when complex numbers are multiplied or divided, their moduli and arguments have much simpler and cleaner relationships than their real and imaginary components. This means that in questions involving multiplication and division of complex numbers, it is often recommended to convert to modulus-argument form and complete the calculation before converting back (if conversion is necessary).

The relationships that students are required to know how to derive are:

 Properties of the Modulus:$Latex formula$ $Latex formula$ $Latex formula$ Properties of the Argument:$Latex formula$ $Latex formula$ $Latex formula$

Students should be able to prove all of these relations. The following example illustrates how to prove the first property in each box.

### Example 12

Let $Latex formula$ and $Latex formula$.

a) Express $Latex formula$ in modulus-argument form

b) Hence show that $Latex formula$ and $Latex formula$

### Solution 12

a) $Latex formula$

$Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$b) From (a), $Latex formula$ as required.
Similarly from (a), $Latex formula$ as required.

The proof for the second property is similar to that of the first. For positive integers $Latex formula$ the third property can be derived using the first property and mathematical induction. For negative integers $Latex formula$ the third property can be derived using $Latex formula$ and the second property. These proofs are left as an exercise to the reader.

The following worked examples illustrate how the above properties can be used in calculations in mod-arg form.

### Example 13

Given $Latex formula$,

i. Find the argument and modulus of $Latex formula$

ii. Find the smallest positive integer $Latex formula$ such that $Latex formula$ is real

### Solution 13

i.

$Latex formula$ (using the property that ) $Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$
$Latex formula$ (using the property that $Latex formula$ )
$Latex formula$ (where using inverse tan is ok because both arguments are in the first quadrant)

$Latex formula$ $Latex formula$ii.

If $Latex formula$ is real then $Latex formula$ for some integer $Latex formula$.
Since $Latex formula$(using the property that $Latex formula$ )we want the smallest integer n such that $Latex formula$ is a multiple of 2. This gives $Latex formula$.
The smallest positive integer $Latex formula$ such that $Latex formula$ is real is $Latex formula$.

### Example 14 (HSC 1997 Q2d)

Let $Latex formula$ and $Latex formula$, so that $Latex formula$.

i. Find $Latex formula$ and $Latex formula$ in the form $Latex formula$

ii. Hence find two distinct ways of writing $Latex formula$ as the sum $Latex formula$, where $Latex formula$ and $Latex formula$ are integers and $Latex formula$.

### Solution 14

i.

$Latex formula$ $Latex formula$ii.

We note two interesting things given in the question: we want to find $Latex formula$, where 65 is in the denominator of all the terms in $Latex formula$ and $Latex formula$, and $Latex formula$. This immediately makes us think to use the moduli of $Latex formula$ and $Latex formula$.

By the properties of the modulus, $Latex formula$ and $Latex formula$
$Latex formula$ so that $Latex formula$.

Similarly $Latex formula$.

### Example 15

i. If $Latex formula$, show that $Latex formula$

ii. $Latex formula$, $Latex formula$ are complex numbers such that $Latex formula$.
If $Latex formula$, $Latex formula$ respectively, where $Latex formula$, show that $Latex formula$ has modulus $Latex formula$.

### Solution 15

i. We wish to prove that $Latex formula$.
Since we know double angle formulae better than half angle formulae, let us start with the RHS and show that it is equal to the LHS.

$Latex formula$ $Latex formula$These groups of terms nearly look like the double angle formulas for $Latex formula$ and $Latex formula$ respectively so we made the necessary adjustments

$Latex formula$ $Latex formula$ $Latex formula$ii. The given expression is ugly, so we want to simplify it as much as possible before plugging in any expressions. We note first that the expression on top factorises:

$Latex formula$

Since it doesn’t look like we can simplify this any more, we use the properties of the modulus to obtain

$Latex formula$

But we know the first term is 1, and we know what $Latex formula$ and $Latex formula$ are from part (i).

$Latex formula$ $Latex formula$ $Latex formula$$Latex formula$ as required.

## Geometric Relationships between points on an Argand Diagram

There are nice geometrical relationships between points representing certain complex numbers. In particular, students are required to recognise the relationship between the point representing a complex number $Latex formula$ and the points representing the complex numbers $Latex formula$, $Latex formula$ ($Latex formula$ real) and $Latex formula$. These can be summarised as follows:

 $Latex formula$$Latex formula$$Latex formula$

Students are not required to know the proofs of these properties. However, the proofs have been included as they help to illustrate some basic principles used in graphing complex numbers and relationships between complex numbers.

### Relationship between Points Representing $Latex formula$ and $Latex formula$

If $Latex formula$ is represented by the point $Latex formula$ then $Latex formula$ is represented by the point $Latex formula$. This is the reflection of $Latex formula$ over the x-axis.

