Download The Complex Number System The need for extending the real

The Complex Number System
The need for extending the real number system is evident when considering solutions of
simple equations. For example, the equation
x2 + 1 = 0
has no real number solutions, for if x is any real number, then x2 ≥ 0 and so x2 + 1 ≥ 1.
We extend the real number system to the system of complex numbers in such a way
that the arithmetic of the real numbers is preserved.
A complex number is an ordered pair of real numbers (a, b). The set of complex numbers
is denoted by C. Just as the real numbers can be viewed as the points on the number line
(real line), the complex numbers can be viewed as the points of the coordinate plane.
• Equality: If z1 = (a1 , b1) and z2 = (a2 , b2) are complex numbers, then z1 = z2
if and only if a1 = a2 and b1 = b2 .
• Addition: If z1 = (a1 , b1 ) and z2 = (a2 , b2) are complex numbers, then
z1 + z2 = (a1 , b1 ) + (a2 , b2 ) = (a1 + a2 , b1 + b2 )
Properties of Addition:
1. z1 + z2 = z2 + z1
(Commutative)
2. (z1 + z2 ) + z3 = z1 + (z2 + z3 )
(Associative)
Let z = (a, b).
3. z + (0, 0) = (0, 0) + z = z (0, 0) is the Additive Identity. (Note: It is conventional to use 0 to represent the additive identity. The context will make it clear
whether we mean the real number 0 or the complex number (0, 0).)
4. z + (−a, −b) = (−a, −b) + z = 0
(−a, −b) = −z is the Additive Inverse
• Subtraction: To subtract z2 from z1 , we add the additive inverse of z2 to z1 .
That is,
z1 − z2 = z1 + (−z2 ) = (a1 , b1) + (−a2 , −b2 ) = (a1 − a2 , b1 − b2 ).
• Multiplication: If z1 = (a1 , b1) and z2 = (a2 , b2) are complex numbers, then
z1 · z2 = (a1 , b1 ) · (a2 , b2) = (a1 a2 − b1 b2 , a1b2 + a2 b1 )
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Properties of Multiplication:
1. z1 · z2 = z2 · z1
(Commutative)
2. (z1 · z2 ) · z3 = z1 · (z2 · z3 )
(Associative)
Let z = (a, b).
3. z · (1, 0) = (1, 0) · z = z
(1, 0) is the Multiplicative Identity
4. If (a, b) 6= 0, then
a
−b
a
−b
z·
,
=
,
· z = (1, 0);
a2 + b2 a2 + b2
a2 + b2 a2 + b2
−b
1
a
, 2
= = z −1 is the Multiplicative Inverse
2
2
2
a +b a +b
z
• Division: To divide z1 by z2 6= 0, we multiply z1 by the multiplicative inverse
of z2 . That is,
z1
−b2
a1 a2 + b1 b2 b1 a2 − a1 b2
1
a2
,
=
,
.
= z1 ·
= (a1 , b1 ) ·
z2
z2
a22 + b22 a22 + b22
a22 + b22
a22 + b22
Note that
(a, 0) + (b, 0) = (a + b, 0) and
(a, 0) · (b, 0) = (ab, 0).
Therefore the set of complex numbers of the form (x, 0) behaves, with respect to addition
and multiplication, in exactly the same way as the set of real numbers. Clearly, there is a
one-to-one correspondence between the set of real numbers and the set of complex numbers
of the form (x, 0). In this sense, then, the complex numbers contain the set of real numbers.
For convenience we denote (x, 0) as simply x.
With this convention and the fact that
(a, b) = (a, 0) + (0, 1) · (b, 0),
we can express the complex number (a, b) as a + (0, 1) · b. If we denote the complex
number (0, 1) by the symbol i, then each complex number (a, b) can be expressed as
a + bi.
The complex number i = (0, 1) has the property that
i2 = (0, 1) · (0, 1) = (−1, 0)
which, by our convention, is −1, that is i2 = −1. Thus, i is a solution of the quadratic
equation
x2 + 1 = 0,
the simple equation which we used to motivate the need for the complex number system.
The complex number −i is also a solution of this equation.
