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Topic 18: Complex Numbers
 

 

1. THE FIELD OF COMPLEX NUMBERS

1.1. Introduction

Until now, you know several sets of numbers, including . In each case, the numbers correspond to points on the real line. There is a new kind of numbers called complex numbers which correspond to points in the plane. The history of complex numbers goes back to the ancient Greeks who decided that no number existed that satisfies . For example, Cardan in 1545 tried to solve the problem of finding two numbers, a and b, whose sum is 10 and whose product is 40.

That is,

Eliminating b we have . Solving this quadratic equation gives

We see that there are no real solutions since the square root of a negative real number does not exist in set of real numbers, but if it is agreed to continue using the numbers then equation (1) and (2) are satisfied. So these are solutions of the original problem but they are not real numbers.

Surprisingly, it was not until the nineteenth century that such solutions were fully understood. The square root of -1 is denoted by i, so that or and .

Many mathematicians contributed to the full development of complex numbers. The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.

1.2. Definition

A complex number is a number that can be put in the form , where a and b are real numbers and . The set of all complex numbers is denoted and defined as

The real number a of the complex number is called the real part of z (denoted by ), and the real number b is often called the imaginary part (denoted by ).

Example

A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number or simply real.

1.3. Complex plane

A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram (or Complex plane or Gauss plane) , representing the complex plane. Re is the real axis (or x-axis), Im is the imaginary axis (or y-axis)

In complex plane we will no longer talk about coordinates but affixes. The affix of point M on the preceding diagram is . It is plotted as a point and position vector on an Argand diagram. is the standard (or algebraic) form of complex number z.

Example

Represent, in same complex plane, the complex numbers

Solution

1.4. Elementary operations

a) Conjuagate and opposite

The complex conjugate of the complex number is denoted and defined by .

The real and imaginary parts of a complex number can be extracted using the conjugate:

Conjugation distributes over the standard arithmetic operations:

The complex number is the opposite of , symmetric of z with respect to 0.

b) Addition and subtraction

Addition of two complex numbers can be done geometrically by constructing a parallelogram. Complex numbers are added by adding the real and imaginary parts of the summands. That is to say,

  or

Similarly, subtraction is defined by

or

c) Multiplication and division

The multiplication of two complex numbers is defined by the following formula:

or

In particular, the square of the imaginary unit is −1:

The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit. Indeed, if i is treated as a number so that di means d times i, the above multiplication rule is identical to the usual rule for multiplying two sums of two terms.

Remark

The powers of imaginary unit: If we continue we return to the same results. Other exponents may be regarded as , and . Thus the following relations may be used:

The inverse (reciprocal) of

Then,

Remark

The product is called the norm of and is noted by . The term noted by is called the modulus of .(We shall talk about modulus later)

Thus,

The division of two complex numbers is defined in terms of complex multiplication, which is described above, and real division. Where at least one of c and d is non-zero:

Consider two complex numbers

Or we can work as follow

And then,

Example

1.5. Matrix representation of a complex number

The complex number can also be represented by 2×2 matrices of the form:

Here the entries a and b are real numbers. The sum and product of two such matrices is again of this form, and the sum and product of complex numbers corresponds to the sum and product of such matrices. The geometric description of the multiplication of complex numbers can also be phrased in terms of rotation matrices by using this correspondence between complex numbers and such matrices.

Moreover, the norm of a complex number expressed as a matrix is equal to the determinant of that matrix:

The conjugate corresponds to the transpose of the matrix

,that is

Example

Consider the complex number .

The matrix representation of this complex number is

1.6. The complex number field

We have seen that the complex numbers can be defined as ordered pairs of real numbers together with the operations:

Now, using these definitions of addition and multiplication, we can verify that the set of complex numbers with these two operations (addition and multiplication) is a field:

We have seen that is a field if the following are verified (for e an identity element for operation *).