Similarly, if $Latex formula$ is represented by the point $Latex formula$ then

$Latex formula$
and $Latex formula$ is represented by the point $Latex formula$, which is again obviously the reflection of $Latex formula$ over the x-axis.

### Relationship between Points Representing $Latex formula$ and $Latex formula$ ($Latex formula$ is real)

If $Latex formula$ is represented by the point $Latex formula$ then $Latex formula$ is represented by the point $Latex formula$.

Similarly, if $Latex formula$ is represented by the point $Latex formula$ then $Latex formula$ is represented by the point $Latex formula$.

This means that the vector $Latex formula$ is the dilation of the vector $Latex formula$ about the origin with factor $Latex formula$.

That is:

• If $Latex formula$, stretch out $Latex formula$ by a factor of $Latex formula$
• If $Latex formula$, stretch out $Latex formula$ by a factor of $Latex formula$ and rotate it by $Latex formula$ about the origin
• If $Latex formula$ then the point representing $Latex formula$ is the point $Latex formula$

### Relationship between Points Representing $Latex formula$ and $Latex formula$

If $Latex formula$ is represented by the point $Latex formula$ then $Latex formula$ is represented by the point $Latex formula$. This means that the vector $Latex formula$ is the vector $Latex formula$ rotated anticlockwise $Latex formula$ radians, or $Latex formula$, about the origin, since its magnitude is the same but its angle of rotation is $Latex formula$ radians larger.

Note that in this case, since the calculations involve multiplying complex numbers, it is a lot simpler to use mode-arg form. If $Latex formula$ were instead represented in Cartesian form, then $Latex formula$ represented by the point $Latex formula$ and $Latex formula$ is represented by the point $Latex formula$.The relationship between these points is not as immediately obvious.

The following example demonstrates how the relationships between the points representing $Latex formula$, $Latex formula$, $Latex formula$($Latex formula$ real) and $Latex formula$ can be used to determine the complex numbers represented by certain points in terms of the complex numbers represented by other points.

#### Example 16

(i) Let $Latex formula$ be a square on an Argand diagram where $Latex formula$ is the origin. The points $Latex formula$ and $Latex formula$ represent the complex numbers $Latex formula$ and $Latex formula$ respectively. Find the complex number represented by $Latex formula$.

(ii) The square is now rotated about $Latex formula$ through $Latex formula$ in an anticlockwise direction to $Latex formula$. Find the complex numbers represented by the points $Latex formula$, $Latex formula$ and $Latex formula$.

#### Solution 16

a) A good approach to an Argand diagram question always begins with a diagram. Diagrams help you to see properties or make links that you otherwise would have missed.

Here, we see that $Latex formula$ is just a special case of a parallelogram, so by the parallelogram rule $Latex formula$ represents the complex number $Latex formula$.
B represents the number $Latex formula$.

b) Again, start with a diagram

We note that rotation by $Latex formula$ is the same as multiplying by $Latex formula$. We can motivate this move from the previously studied relationships: rotation by $Latex formula$ is the same as multiplying by $Latex formula$

$Latex formula$ represents $Latex formula$, $Latex formula$ represents $Latex formula$ and $Latex formula$ represents $Latex formula$

## Representing Complex Numbers as Vectors on an Argand Diagram

We showed before that there is a 1-1 correspondence between the complex numbers and the points in the x-y plane, so that each complex number can be represented by a unique point $Latex formula$ in the complex plane. This means that there is also a 1-1 correspondence between the complex numbers and the vectors $Latex formula$, where $Latex formula$ is the origin and $Latex formula$ is a point in the complex plane.

We represent each complex number $Latex formula$ by the vector $Latex formula$, where $Latex formula$. From the definition of the modulus and argument, we see that this means that:

• The length of the vector $Latex formula$ is the distance from $Latex formula$ to the origin, which is the modulus
i.e.$Latex formula$
• The direction of the vector $Latex formula$ is defined by the angle from the positive x axis to the ray $Latex formula$, which is the argument. This means that the direction of the vector $Latex formula$ is uniquely defined by $Latex formula$.

So we see that this vector representation is similar to the Argand diagram representation. What this representation adds to the point representation is that vectors can be moved about, whereas points are fixed in the plane. Figure 3 illustrates this concept. This means that a lot of geometric properties (and hence algebraic properties) can be determined by rearranging the vectors.

