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Complex Numbers

- GE 111

COMPLEX NUMBERS

- Because x2 must be greater than 0 for every real

number, x, the equation - x2 -1
- Has no real solutions. To deal with this problem,

Mathematicians of the eighteenth century

introduced the imaginary number - Which they assumed had the property
- But otherwise could be treated as a real number.

Expressions of the form where a and b are real

numbers, were called complex numbers, and these

were manipulated according to the standard rules

of arithmetic with the added property that By

the beginning of the nineteenth century it was

recognized that a complex number, Could be

regarded as an alternate symbol for the ordered

pair (a,b), of real numbers and that operations

of addition, subtraction, multiplication, and

division could be defined on these ordered pairs

so that the familiar laws of arithmetic hold and

i2 -1. Thus A complex number is an ordered

pair of real numbers, denoted either by (a,b) or

abi.

Examples of complex numbers in both notations

- For simplicity, the last three complex numbers

would usually be abbreviated as - Geometrically, a complex number can be viewed

either as a point or a vector in an xy plane.

Figure 2

Figure 1

Some complex numbers are shown as points (Figure

1) and as vectors (Figure 2). It may also be

convenient to use a single letter, such as z, to

denote a complex number. Thus we might write The

real number, a, is called the real part of z and

the real number, b, is called the imaginary part

of z. These numbers are sometimes denoted Re(z)

and Im(z), respectively. Therefore When

complex numbers are represented geometrically, in

an xy-coordinate system, the x-axis (horizontal)

is called the real axis, the y-axis the imaginary

axis, and the plane is called the complex

plane. Recall, that we say that 2 vectors are

defined to be equal if they have the same

components, so we define two complex numbers to

be equal if both their real and imaginary parts

are equal.

Two complex numbers, abi and cdi, are defined

to be equal (i.e. abicdi) if ac and

bd. Note that all real numbers are special

cases of complex numbers.... the imaginary

component is zero. Geometrically, the real

numbers correspond to points on the real

axis. If the real part of a complex number is

zero and the imaginary portion is non zero, these

points lie on the imaginary axis and these

numbers are considered purely imaginary (called

pure imaginary numbers). Addition of complex

numbers Complex numbers are added by adding

their real parts and adding their imaginary

parts (abi)(cdi)(ac)(bd)i Subtraction of

real numbers Similar to addition (but in the

opposite sense) subtraction is performed by

subtracting like parts (abi)-(cdi)(a-c)(b-d)i

Multiplication by a scalar

- Both components of the complex number are

multiplied by the scalar - k(abi)(ka)(kb)i (if k is real)
- k(abi)(-kb)(ka)i (if k is purely imaginary)
- Graphically

Figure 3

Figure 4

Recall that when adding vectors graphically, the

vectors are placed head to tail, while

subtraction is performed by placing consecutive

vectors head to head. Note that the vector

z1-z2 followed by the (tail of) vector z2

results in z1. This makes sense algebraically,

as well (z1-z2z2z1).

K gt 0

K lt 0

Figure 6

Figure 5

- Multiplying a complex number (vector) by a scalar

simply changes the amplitude of the (vector)

complex number, as long as the scalar is greater

than zero (Figure 5). If the scalar is less than

zero, the (vector) complex number is positioned

on the opposite side of the origin. This is often

referred to as a 180 degree phase shift. - To this point, there have been parallels between

complex numbers and vectors in 2-Dimensional

space.

However, lets now consider multiplication of

complex numbers, an operation without a vector

analog in 2-D space. When calculating products

of complex numbers, follow the usual rules of

algebra, but treat i2 as -1. Other

properties of complex arithmetic

Modulus, Complex Conjugate, Division

If zabi is any complex number, then the

conjugate of z, denoted by is defined by In

words, is obtained by reversing the sign of

the imaginary part of z. Geometrically, z is the

reflection of about the real axis

Figure 7. A complex number and its conjugate.

It is interesting to note that if and

only if z is a real number.

If a complex number is viewed as a vector in 2-D

space, then the norm or length of the vector is

called the modulus (or absolute value) of z. The

modulus of a complex number zabi, denoted by

z, is defined by

Note that if b0, then za is a real number, and

so the modulus of a real number is simply its

absolute value. It is for this reason, that the

modulus of z is called the absolute value of z.

Example Prove that

Division of complex numbers is typically

considered the opposite of multiplication. Thus,

if

then the definition of should be

such that

Thus

Let

equating real and imaginary parts gives two

simultaneous equations

Using Cramers Rule

So for

Although this may seem complex, this is merely

the original quotient multiplied by the complex

conjugate of the denominator

in the form

Example Express

Properties of Complex Conjugates

Polar Form

If zxiy is a nonzero complex number, r z and

indicates the angle from the positive real axis

to the vector z, then as suggested in the figure

on the next

The projection of the vector on the X axis is

and the projection of the vector on the Y axis is

Such that zxiy can be written as

or

r is the amplitude (modulus) of the complex

number and is the angle between the vector and

the x axis, (arg(z) or phase angle)

Note that the angle, , can be determined using

However, care must be taken in the calculation of

as it will depend on the quadrant location of

the complex number as illustrated in the graph

below

Both quadrants II and IV produce negative numbers

in the calculation of the tan-1 function but the

calculated angle is for quadrant IV. Similarly,

quadrant I and III produce positive numbers for

the tan-1 calculations but the result applies

only to quadrant I. Hence the need to add 180

deg (p rad) to the angle values of the complex

numbers when located in quadrants II or

III. Calculate r and for z -2 3i, -3 4i

and 1 3i

Complex Exponentials

Eulers Formula can also be used as another form

for expressing the phase angle of a complex

number and is given by

Other relationships with exponentials and complex

numbers

Eulers formula allows us to envision the

geometrical implications of complex

multiplication more easily.

For example, the complex number with amplitude r1

and phase 1 multiplied by

a second complex number with amplitude r2 and

phase 2 can be calculated by

Similarly

This means that the result of these operations is

another complex number, whose amplitude is the

product of the 2 amplitudes and the phase angle

is the sum of the 2 phase angles. Consider the

example problem on16 done using polar

coordinates C A/B A 3 4i B 1

2i

- Note that on16 our answer for A/B was
- A/B -1 2i
- The modulus and phase angle for this complex

number are - Which agrees with previous
- Compute using polar method
- D AB/C where A -7i B 2 3i C 4 5i

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