Download THE COMPLEX NUMBERS

COMPLEX NUMBERS
INDEX
1.
2.
3.
4.
5.
6.
7.
The history of the complex numbers;
The imaginary unit I ;
The Algebraic form;
The Gauss plane;
The trigonometric form;
The exponential form;
The applications of the complex numbers.
School year 2007-08
Class 3 G
Teacher : Rosella Natalini
1. The history of complex numbers
Complex numbers, of completely different nature from the numbers that we are
accustomed to know, allow getting to the solution of an apparently impossible problem, like
the problem of the extraction of the square root of a negative number.
The most ancient text, where the negative number’s root is cited, is by Eron of Alessandria
in the first century A.D.
But the problem about the extraction of a square root of a negative number appeared more
frequently in the study of the equations of the second and third degree by Niccolò Tartaglia
and Girolamo Cardano, two famous algebraists of the Renaissance.
These two algebraists found a formula to determine the solutions of the third degree
equation, which has the following form
x 3 + px + q = 0 .
In this formula, the calculation of the square root of a negative number was often essential
to find real solutions.
This fact was extraordinary at that time because nobody believed in the real existence of
these numbers, which were only regarded as some tricks used to arrive at the solution of
the equations.
In the definition of the complex numbers, we introduce a
particular unit, called imaginary unit, indicated as ‘ i ’, which
has the following property;
i 2 = −1
Thanks to the introduction of the imaginary unit, it is possible
to calculate the square root of any negative number, writing it
as the product of its opposite multiplied by − 1 .
An imaginary number is the product of a real number
multiplied by the imaginary unit; the term ”imaginary” was
used, for the first time, by Cartesio.
In the XVIII century, great mathematicians worked on the complex numbers’ theory, in
particular De Moivre(1739), who found the famous formula for the calculation of the
powers: (cos ϑ + senϑ ) n = cos nϑ + sin nϑ and Euler (1748), who introduced the
exponential notation for the complex numbers: ρe iϑ = ρ (cos ϑ + i sin ϑ )
But it was only thanks to the introduction of the geometric interpretation of the complex
numbers that mathematicians accepted these numbers.
Since the second half of the seventeenth century, the mathematicians John Wallis had
interpreted imaginary numbers on a perpendicular straight line; however, nobody had had
the idea to considerate the real part a and the coefficient b of the imaginary part of the
complex number
a + ib , as the coordinates of the points of a plane.
It was only in the 1797 that the Danish mathematician Caspar Wessel had this intuition.
However, his work remained unknown like that one of the French mathematicians Jean
Robert Argand in the 1806.
It was Gauss, about thirty years later, who gave full dignity to
complex numbers, with their geometric interpretation on a
plane, called exactly Gaussian plane.
Friedrich Gauss
2. The imaginary unit i
To form complex numbers you have to refer to the imaginary unit, with the symbol i that it
is characterized by the following property : i 2 = −1 .
In order to preserve the formal property of the operations with this new number you have
to consider also (−i ) 2 = −1 .
This means than we can identify the square root of -1 with ± i :
−1 = ±i
Now it is possible to calculate the square root of a negative number writing this as a
product of its opposite by -1.
EXAMPLE
− 16 = 16(−1) = 16 ⋅ − 1 = ±4i
Number as ± 4i are imaginary numbers.
An imaginary number is the product between a real number and the imaginary unit.
With the imaginary numbers it is possible to resolve a 2 nd grade equation as ax 2 = b being
a and b of opposite sign.
EXAMPLE
x 2 + 25 = 0
It has got as a solution x = ± − 25 that is x = ±5i .
To operate with imaginary numbers, you have to use the imaginary unit (i) such as a letter
of algebraic computation remembering that i 2 = −1 .
EXAMPLES
1)
2i − 4i = −2i
2)
− 3i ⋅ 4i = −12i 2 = −12 ⋅ (i ) 2 = −12 ⋅ (−1) = 12
3)
20i / 5i = 4
(2i ) 5 = 2 5 ⋅ i 5 = 32 ⋅ i 3 ⋅ i 2 = 32 ⋅ (i ) 2 ⋅ i (−1) = 32 ⋅ (−1) ⋅ i (−1) = 32i
4)
The properties of imaginary numbers are:
•
The powers of i are cyclic of period 4; in fact we have i 1 = i ; i 2 = −1 ; i 3 = i 2 ⋅ i = −i
and so on.
•
The sum and difference of two or more imaginary numbers is also an imaginary
number.
•
The product and quotient of two imaginary numbers is a real number.
•
The result of the sum or of the difference between a real number and an imaginary
isn’t a real number or an imaginary one.
3. The Algebraic form
The definition of complex number is each equation such as:
a+ib
a, b ∈ R.
Where a is the real part of the complex number, b is the coefficient of the imaginary part, C
is the set of complex numbers.