 Figure 3 Although vectors $Latex formula$, $Latex formula$ and $Latex formula$ all represent the complex number $Latex formula$ and the vectors $Latex formula$ and $Latex formula$ both represent the complex number $Latex formula$, they are placed in different positions on the complex plane. In particular, vectors $Latex formula$ and $Latex formula$ are placed tail to tail, whilst vectors $Latex formula$ and $Latex formula$ are placed tip to tail. These vectors are therefore easier to relate than say, $Latex formula$ and $Latex formula$.

## Constructing Vector Representations of Complex Numbers

Given the vector representations of two complex numbers $Latex formula$ and $Latex formula$, students are required to construct the vectors $Latex formula$, $Latex formula$, $Latex formula$ and $Latex formula$. This can be done by considering the geometrical properties of the vectors.

Two vectors, $Latex formula$ and $Latex formula$, can be added either by using the triangle method or by using the parallelogram method. Each method is equally viable and in different situations one method will be more useful than the other.

The triangle method, or the tip-to-tail method, involves placing the tip of the vector $Latex formula$ at the tail of the vector $Latex formula$. The vector pointing from the remaining tail to the remaining tip, i.e. from the tail of $Latex formula$ to the tip of $Latex formula$, gives the vector $Latex formula$.
The parallelogram method involves placing the tails of each vector at the same spot. If the parallelogram with $Latex formula$ and $Latex formula$ as two of its adjacent sides is constructed, the vector along the diagonal from the tails to the opposite vertex gives the vector $Latex formula$.

Note that the vector $Latex formula$ is the vector with the same magnitude as $Latex formula$ but in the opposite direction.

Note also that subtracting a vector $Latex formula$ is the same as adding the vector $Latex formula$.

### Constructing the Point Representing $Latex formula$ from the Points Representing $Latex formula$ and $Latex formula$

Given the points representing $Latex formula$ and $Latex formula$, the vectors representing $Latex formula$ and $Latex formula$ with tail at the origin can be obtained by joining the points with the origin. Since the vectors representing $Latex formula$ and $Latex formula$ are placed tail to tail, the parallelogram method can then be used to construct the vector representing $Latex formula$ with tail at the origin. The tip of this vector gives the point representing $Latex formula$.

 Students must include:· Vectors representing $Latex formula$ and $Latex formula$ with tail at the origin and corresponding labels Lines indicating construction of parallelogram  Vector representing $Latex formula$ with label

### Constructing Vectors Representing $Latex formula$ and $Latex formula$ from Vectors Representing $Latex formula$ and $Latex formula$

Given the vectors $Latex formula$ and $Latex formula$ representing $Latex formula$ and $Latex formula$ respectively, vectors representing $Latex formula$ and $Latex formula$ with tip at the origin are given by $Latex formula$ and $Latex formula$ respectively. Since $Latex formula$ and $Latex formula$ are placed tip to tail, the triangle method can then be used to construct a vector representing $Latex formula$ by adding $Latex formula$ and $Latex formula$. The same method holds for constructing a vector representing $Latex formula$.

 Students must include:· Vectors representing $Latex formula$ and $Latex formula$ or $Latex formula$ and $Latex formula$ with corresponding labels. Clear tip-to-tail vector representation of $Latex formula$ with label.

### Constructing the Point Representing $Latex formula$ from the Points Representing $Latex formula$ and $Latex formula$

Suppose the points representing $Latex formula$ and $Latex formula$ are given by $Latex formula$ and $Latex formula$. Denote by $Latex formula$ the point $Latex formula$. Use equal angles to construct the similar triangles $Latex formula$ and $Latex formula$. Then $Latex formula$ is the point representing $Latex formula$.

 Students must include: Indication of two pairs of equal angles. Coordinates of point $Latex formula$. Labels of $Latex formula$ or $Latex formula$ for each respective vector.

To see why this is true, consider $Latex formula$ and the angle from the positive x axis to the ray $Latex formula$, $Latex formula$.

By similar triangles, $Latex formula$ so that $Latex formula$.

Again by similar triangles, $Latex formula$
so that $Latex formula$.

Together, these tell us that $Latex formula$ is the point representing $Latex formula$.

### Triangle Inequality

Students are expected to be able to prove the triangle inequality for the moduli of complex numbers:

 $Latex formula$

The geometrical proof of this given in the following worked example and demonstrates one application of vector representations.

#### Example 17

Given two arbitrary complex numbers $Latex formula$ and $Latex formula$, prove that $Latex formula$ (where $Latex formula$ is the modulus of $Latex formula$).

#### Solution 17

 Let the two vectors $Latex formula$ and $Latex formula$ represent $Latex formula$ and $Latex formula$ respectively. By the triangle method, the vector $Latex formula$ represents the complex number $Latex formula$.$Latex formula$$Latex formula$ by the geometric triangle inequality$Latex formula$

## Worked Examples involving Vector Representations of Complex Numbers

HSC questions do not simply ask students to draw a memorised diagram. Students are often required to synthesise the information embedded in the above techniques, such as by identifying the third side of the triangle $Latex formula$, $Latex formula$ as the vector $Latex formula$, or even using algebra to determine other geometric properties. The following examples illustrate a few applications of the techniques presented in this section.