Now,

a) Is an abelian group?

is closed under addition:
• Addition is associative:


• Addition is commutative:


• Identity element

Remember that addition is commutative. Then, additive identity is (0,0)
• Inverse element

Remember that addition is commutative. Then, additive inverse is

Thus, is an abelian group.

b) Is an abelian group?

is closed under multiplication:
• Multiplication is associative:


• Multiplication is commutayive:


• Identity element

Remember that multiplication is commutative. Then, multiplicative identity is (1,0)
• Inverse element

Remember that multiplication is commutative. Then, multiplicative inverse of is

Thus, is an abelian group.

The next is to check if multiplication is distributive over addition.

Remember that the multiplication is commutative. Then, multiplication is distributive over addition.

Hence, is a field.

Unlike the reals, set of complex numbers is not an ordered field, that is to say, it is not possible to define a relation . That is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so precludes the existence of an ordering on .

2. CALCULATION IN THE FIELD OF COMPLEX NUMBERS

2.1. Square root of a complex number

A complex number is a square root of the complex number if

Also

By addition and substraction, the equation (1) and (3) give;

and

Remark

If , the signs cannot be taken arbitrary because (2) shows that the product xy has sign of b. Then, if b>0, we take the same signs and if b<0 we take different signs.

Example

Find the square roots of

Solution

Let the square root be

Since the product xy is negative, we take the different signs. Then,

Example

Find the square roots of

Solution

Let the square root be

Since the product xy is positive, we take the same signs. Then,

2.2. Quadratic equations in complex numbers

Let a, b and c be real numbers; for and , the equation has

- two distinct real root if Δ>0
- a double real root if Δ=0
- two conjugate complex roots if Δ<0

Example

Solve

Solution

2.3. Polynomials in complex numbers

The process of finding the roots of a polynomial in set of complex numbers is similar for the case of real number remembering that the square root of a negative real number exist in set of complex numbers. We need the important result known as the Fundamental Theorem of Algebra:

Every polynomial of positive degree with coefficients in the system of complex numbers has a zero in the system of complex numbers. Moreover, every such polynomial can be factored linearly in the system of complex numbers.

Example

Factorize the polynomial

Solution

1 is one of the 4 roots. Using Horner’s method we have

and

i is also a root

and

Now, is factorized as follow:

Then and finally

3. POLAR FORM OF A COMPLEX NUMBER

3.1. Modulus of a complex number

The absolute value (or modulus or magnitude) of a complex number is

If z is a real number (i.e., ), then . In general, by Pythagoras' theorem, r is the distance of the point P representing the complex number z to the origin.

Example

Find the modulus of the following complex numbers

Solution

1)               2)

3)            4)

3.2. Argument of a complex number

The argument or phase of z is the angle of the radius OP with the positive real axis, and is written as . As with the modulus, the argument can be found from the rectangular form .

The value of must always be expressed in radians. It can change by any multiple of and still give the same angle. Normally, as given above, the principal argument in the interval is chosen. Values in the range are obtained by adding if the value is negative. The polar angle for the complex number 0 is undefined, but arbitrary choice of the angle 0 is common.

The following relations are true:

Example

Find the argument of the following complex numbers

1)                 2)                   3)

Solution

1)             2)           3)

3.3. Polar form of a complex numbers

From the relations

and

we can write and

and hence,

This form is called polar form or modulus-argument form.

Using the cis function, this is sometimes abbreviated to

Example

Put the complex number in polar form

Solution

The polar form is

Example

Put the complex number in polar form

Solution

The polar form is

Notice

Having a polar form of a complex number you can get its corresponding algebraic form.

Example

Convert the complex number in algebraic form

Solution

3.4. Operations in polar form

Multiplication, division and powers

Formulas for multiplication, division and powers are simpler in polar form than the corresponding formulas in Cartesian coordinates.

Given two complex numbers then

Thus, the formula for multiplication is . With the proviso that may have to be added to, or substracted from , if is outside the permitted range of the principal argument.

