Complex
Analysis. FreeTutorial
5.1
Arithmetic operations with complex numbers
5.1.1
Addition
5.1.2
Subtraction
5.1.3
Multiplication
5.1.4
Division
5.2
Properties of the complex numbers
5.2.1
Proof of the properties of the complex
numbers
5.1 Arithmetic operation with
complex numbers
z1
= x1+
y1i,
z2
= x2+
y2i.
Assuming that the ordinary rules of arithmetic
apply to complex numbers we can find:
Addition
z1
+ z2
= (x1+y1i)
+ (x2+y2i)
= (x1+x2)
+ i(y1+y2)
(1.10)
Subtraction
z1
 z2
= (x1+y1i)
 (x2+y2i)
= (x1x2)
+ i(y1y2)
Multiplication
z1z2
= (x1+y1i)(x2+y2i)
= x1x2
+ y1x2i
+ x1y2i
+y1y2i^{2}
z1z2
= (x1x2
 y1y2)
+ (y1x2
+ x1y2)i
(1.11)
Example:
z1
= 0 + 1i
= i;
z2=
0 + 1i
= i.
z1z2
= i^{2}
= (0+1i)(0+1i)
= (0 1)
+ (0+0)i
= 1
Division
z1/z2
= (x1+y1i)/(x2+y2i)
is a complex number, provided that z2
= x2+
y2i
0.
(1.12)
x2^{2}+
y2^{2}
0.
To get the equation (1.12), we force the
complex denominator to be real by multiplying
both numerator and denominator by the
number x2
y2i.
=
= .
(1.13)
After multiplication (x1+y1i)(x2
y2i)
we get the formula (1.12).
Example
13:
There are two numbers:
z1
= 5 
3i,
z2
= 1 +
4i.
Addition:
z1
+ z2
= (5 
3i)
+ (1 +
4i)
= (5 + 1)
+ (3 + 4)i
= 6 + i
Subtraction:
z1
 z2
= (5 
3i)
 (1 +
4i)
= (5  1)
+ (3  4)i
= 4  7i
Multiplication:
z1z2
= (5 
3i)(1
+
4i)
= 5 
3i
+ 20i
 12i^{2}
= 17 + 17i
Division:
z1/z2
= (5 
3i)/(1
+
4i)
= {(12
+5) + i(3
 20)}/(16
+ 1) =  7/17
 i(23/17)
Example
14:
z1
= 1+
i,
z2
= 1
i.
z1/z2
= (1
+ i)/(1
i)
= (1
+ i)^{2}/{(1
i)(1+
i)}
= (1 + 2i
+ i^{2})/(1
 i^{2})
= 2i/2
= i.
5.2 Properties of the complex
numbers
1. Commutative law for addition:
z1
+ z2
= z2
+ z1
Proof
2. Commutative law for multiplication:
z1z2
= z2z1
Proof
3. Associative law for addition:
z1
+ (z2
+ z3)
= (z1
+ z2)
+ z3
4. Associative law for multiplication:
z1(z2z3)
= (z1z2)z3
5. Multiplication is distributive
with respect to addition:
z1(z2
+ z3)
= z1z2
+ z1z3
6. The product of two complex
numbers is zero if and only if at least
one of the factors is zero.
7. Additive Inverses:
Any complex number z
has a unique negative –z
such that z
+ (–z)
= 0. If z
= x
+
yi,
the negative –z
= – x
–
yi.
8. Multiplicative Inverses:
Any nonzero complex number z
= x
+
yi
has a unique inverse 1/z
such that z(1/z)
= 1.
The number 1/z
is called the reciprocal
of the complex number z.
1/z
= .
(1.14)
9. Additive Identity.
There is a complex number w
such that z
+ w
= z
for all complex numbers z.
The number w
is the ordered pair (0,
0).
10. Multiplicative Identity.
There is a complex number
such that z
= z
for all complex numbers z.
The ordered pair (1, 0)
= 1 + 0i
is the unique complex number with this
property.
5.2.1
Proof of the properties of the complex
numbers
None of these properties is difficult
to prove. Most of the proofs use the corresponding
facts in the real number system:
Real numbers are commutative under addition
x
+
y
= y
+
x.
Real numbers are commutative under multiplication
x·y
= y·x.
1. Proof
of the commutative law for addition
Let us prove that z1
+ z2
= z2
+ z1.
z1
= x1
+
iy1;
z2
= x2
+ iy2.
By definition of addition of complex
numbers (1.10) z1
+ z2
= (x1
+
x2)
+ i(y1
+ y2).
x1,
x2,
y1,
y2
are all real and it doesn't matter what
order real numbers are added up in.
By the commutative law for addition
for real numbers
(x1
+
x2)
= (x2
+ x1)
and (y1
+ y2)
= (y2
+ y1).
z1
+ z2
= (x1
+
x2)
+ i(y1
+ y2)
= (x2
+ x1)
+ i(y2
+ y1)
= z2
+ z1
2. Proof
of the commutative law for multiplication
Let us prove that z1z2
= z2z1.
z1
= x1
+
iy1;
z2
= x2
+ iy2.
By definition of multiplication of complex
numbers (1.11) z1z2
= (x1x2
 y1y2)
+ (y1x2
+ x1y2)i
and
z2z1
= (x2x1
 y2y1)
+ (y2x1
+ x2y1)i.
x1,
x2,
y1,
y2
are all real.
By the commutative law
for multiplication for real numbers
x1x2
= x2x1
and
y1y2
= y2y1.
It means that
x1x2
 y1y2
= x2x1
 y2y1
and
y1x2
+ x1y2
= y2x1
+ x2y1.
Such way
z1z2
= z2z1.
by
Tetyana Butler