### Example 18

Let $Latex formula$ and $Latex formula$ be two complex numbers, where $Latex formula$ and $Latex formula$ is defined by $Latex formula$ and $Latex formula$

(i) On an Argand diagram plot the points $Latex formula$ and $Latex formula$ representing the complex numbers $Latex formula$ and $Latex formula$ respectively.

(ii) Plot the points $Latex formula$ and $Latex formula$ represented by the complex numbers $Latex formula$ and $Latex formula$ respectively. Indicate any geometric relationships between the four points $Latex formula$, $Latex formula$, $Latex formula$ and $Latex formula$.

### Solution 18

Graphs for (i) and (ii):

(i) Since $Latex formula$ and $Latex formula$ are given in Cartesian and mod-arg form respectively, the points were plotted using those respective forms. This is fine, as long as for the Cartesian form both the x and y coordinates are specified, and for the mod-arg form both the modulus and argument are specified.

(ii) By the triangle rule for vector subtraction the vector $Latex formula$is equal to $Latex formula$. Therefore the point $Latex formula$ is such that $Latex formula$, that is, $Latex formula$ is a parallelogram.
To obtain $Latex formula$ rotate point $Latex formula$ $Latex formula$about the origin. This means that $Latex formula$ and $Latex formula$. The diagram looks like so:

Geometric relationships:
$Latex formula$ is a parallelogram
$Latex formula$ and $Latex formula$

### Example 19

 The diagram shows a complex plane with origin $Latex formula$. The points $Latex formula$ and $Latex formula$ represent arbitrary non-zero complex numbers $Latex formula$ and $Latex formula$ respectively. Thus the length of $Latex formula$ is $Latex formula$.(i) Use the diagram to show that $Latex formula$.(ii) Construct the point $Latex formula$ representing $Latex formula$. What can be said about quadrilateral $Latex formula$?(iii) If $Latex formula$, what can be said about the complex number $Latex formula$?

### Solution 19

(i) The proof follows the same lines as the proof for $Latex formula$.
From the diagram, $Latex formula$, $Latex formula$ and $Latex formula$ are the sides of a triangle. Therefore by the geometric triangle inequality, $Latex formula$.

(ii) The point $Latex formula$ can be obtained using the parallelogram rule for addition:

The quadrilateral $Latex formula$ is a parallelogram.

We guess that the previous parts of the question should help us to answer this question.
From the diagram, $Latex formula$ and $Latex formula$ are the diagonals of $Latex formula$, which is a parallelogram.
This means that if $Latex formula$ then $Latex formula$ will be a rectangle.
This tells us that the angle from $Latex formula$ to $Latex formula$ is a right angle.
We know from our construction of $Latex formula$ from $Latex formula$ that right angles involve multiplying by $Latex formula$, so we think that $Latex formula$ will be some multiple of $Latex formula$.
Checking this using args (since we have information about args): $Latex formula$ so that $Latex formula$ is purely imaginary, of the form $Latex formula$ where $Latex formula$ is positive.

### In an Argand Diagram, $Latex formula$, $Latex formula$ and $Latex formula$ represent the complex numbers $Latex formula$, $Latex formula$, $Latex formula$ and $Latex formula$ respectively.

(i) Describe the point which represents $Latex formula$.

(ii) Deduce that if $Latex formula$, then $Latex formula$ is a parallelogram.

### Solution 20

(i) First, we draw a diagram. Since we know how to find the point that represents $Latex formula$ we find that, call it $Latex formula$ and then we know that the midpoint of $Latex formula$ is the point that we want.

However, since we obtained $Latex formula$ using the parallelogram rule, we know that $Latex formula$ is a parallelogram so that $Latex formula$ and $Latex formula$, its diagonals, bisect each other. Therefore if $Latex formula$ is the midpoint of $Latex formula$ then $Latex formula$ is also the midpoint of $Latex formula$ and the point that we want is the midpoint of $Latex formula$.
The point that represents $Latex formula$ is the midpoint of $Latex formula$.

(ii) We want to use part (i) to give us some information about $Latex formula$ and $Latex formula$. From part (i) the point that represents $Latex formula$ is the midpoint of $Latex formula$. Since we want to relate $Latex formula$ with $Latex formula$ we note that similarly the point that represents $Latex formula$ is the midpoint of $Latex formula$.
From part (i) we see that if $Latex formula$, the midpoints of $Latex formula$ and $Latex formula$ are the same point. This means that if $Latex formula$ and $Latex formula$ are the diagonals of $Latex formula$, then the diagonals of this quadrilateral bisect each other so that $Latex formula$ is a parallelogram.