−
if a=0 and b ≠ 0 the number you obtain is imaginary;
−
if b=0 the number is real;
−
two complex numbers are equivalent if both the imaginary part and the real part are
equivalent: a+ib = a+ib;
−
two complex numbers are complex conjugate if they have got opposite imaginary parts:
a+ib conjugate a-ib;
−
two complex numbers are opposite if a and b are opposite: a+ib is opposed to -a-ib.
Operations with complex numbers
To operate with complex numbers we use the same rules to operate with polynomials,
using i2 = -1.
Sum
(a + ib) + (c + id) = (a +c) + i(b + d)
Example : (8 + 2i) + (4 - 6i) = (8 + 4) + i(2 - 6) = 12 – 4i
Multiplication
(a + ib)(c + id) = ac + bci + adi + bdi2 = (ac - bd) + i(bc + ad)
Example: (3 + 6i)(4 + 9i) = 12 + 24i + 27i + 54i2 = (12 - 54) + i(24 + 27) = - 42 + 51i
Division
a + ib c − id (a + ib)(c − id ) ac + bd + i (bc − ad )
*
=
=
c + id c − id
c 2 − d 2i 2
c2 + d 2
Example:
12 + 2i 4 − 4i (12 + 2i )(4 − 4i ) 48 − 48i + 8i − 6i 48 − i (48 + 8 − 6) 48 − 50i
=
=
=
=
*
4 + 4i 4 − 4i
16 + 16
32
32
16 − 16i 2
Resolution of the equations in C
With complex numbers we can solve 2nd degree equations also with negative discriminant.
If the solutions of a 2nd degree equations are not real, they're 2 conjugate complex
numbers.
The solutions of a 2nd degree equations may be:
- different and real if
Δ>0
- equivalents and real if Δ=0
- complex conjugate if
Δ<0
Fundamental theorem of algebra
Each N degrees algebraic equation admits, in C, N solutions if each of those solutions is
counted with its multiplicity.
4. The Gaussian plane
Gauss’s plane is a Cartesian plane where we can represent the complex numbers a + ib
as pairs of real numbers (a,b) where “a” is the real part and “b” is the imaginary part : so
the P point of coordinates (a ,b) represents the complex a + ib , that can be represented by
the OP vector.
On the Gaussian plane we can also represent the addition of complex numbers
In the picture below we can see the addition of complex numbers Z=1 +5i and W=G + i
Z +W = 7 + 6i
Z+W
Z
W
5. The trigonometric form
On the Cartesian plane we see that the complex number z has two coordinates: A is
represented on the axis of the abscises and B is represented on the axis of ordinates.
The segment that starts from the centre and goes to the complex number z is a vector that
we call ρ .So we can say that z = (ρ, ϑ ).
We can conclude that sin ϑ = b / ρ and cos ϑ = a / ρ
z = a + ib = ρcos ϑ + iρsin ϑ = ρ(cos ϑ + isin ϑ ).
we have:
The notation z = ρ(cos ϑ + isin ϑ ) is called trigonometric form of the complex number z.
ρ is the module of z and it is shown also by the symbol |z| , ϑ is the argument or the
anomaly.
How we can pass from algebraic form to trigonometric form
ρ=
a2 + b2
sin ϑ =
b
ρ
b
=
cos ϑ =
a2 + b2
a
ρ
=
a
a2 + b2
tan ϑ =
b
a
Multiplication
The product of the two complex number has as a module the product of the modules and
as an argument the addition of the arguments.
Formula:
Example:
z1*z2= ρ1 ⋅ ρ 2[cos(ϑ1 + ϑ 2) + i sin (ϑ1 + ϑ 2)]
⎡ ⎛π ⎞
⎛ π ⎞⎤
z1 = 3⎢cos⎜ ⎟ + i sin ⎜ ⎟ ⎥
⎝ 6 ⎠⎦
⎣ ⎝6⎠
⎡ ⎛π ⎞
⎛ π ⎞⎤
z 2 = 2 ⎢cos⎜ ⎟ + i sin ⎜ ⎟ ⎥
⎝ 4 ⎠⎦
⎣ ⎝4⎠
⎡ ⎛π π ⎞
⎛ π π ⎞⎤
z1 * z 2 = 3 ⋅ 2 ⋅ ⎢cos ⎜ + ⎟ + i sin ⎜ + ⎟ ⎥ =
⎝ 6 4 ⎠⎦
⎣ ⎝6 4⎠
5
5 ⎞
⎛
= 6⎜ cos π + i sin π ⎟
12
12 ⎠
⎝
Division
The quotient of two complex numbers is the complex number that has as a module the
quotient of modules and as an argument the difference of the arguments of the given
numbers.
