complex numbers - nqt education · pdf file- the problem with this system is that the...

COMPLEX NUMBERS

Assumed Knowledge from Extension 1Make sure you are familiar with the following before proceeding

• Solving Quadratic Equations

• Absolute Values

• Preliminary Trigonometry

• Locus

• Circle Geometry

Where are we up to?

• Complex Numbers and Quadratic Equations

• The Argand Diagram

• Modulus-Argument Form

• Vector Representation

• Locus Problems

• Roots of Unity

1. Introduction to complex numbers

1.1 Number Systems

Throughout your progression in primary to high school mathematics you were introduced todifferent number systems which eventually led to the commonly used real number system:

• Natural Numbers N = {0, 1, 2, 3, ....} - the problem with this system is that the operationof say 1 − 2 gives no solution in the natural number set, so we introduce the concept ofnegative numbers

• Integers Z = {....,−2,−1, 0, 1, 2, ....} - the problem with this system is that the operationof say 1 ÷ 2 gives no solution in the integer set, so we introduce the concept of fractions

• Rational Numbers Q ={mn

: m ∈ Z, n ∈ Z}

- the problem with this system is that some

numbers cannot be written as a fraction (e.g. π), so we introduce the concept of irrationalnumbers

• Real Numbers R ={Q ∪ Q

}- the problem with this system is that the operation of say√

−1 has no solution in the real number set, so we introduce the concept of IMAGINARYNUMBERS

• Complex Numbers C - is the union of real numbers and imaginary numbers. We will nowlook at this number system in detail

Copyright - The information and materials on this worksheet are copyright NQ Corp Pty Ltd

1.2 Imaginary numbers

These are based on the following definition of the imaginary unit i where:

i2 = −1

In other words we are proposing there exists this unusual ’thing’ called i such that when it issquared we get negative one. All imaginary numbers exist in the form iy where y is a real num-ber. In other words, any real number multiplied by i becomes an imaginary number e.g. 2i, i

2, 7i.

They have no real physical meaning, as they are ’imaginary’.

1.3 Complex numbers

These are the UNION (combination) of real numbers and imaginary numbers. Therefore, allcomplex numbers (commonly called z) can be written in the form

z = x+ iy

where:x is the REAL PART of z or Re(z)y is the IMAGINARY PART z or Im(z)

If Re(z) = 0, then we say that z is purely imaginary (real part ’does not exist’)If Im(z) = 0, then we say that z is purely real (imaginary part ’does not exist’)

The complex number system has no ordering. In other words, we cannot say one complex numberis greater than or less than another complex number.

1.4 Arithmetic involving complex numbers

Addition, subtraction and multiplication on the set of complex numbers are calculated usingthe standard arithmetic procedures we are used to doing in real numbers and algebra.

e.g. Addition: (1 + 2i) + (2 + 6i) = ..............................

e.g. Subtraction: (1 + 2i)− (2 + 6i) = ..............................

e.g. Multiplication: (1 + 2i)(2 + 6i) = ............................................................................................

Notice that the final answer is often written with their respective real and imaginary parts inthe form x+ iy. Division of complex numbers involves a bit more effort to get it in the form x+ iy,so before we look at division we need to first look at complex conjugates.

1.5 Complex conjugates

If z = x + iy, then we define the complex CONJUGATE of z as z = x − iy. This allows usto derive a very useful property. Note that:

zz = (x+ iy)(x− iy)

= ............................

= ............................


Note that this number is purely real (i.e. no imaginary parts) and is always non-negative.

e.g. If z = 3 + 2i then find z, hence find the value of zz.

1.6 Division involving complex numbers

The idea of complex conjugates is needed to divide two complex numbers and simplify themto the form x+ iy. The first step is to make the denominator REAL (sometimes called ’realis-ing’ the denominator). In order to do this we needed to multiply the numerator and denominatorby the conjugate of the denominator, because this converts the denominator into a real number(as shown in the previous part) rather than a non-real number. It is very similar to the techniqueof rationalising the denominator.

e.g. Simplify3 + 2i

1 + 4iin the form x+ iy.