## De Moivre’s Theorem

De Moivre’s Theorem states that for all integers $Latex formula$:

 $Latex formula$

This result is equivalent to $Latex formula$.
The result is trivial for $Latex formula$ and can be proven for positive integers $Latex formula$ by mathematical induction, using the expansions of $Latex formula$ and $Latex formula$. It can then be extended to the negative integers using complex division and $Latex formula$. The proof is left as an exercise to the reader.

The most obvious application of De Moivre’s theorem is in calculating integer powers of complex numbers. For any complex number $Latex formula$, De Moivre’s theorem tells us that $Latex formula$. This means that integer powers of a complex number can be easily calculated using De Moivre’s theorem and the modulus-argument form of the number. The following example illustrates how to calculate the integer power of a complex number.

### Example 21

Express in modulus-argument form:

(i) $Latex formula$

(ii) $Latex formula$, where $Latex formula$ is a positive integer.

### Solution 21

(i) $Latex formula$
$Latex formula$(where we use $Latex formula$ since we see that $Latex formula$ is in the 2nd quadrant)

$Latex formula$(ii) $Latex formula$

$Latex formula$$Latex formula$ by de Moivre’s theorem

De Moivre’s theorem is actually a lot more powerful than it looks. As was just demonstrated, directly comparing the two sides of the equation provides a direct formula for powers of complex numbers. However, the real and imaginary components can also be compared separately. Doing so provides an efficient way of calculating $Latex formula$ and $Latex formula$ as polynomials in $Latex formula$ and $Latex formula$. The following example illustrates how this is done.

### Example 22

se De Moivre’s thorem to express $Latex formula$ and $Latex formula$ in terms of $Latex formula$ and $Latex formula$. Hence show that $Latex formula$

### Solution 22

Since we want to express $Latex formula$ and $Latex formula$ in terms of $Latex formula$ and $Latex formula$, it makes sense to use De Moivre’s theorem for $Latex formula$.
$Latex formula$(by De Moivre’s theorem)

$Latex formula$

$Latex formula$

Comparing real and imaginary components:

$Latex formula$

$Latex formula$ $Latex formula$Since we want our answer to be in terms of $Latex formula$ we note that $Latex formula$ so we divide through by $Latex formula$ to put all the powers of $Latex formula$ on the bottom of each term.

$Latex formula$
$Latex formula$

$Latex formula$ as required.

De Moivre’s theorem is also useful because it provides an efficient way of expressing large powers of $Latex formula$ and $Latex formula$ as a linear combination of the sines and cosines of multiples of $Latex formula$. This is because if we let $Latex formula$:

• $Latex formula$
• $Latex formula$
• $Latex formula$
• $Latex formula$
• $Latex formula$

The next example demonstrates how De Moivre’s theorem can be used to calculate large powers of $Latex formula$ and $Latex formula$.

### Example 23

If $Latex formula$ show that $Latex formula$ and hence show that

$Latex formula$

### Solution 23

Let $Latex formula$.

Then $Latex formula$

$Latex formula$ (by De Moivre’s theorem)

$Latex formula$ $Latex formula$Students might then be tempted to go straight for $Latex formula$. This, unfortunately, yields two powers of $Latex formula$ and only one cosine term. It seems better to express $Latex formula$ in terms of multiple powers of $Latex formula$ that can then be grouped together and converted to cosines of multiples of $Latex formula$. The following method is the general method for calculating large powers of $Latex formula$ and $Latex formula$.

From this, we know that $Latex formula$

$Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$$Latex formula$

## The $Latex formula$ Roots of Unity

The nth roots of unity are the complex numbers $Latex formula$ such that $Latex formula$. These can be determined using de Moivre’s theorem:

Let $Latex formula$. Then $Latex formula$. Since two non-zero complex numbers can only be equal if their moduli are equal, $Latex formula$ so that $Latex formula$. This gives $Latex formula$, $Latex formula$ and $Latex formula$, $Latex formula$. This yields $Latex formula$ distinct values for $Latex formula$ corresponding to $Latex formula$. Any other value of $Latex formula$ yields a value of $Latex formula$ that gives the same sine and cosine values as one of the n values of $Latex formula$ specified above.

Therefore the n nth roots of unity are given by $Latex formula$.