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We now have two ways of representing a complex number z: as an ordered pair
z = (a, b), which is useful in viewing the complex numbers geometrically as points in
the coordinate plane, and as z = a + bi which is useful in arithmetical calculations. In
particular,
• Addition: (a1 + b1 i) + (a2 + b2 i) = (a1 + a2 ) + (b1 + b2 )i and
• Multiplication:
(a1 + b1 i) · (a2 + b2 i) = (a1 · a2 + b1 i · b2 i) + (a1 · b2 i + a2 · b1 i)
= (a1 a2 + b1 b2 i2 ) + (a1 b2 + a2 b1 )i
= (a1 a2 − b1 b2 ) + (a1 b2 + a2 b1 )i
The a+bi representation of a complex number provides a clear explanation of the definitions
of addition and multiplication of ordered pairs given above.
Let z = a + bi be a complex number. The real number a is called the real part of z
and the real number b is called the imaginary part of z.
The complex conjugate of z = a + bi, denoted z, is the complex number z = a − bi.
Some of the properties of the complex conjugate are:
(i) z + z = 2a.
(ii) z = z.
iii) z · z = a2 + b2 .
(iv) z1 + z2 = z1 + z2 .
(v) z1 · z2 = z1 · z2 .
The modulus or norm of z = a + bi, denoted by |z|, is the nonnegative real number
p
|z| = (z · z)1/2 = a2 + b2 .
√
Note that if z = a is a real number, then |z| = a2 = |a|, the absolute value of a, which
is the distance from the point a to the origin on the number line. In general, the modulus
of the complex number z is the distance from the point (a, b) to the origin (0, 0) in the
coordinate plane. Also, if |z| = 1, then a2 + b2 = 1 and so it is possible to choose a real
number θ such that a = cos θ, b = sin θ, and
z = cos θ + i sin θ.
If z = a + bi 6= 0, then
a
b
z = |z|
+
i .
|z| |z|
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Since a/|z| + b/|z| i is a complex number of modulus 1, we can represent z as
z = |z|(cos θ + i sin θ)
by our remarks above. This representation is known as the polar form of z. A justification
for deriving the polar form of a complex number is the simplification of the operation of
multiplication of complex numbers. In particular, if
z = |z|(cos θ + i sin θ) and
w = |w| cos φ + i sin φ)
are complex numbers in polar form, then by the addition formulas for sine and cosine, we
have
z · w = |z||w| [cos (θ + φ) + i sin(θ + φ)] .
The study of linear differential equations with constant coefficients requires an extension
of the exponential function exp(x) = ex to a complex-valued function of a complex variable.
A complete treatment of the mathematical details of this extension is beyond the scope of
this discussion. However, if x and y are real numbers and z = x + yi, and if we define
exp(z) = ex+yi = ex eyi = ex (cos y + i sin y),
then this new function agrees with the real exponential function when y = 0. This definition
implies that
eiθ = cos θ + i sin θ
(E)
for all real numbers θ. This result is known as Euler’s Formula. With this result, the polar
form of a complex number can be written as
z = |z|eiθ .
The plausibility of (E) can be demonstrated as follows. Recall from Calculus II that
X xk
xn
x2 x3
+
+···+
+···=
.
2!
3!
n!
k!
∞
ex = 1 + x +
k=0
If we replace x by iθ, we get
eiθ = 1 + (iθ) +
(iθ)2 (iθ)3 (iθ)4 (iθ)5
+
+
+
+··· .
2!
3!
4!
5!
By using the facts that
(iθ)2 = i2 θ2 = −θ2 ,
(iθ)3 = i3 θ3 = −iθ3 ,
(iθ)4 = i4 θ4 = θ4 ,
(iθ)5 = i5 θ5 = iθ5
and so on, it follows that
(iθ)2 (iθ)3 (iθ)4
+
+
+···
2!
3!
4!
θ2
θ3
θ4
θ5
= 1 + iθ −
−i +
+i +···
2!
3!
4!
5!
θ2
θ4
θ3
θ5
= 1−
+
+··· +i θ −
+
+···
2!
4!
3!
5!
eiθ = 1 + (iθ) +
4
Again from Calculus II, the series in the parentheses are the Taylor series expansions of
cos θ and sin θ, respectively. Thus,
eiθ = cos θ + i sin θ.
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