Similarly,

The formula for division is given by With the proviso that may have to be added to, or substracted from , if is outside the permitted range of the principal argument.

This also implies power of a complex number z:

From the power of a complex number if r=1, we have De Moivre's theorem

Example

Consider the complex number

a) Put z in algebraic form and trigonometric form.

b) Deduce the exact value of and .

Solution

a) Algebraic form

Trigonometric form

or let and

Here we need to add because the value is negative and is not in the desired interval.

That is

Then the trigonometric form of z is

b) We equate the trigonometric form of z and algebraic form of z:

Thus,

3.5. Exponential form of a complex number

Consider the series

Replacing x by in relation (1) gives

Using relations (2) and (3) we get . If we consider the the complex number , its exponential form is .

Generally,

The formulas for product, quotient and power become

Example

1)                       2)

EULER’s formulae

From exponential form of a complex number, we can find real part and imaginary part as follow:

(1)+(2) gives

(1)-(2) gives

The formulas are called the Euler’s formulas.

The Euler’s formulas are used to linearize trigonometric expressions. This method is called linearization. We will see this in applications of complex numbers.

4. THE NTH ROOT OF A COMPLEX NUMBER

4.1. Nth roots of a complex number

A complex number is the nth root of if , means that

Thus,

Example

Determine the 4th roots of -4

Solution

Then the 4th roots of -4 are

4.2. SPECIAL CASE: nth roots of unity

Here z=1 and

Then

And then, the nth roots of unit are given by

This shows that the first root among the nth roots of unit is always 1.

Notice

1. The nth roots of unit can be used to find the nth roots of any complex number known one of these roots. If one of the nth roots of a complex number z is known, the other roots are found by multiplying that root and nth roots of unit.

2. The sum of nth roots of unit is zero.

     In fact,

The sum is

Multiplying both sides by gives

(1) - (2) gives

But , then

Thus, the sum of nth roots of unit is zero.

Example

Find cube roots of unit

Solution

Example

Using cube roots of unit, find the cube root of -27, known that -3 is one of the roots.

Solution

We have one of the cubic roots of -27, which is -3. We have seen that the cube roots of units are

Then, cubic roots of -27 are

4.3. Graphical representation of nth roots of a complex number

The n roots of a complex number are equally spaced around the circumference of a circle of centre 0 in the complex plane. If the complex number for which we are computing the n nth roots is , the radius of the circle will be and the first root correspond to will be at an amplitude of .

This root will be followed by the n-1 remaining roots at equal distances apart. The angular amplitude between each root is .

Now, if z=1 the radius of the circle is 1. Thus, the n roots of unit are equally spaced around the circumference of a unit circle (circle of centre o and radius 1) in the complex plane.

Example

Represent graphically the 4th roots of

Solution

The roots are given by

This is

The circle will have radius 2.

Example

Represent graphically the nth roots of unit for n=2,3 and 4.

Solution

a)

b)

c)

We can see that the nth roots of unit for n>2 are the vertices of a regular polygon inscribed in a circle of centre 0 and radius 1.

5. APPLICATIONS

5.1. Trigonometric applications

a) Trigonometric numbers of a multiple of an angle

We have seen that De Moivre’s formula is given by

By Newton binomial, we have

Relations(1) and (2) are equivalent. Then,

Recall that two complex numbers are equal if they have the same real parts and same imaginary parts. Thus from (3), we have

In general,

Where

Example

Express in terms of

Solution

Method 1 (use of De Moivre’s formula and binomial expansion)

By De Moivre’s formula:

By binomial expansion:

Then,

This gives

Then

Method 2 (use of the general formulae)

Example

Express in terms of

Solution

But,

Now,

Dividing every term by gives

b) Linearization of trigonometric expressions

As we have seen it, the formulas to be used in linearization of trigonometric expressions are Eurer’s formulas:

Example

Linearize

Solution

Example

Linearize

Solution

5.2. Geometric applications

 

6. References

 

 
 

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