 $Latex formula$ Roots of Unity:$Latex formula$where $Latex formula$

This shows that nth roots of unity can be expressed on the Argand diagram as $Latex formula$ vectors of length 1 radiating from the origin, spaced equally around the unit circle. This means that they form the vertices of a regular n-gon. Students will often be asked to demonstrate their understanding of this through graphing the nth roots on an Argand diagram. For example, the cube roots of unity can be represented as the vertices of an equilateral triangle as follows:

 Students must include: Indication of angles between adjacent vectors. Indication that each vector is of length 1 (either using equal length markings or by lightly drawing in the unit circle) The relevant marking $Latex formula$ at each point.

The roots can be found in the form $Latex formula$ by converting from modulus-argument form.

The nth roots of $Latex formula$ can similarly be determined using de Moivre’s theorem:
Let $Latex formula$. Then $Latex formula$.
$Latex formula$, $Latex formula$ and $Latex formula$ and $Latex formula$, $Latex formula$.

 $Latex formula$ $Latex formula$ Where $Latex formula$

Again, the roots must be equally spaced around the unit circle. However, they look a little different to the nth roots of unity. One way to think about it is that whereas nth roots of unity begin at 1 and move around the circle anticlockwise with a step angle of $Latex formula$, nth roots of unity begin at -1 and move around the circle anticlockwise with a step angle of $Latex formula$. Another way to think about it is that each nth root of -1 is of the form $Latex formula$, which is one of the nth roots of unity rotated anticlockwise about the origin by an angle of $Latex formula$.

## Polynomials and the $Latex formula$ Roots of $Latex formula$

One of the more difficult applications of complex number theory involves using the algebra and ideas expressed in de Moivre’s theorem and the derivation of expression for the nth roots of $Latex formula$ to solve polynomials. These questions can usually be broken down into little steps by using complex number theory and ideas involved in solving polynomials, such as sum and product of roots. The nice thing is that HSC questions will often help students to separate the question into simple steps, or even present the steps and links as part of the problem statement. However, since this is not always the case, students should acquaint themselves with the various processes involved in breaking down questions.

Often, smaller steps can be obtained by considering the type of information given to them in the problem statement and the type of information that they needs to be extract. The following examples illustrate how this can be done.

### showing that the nth roots of unity can be written as powers of one of its roots, dividing through by $Latex formula$ to get an equation whose roots are all complex and then using a relation to do with polynomials or complex numbers with magnitude 1 to get some new information.

Let $Latex formula$

(i) Show that $Latex formula$ is a solution of $Latex formula$, where $Latex formula$ is an integer.

(ii) Prove that $Latex formula$

(iii) Hence show that $Latex formula$

### Solution 24

(i) To show that $Latex formula$ is a solution of $Latex formula$, all we have to do is plug it in and show that $Latex formula$.

$Latex formula$ $Latex formula$ $Latex formula$$Latex formula$ by De Moivre’s Theorem

$Latex formula$ $Latex formula$ $Latex formula$ $Latex formula$Hence $Latex formula$, where $Latex formula$ is an integer, is a solution of $Latex formula$.

(ii) Method 1:
From part (i), we know that $Latex formula$ is a solution to the equation $Latex formula$ for all integers $Latex formula$. Furthermore, if we take $Latex formula$ then $Latex formula$ so that $Latex formula$ and there are $Latex formula$ distinct solutions to the equation. These are therefore the 9 roots of the equation $Latex formula$. Since the sum of the roots is the coefficient of $Latex formula$ in $Latex formula$, this means that the sum of the roots is 0.
That is, $Latex formula$

$Latex formula$
Method 2:
From part (i), we know that W is a solution to the equation $Latex formula$.
This means that $Latex formula$.
Since $Latex formula$ is complex and $Latex formula$, $Latex formula$ so that $Latex formula$, which implies that $Latex formula$.
Both these methods can be used for this question. The first approach is usually used when something is known about all the roots, so that it is possible to sum them up. The second is used when something is known about one root and so that one of the factors in the factorised expression can be cancelled.

(iii) We want to relate cosines, which are real numbers, with powers of $Latex formula$, which are complex numbers. To do this, we use one of the niftier consequences of de Moivre’s theorem:
$Latex formula$ and $Latex formula$.
So now we know that if we can express the left hand side of required result in terms of cosines of multiples of $Latex formula$
, where $Latex formula$ i.e. $Latex formula$, we can then express those as powers of $Latex formula$ and use the results from the previous parts. This gives us three steps.
Step 1: Express $Latex formula$ as cosines of multiples of $Latex formula$

$Latex formula$

Step 2: Evaluate cosines of multiples of $Latex formula$ in terms of $Latex formula$.