Formula:
z1
p1
=
⋅ [cos (ϑ1 − ϑ 2 ) + i sin (ϑ1 − ϑ 2 )]
z2
p2
Example:
⎡ ⎛3 ⎞
⎛ 3 ⎞⎤
z1 = 6 ⎢cos⎜ π ⎟ + i sin ⎜ π ⎟ ⎥
⎝ 2 ⎠⎦
⎣ ⎝2 ⎠
⎡ ⎛π ⎞
⎛ π ⎞⎤
z 2 = 2 ⎢cos⎜ ⎟ + i sin ⎜ ⎟ ⎥
⎝ 6 ⎠⎦
⎣ ⎝6⎠
z1 6 ⎡ ⎛ 3
π⎞
π ⎞⎤
⎛3
= ⎢cos ⎜ π − ⎟ + i sin ⎜ π − ⎟ ⎥ =
6 ⎠4
2
6 ⎠⎦
z 2 2 ⎣ ⎝4 2
⎛
⎞ ⎝
= 3⎜ cos π + i sin π ⎟
3
3 ⎠
⎝
Power
Formula: [ρ ⋅ (cos ϑ + i sin ϑ )] = ρ n ⋅ (cos nϑ + i sin⋅ nϑ )
n
2
Example:
⎡ ⎛
2
2 ⎞
π
π ⎞⎤
⎛
⎢3 ⋅ ⎜ cos 3 + i sin 3 ⎟ ⎥ = 9 ⋅ ⎜ cos 3 π + i sin 3 π ⎟
⎝
⎠
⎠⎦
⎣ ⎝
6. The exponential form
First of all we define the exponential function with imaginary exponent
e bi = cos b + i sin b
Then applying the law of the powers we can write that
e a + bi = e a (cos b + i sin b) .
If b=0 we obtain as a particular case the exponential function with real exponent:
e a (cos 0 + i sin 0) = e a
All the formal laws of the powers apply also to this type of exponential function.
Euler found some important relations between the exponential and trigonometric functions
which were called Euler’s formulas. If we consider a=0 we obtain the first two formulas:
e bi = cos b + i sin b
e − bi = cos b − i sin b
and
If we add and subtract them we obtain the two other relations:
e bi + e − bi
cos b =
2
e bi + e − bi
sin b =
2i
e
These formulas create a connection between the goniometric form of complex numbers
and the defined exponential function. In fact, applying Euler’s first formula, we can write
the complex number z = ρ (cos ϑ + i sin ϑ ) as
z = ρe iϑ
This is the exponential form of the complex number z with module ρ and argument ϑ .
Operations such as the multiplication, power and division of complex numbers in
exponential form are easily deduced from the definition.
ϑi
If z1 = ρ1e 1 e z 2 =
ρ 2 ≠ 0 will be
ρ 2 eϑ i , the product will be z1 z 2 = ρ1 ρ 2 e (ϑ +ϑ )i , the quotient of
2
1
2
z1 ρ1 (ϑ1 −ϑ2 )i
n
n nϑi
=
e
and the power will be z = ρ e .
z2 ρ2
EXAMPLES
Algebraic form z = 2 + 2i
Trigonometric form z = 2(cos
π
π
+ isen )
4
4
π
Exponential form
Algebraic form
z = 2e 4
i
z = −1
Trigonometric form z = (cos π + isenπ )
Exponential form
z = e πi
7. The applications of the complex numbers
Complex numbers find applications in various fields such as Mathematics, Physics and
Engineering.
MATHEMATICS
complex numbers are used in Complex Analysis, in Geometry or in
other secondary applications, as:
•
Number theory: this theory uses the Complex Analysis to face problems on the
complex numbers
•
Improper integrals: some of these can be solved with the theorem of the residues of
Complex Analysis
•
Differential equations: they are solved finding the complex roots of a polynomial
associated to the equation
•
Fractals: a fractal is a geometric object that repeats itself in its structure equally on
different scales, so it doesn't change aspect. Fractals start with a complex number.
Each number produced gives a value for each pixel on the screen. Common
fractals are based on the Julia Set and the Mandelbrot Set.
Julia Set
PHYSICS
•
Mandelbrot Set
they are used in some sectors as:
Fluid dynamics: complex numbers are used for describing the potential flow in two
dimensions
•
Quantum mechanics: in this field, complex numbers are essential because the
theory is developed in Hilbert space of infinite dimension derived by C, complex
numbers’ set
• Relativity: complex numbers are used
because some formulas of the metric
space become simpler if the temporal
variable is supposed as an imaginary
variable.
General Relativity
ENGINEERING
complex numbers are used in the analysis of the signals and in all the
fields where the periodic signals are studied. In Electric Engineering and Electronics they
are used for pointing out voltage and current. With the complex numbers you can sum up
the analysis of resistance, capacity and inductance in only one entity, called impedance.
They can, besides, express some relations that take into consideration the frequencies
and the behaviour of the components, according to frequency.