3 + 2i

1 + 4i=

3 + 2i

1 + 4i× 1− 4i

1− 4i

= ......................................................

= ......................................................

= ......................................................

BRIEF EXERCISES - express the following in the form x+ iy

1. (3 + 2i) + (4− 5i)

2. (2 + 3i)− (3− 2i)


3. (1 + i)(2 + i)

4.3− 4i

1 + 2i

2. Quadratic Equations Involving Complex Numbers

2.1 Equality of complex numbers

Suppose z = a + ib and say ω = c + id. If z = ω then we can say that a = c and b = d.In other words Re(z) = Re(ω) (the real parts are equal) and Im(z) = Im(ω) (the imaginary partsare equal).

e.g. If (2 + i)2 = x+ iy, find the values of x and y.

e.g. If1 + i

1− i= x+ iy, find the values of x and y.


e.g. If1

2+

√3

2i = cos θ + i sin θ for 0 < θ < π

2, find the value of θ.

(Note that we are using radian measures of angles rather than degrees. The conversion fromdegrees to radians uses π radians = 180 degrees. You will be shown the reason for this in moredetail in the latter part of the Extension 1 course, but for now just accept the result.)

2.2 Square root of complex numbers

Brief Exercise - Suppose√a+ ib = x + iy, find an expression for a and find another ex-

pression for b in terms of x and y.

Suppose we want to find say√

3 + 4i in the form x + iy (keep in mind that x and y are realnumbers). Using the same method employed above (here a = 3 and b = 4), notice that we havea set of simultaneous equations to solve for x and y. Since one of the equations is quadratic, weexpect TWO pairs of solutions which means there are TWO ways to write

√3 + 4i in the form

x+ iy. We solve this problem as follows:√

3 + 4i = x+ iy⇒ 3 + 4i = (x+ iy)2


BRIEF EXERCISES - express the following in the form x+ iy

1.√

24 + 10i


2.√

15 + 8i

3.√−21 + 20i


4.√i

2.3 Solving quadratic equations with real coefficients

Many quadratic equations you may have encountered may have had no real solution becauseyou end up square rooting negative numbers. This does not necessarily mean that it is impossibleto have a solution. It just means that the solutions that exist are not in the real number system.Using the knowledge of complex numbers, it is now possible to solve EVERY quadratic equation.In other words, whilst the solutions of a quadratic equation may not exist in the real numbersystem R, they will always exist in the complex number system C.


e.g. Solve x2 + 2 = 0 if

a) x ∈ R

x2 = −2 gives no real solutions for x

b) x ∈ C

x2 = −2

x = ±√−2

x = i±√

2

Note that there are always two solutions in the complex number system. We can now solve equa-tions which were otherwise unsolvable in the real number system.

e.g. Solve x2 + 2x+ 5 = 0 for x ∈ C

e.g. Solve 2z2 − 2z + 1 = 0 for z ∈ C


2.4 Solving quadratic equations with non-real coefficients

The complex number system also allows us to invent quadratic equations with complex coefficients.These are solvable using the same methods as solving any real quadratic equation. However, theyoften require a bit more effort to simplify to the form x + iy because we need to evaluate thesquare root of a complex number.

e.g. Solve 3z2 − 3z − i = 0 for z ∈ C

z =3±

√(−3)2 − 4(3)(−i)

2(3)

=3±√

9 + 12i

6

The√

9 + 12i needs to be simplified so that z can be written in the form x + iy. We employ thestandard procedure for finding the square root of the complex number.

Let√

9 + 12i = x+ iy

⇒ 9 + 12i = (x+ iy)2


BRIEF EXERCISES - solve the following quadratic equations expressing your solutions inthe form x+ iy

1. z2 + 1 = 0

2. u2 − 2u+ 3 = 0

3. 3y2 + 3y + 1 = 0

4. x2 − (4 + i)x+ 9 + 7i = 0


5. a2 − (1− 4i)a− 5 + i = 0


complex numbers - nqt education · pdf file- the problem with this system is that the...

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