$Latex formula$Step 3: Simplify this expression using results from previous parts until we get to the required result

$Latex formula$What we’ve done here is make the result easier to expand (we think that we will have to expand, since part (ii) looks very much like an expanded result).
In the first equality we multiply through by enough powers of $Latex formula$
to make each factor have only non-negative powers. Of course, this means that we have to divide through by them again, and hence we get $Latex formula$ at the front.
In the second equality we use the fact from part (i) that $Latex formula$
to make the powers in the factors as small as possible, and to make the power out front a positive power.

$Latex formula$Now it’s just a matter of expanding and hoping for the best. We expand and lo and behold, we get the long side of the equation in part (ii) and we can just replace it with $Latex formula$ and everything falls out.

### This problem demonstrates a type of problem that many students will shy away from: one involving not 1, not 2, but n variables. The trick is to deal with each variable separately, or to work patiently through sums, preferably using summation notation. Students will see that this question, as scary as it looks, is actually a lot easier than the previous one.

Let the points $Latex formula$, $Latex formula$…,$Latex formula$ represent the nth roots of unity, $Latex formula$, $Latex formula$, …, $Latex formula$, and suppose $Latex formula$ represents any complex number $Latex formula$ such that $Latex formula$.

(i) Prove that $Latex formula$.

(ii) Show that $Latex formula$ for $Latex formula$.

(iii) Prove that $Latex formula$

### Solution 25

(i) We want to know the value of $Latex formula$, where $Latex formula$, $Latex formula$, …,$Latex formula$ are the nth roots of unity. This in itself already tells us something: they are the n solutions to some equation, and we want to find their sum. If this equation is a polynomial, we know right away what the sum of the solutions is. So let’s consider this equation.
Since $Latex formula$, $Latex formula$, …,$Latex formula$ are the nth roots of unity, they are the n solutions to the equation $Latex formula$. This means that their sum is given by the coefficient of the linear term in $Latex formula$, which is 0.

$Latex formula$(ii) To determine what $Latex formula$ is, we first have to take a look at what is meant by these points. $Latex formula$ represents the complex number $Latex formula$. $Latex formula$ represents the complex number $Latex formula$. Therefore by the triangle method for vector subtraction, $Latex formula$ represents $Latex formula$ and $Latex formula$.
$Latex formula$ (since $Latex formula$ represents $Latex formula$)
$Latex formula$(we separate it out because then one factor is in the RHS and the other is nearly)
$Latex formula$(one more property of conjugates gives us the expression on the RHS)

(iii) Before we dive into summation, let’s see if we can simplify each term in the expression any more.

$Latex formula$
$Latex formula$

$Latex formula$$Latex formula$
Summing over all n,

$Latex formula$$Latex formula$(since $Latex formula$ is independent of $Latex formula$)
$Latex formula$(since sum of conjugates is conjugate of sum, and the sum is 0 by part (i))
$Latex formula$ (again since the sum is 0)

$Latex formula$

## Sketching Curves and Regions in the Complex Plane

Sometimes it is useful to represent the complex solutions to a given equation or inequality in graphical form. In particular, there are a few standard curves that students are expected to know how to sketch. In the HSC, the complex equations describing these curves are often combined or turned into inequalities to create more involved questions. However, these curves and regions are quite simple to sketch if students have a firm understanding of the elementary curves that they are comprised of.

When determining the nature of the following standard curves, a number of algebraic and geometric approaches can be taken.If using an algebraic approach, students should ensure that they are able to obtain a Cartesian equation for the locus, and able to translate this equation into a geometric description.If using the geometric approach, students should ensure that they are able to translate the geometric properties describing the locus into algebraic form, through a Cartesian equation.

### Algebraic approach

The solutions of these equations are given by lines parallel to the y axis and x axis respectively:
If $Latex formula$, $Latex formula$.If $Latex formula$.

$Latex formula$

### Solutions to Equations of the Form $Latex formula$

#### Algebraic approach

Since the equations involve the subtraction of complex numbers,the simpler algebraic approach should involve expressing the complex numbers as the sum of their real and imaginary parts.

Let $Latex formula$, $Latex formula$ and $Latex formula$.

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$

$Latex formula$which is of the form

$Latex formula$, a straight line!

#### Geometric approach

This seems like a remarkably simple result for some not-so-simple equations. However it is quite easily explained using a geometric interpretation. The solutions describe the locus of points that are equidistant from two fixed points in the plane: the point $Latex formula$ representing $Latex formula$ and the point $Latex formula$ representing $Latex formula$. If a possible value of $Latex formula$ is represented by the point $Latex formula$, then $Latex formula$ is isosceles with $Latex formula$ so that the altitude from $Latex formula$ to $Latex formula$ bisects $Latex formula$. That is, $Latex formula$ lies on the perpendicular bisector of $Latex formula$, which is a straight line. Furthermore, any point on the perpendicular bisector satisfies $Latex formula$ so that the locus of points is the perpendicular bisector.

To plot the perpendicular bisector of $Latex formula$, all that is required is to find a point on the line and the gradient of the line.
A point on the line is given by the midpoint of $Latex formula$. The gradient is the one perpendicular to the gradient of $Latex formula$.
These give $Latex formula$.

 Students must include: Equation of line in Cartesian form At least two points on the line (e.g. intercepts)

### Solutions to Equations of the Form $Latex formula$

#### Algebraic approach

Again, since the equations involve the subtraction of complex numbers, the simpler algebraic approach should involve expressing the complex numbers as the sum of their real and imaginary parts.

Let $Latex formula$ and $Latex formula$.

$Latex formula$

$Latex formula$

$Latex formula$

The locus of $Latex formula$ therefore traces out a circle in the complex plane center $Latex formula$ and radius $Latex formula$.

#### Geometric approach

The solutions describe the locus of points that are a fixed distance $Latex formula$ from the point representing the complex number $Latex formula$. Thus the locus of $Latex formula$ traces out a circle in the complex plane center $Latex formula$ and radius $Latex formula$. This has equation $Latex formula$.

 Students must include: Coordinates of center of circle. Some indication of the size of the radius (e.g. using intercepts, arbitrary point on circle, radius marking)

### Solutions to Equations of the Form $Latex formula$

The graph is a ray starting with an open circle on the point $Latex formula$ representing the complex number $Latex formula$ and pointing in a direction that is $Latex formula$ anticlockwise from the positive x-axis.

 Students must include:  Open circle at the point A representing $Latex formula$ Coordinates of A. An indication that the angle between the ray AP and the positive $Latex formula$-axis is $Latex formula$.

#### Geometric approach

Consider the vector representation of complex numbers, where $Latex formula$ is the angle from the positive x-axis to the ray $Latex formula$. Since $Latex formula$ is the vector pointing from the point $Latex formula$ representing $Latex formula$ to the point $Latex formula$ representing $Latex formula$, if this makes an angle of $Latex formula$ with the positive x-axis then the ray $Latex formula$ is $Latex formula$ radians anticlockwise from the positive x-axis. Similarly if $Latex formula$ is represented by an arbitrary point $Latex formula$ on the ray with tail $Latex formula$ and angle $Latex formula$ from positive x-axis, then $Latex formula$ as required. At $Latex formula$ or $Latex formula$ $Latex formula$ is undefined so that there is an open circle at that point.

### Equations involving both $Latex formula$ and $Latex formula$

Students will often be asked to graph equations involving both $Latex formula$ and $Latex formula$. Cartesian equations can usually be obtained by substituting $Latex formula$ and $Latex formula$.

## Worked Examples involving Sketching Curves and Regions in the Complex Plane

Sketching the solutions to inequalities is not any more difficult than sketching the solutions to equalities. The equality case creates a curve, or boundary, that splits the complex plane into two sections. The inequality sign means that everything on one side of the boundary is part of the region and everything on the other side is not.

The following examples illustrate how compositions of elementary curves can be used to graph regions in the complex plane that are described by a number of equations and inequalities.

### Example 26

Sketch the region where the inequalities $Latex formula$ and $Latex formula$

### Solution 26

The trick with sketching composite regions is to sketch each region separately, and see which parts of the plane lie in both regions.

 $Latex formula$ $Latex formula$describes the set of points 5 units from the point $Latex formula$. This is a circle in the complex plane with center $Latex formula$ so that it is in the shaded region. Students should note the common mistake of taking the center to be the point (3,-1). Remember that it is usually the quantity $Latex formula$ that is involved in these equations.

 $Latex formula$describes the set of points equidistant from the points $Latex formula$ and $Latex formula$. This is the perpendicular bisector of $Latex formula$ and $Latex formula$. It is the line through $Latex formula$ perpendicular to the x-axis, which is the y-axis.$Latex formula$describes the set of points closer to $Latex formula$ than $Latex formula$. This is the region to the left of the y-axis. To check which region should be shaded in, pick a point on one side of the boundary. In this case, the point $Latex formula$ on the right of the boundary yields $Latex formula$ so that it is not in the shaded region.

Taking the intersection of these two regions yields:

### Example 27

On an Argand diagram, shade in the region determined by the inequalities $Latex formula$ and $Latex formula$.

### Solution 27

 $Latex formula$ Note the open circle at $Latex formula$ $Latex formula$ $Latex formula$ and $Latex formula$\frac{\pi }{6}\le argz\le \frac{\pi }{4}$Latex formula$