# Algebraic form.

A complex number is an expression of the form , where are real numbers and is a number'' such that . The number is called the real part of ( ) and is the imaginary part (or imaginary coefficient) of ( ).

The set of all complex numbers is denoted . We denote .

Properties:

1. Equality: and .
2.    Im .
3. If , is called pure imaginary.

Example 1.1.1
• is pure imaginary.
• is real.

Definition 1.1.2 (Operations)   Let and .
2. Multiplication: .

Example 1.1.3 (Maple writing)   : Z1 := 2 + 5*I:

Z2 := 3 - 7*I:

z[1] = Z1;

z[2] = Z2;  ;

z[1]+z[2]=Z1+Z2;

z[1]*z[2] = Z1*Z2;

These operations have the same algebraic properties as the corresponding operations in (associativity, commutativity, etc.; please prove ...). Thus, the classical formulas (such as Newton's binomial) are also true in .

 . . . . .

Example 1.1.4

• .
• .
• .
• .
• .

It is possible to use Maple to verify these laws (associativity, commutativity, etc.). Here a a few examples:

• : Associativity: Z1 := x[1] + I*y[1]: z1  = Z1;

Z2 := x[2] + I*y[2]: z2  = Z2;

Z3 := x[3] + I*y[3]: z3  = Z3;

w1 := Z1*(Z2 + Z3):

w2 := Z1*Z2 + Z1*Z3:  ;

z1*(z2 + z3)  = w1;

z1*z2 + z1*z3  = w2;

w1 := expand(w1):

w2 := expand(w2):  ;

z1*(z2 + z3)  = w1;

z1*z2 + z1*z3  = w2;  ;

Does z1*(z2 + z3) = z1*z2 + z1*z3 ¿;

evalb(w1 = w2);

• Commutativity: Z1:='Z1': Z1 := x[1] + I*y[1]:

Z2:='Z2': Z2 := x[2] + I*y[2]:

z[1] = Z1;

z[2] = Z2;  ;

z1 + z2 = Z1 + Z2;

z2 + z1 = Z2 + Z1;  ;

Does z1 + z2 = z2 + z1 ¿;

Z1+Z2 = Z2+Z1;

evalb(Z1+Z2 = Z2+Z1);

Definition 1.1.5   Let , where . The complex conjugate of is the number .

Example 1.1.6

• .
• .
• .

Example 1.1.7 (With Maple)   Z1 := 3 + 2*I:

z[1] = Z1;

conjugate(z[1]) = conjugate(Z1);      ;

Z2 := -7 + 5*I:

z[2] = Z2;

conjugate(z2) = conjugate(Z2);

Proposition 1.1.8   :
(i)
.
(ii)
.

Proof. Denote and , where are real numbers. Then:
(i)
, thus:

(ii)
, thus:

The following proposition is very simple; we leave the task of proving it to the reader.

Proposition 1.1.9   Let . Then:
1. .
2. is pure imaginary if, and only if, .

Proof. Denote , where and are real numbers. Then we have:
1. .
2. is pure imaginary if, and only if, , that is . In this case, .

Example 1.1.10   Find all the complex numbers such that is a real number.

Denote , where and are real numbers. Then we have:

Now, is a real number if, and only if, , i.e. or .

Let us consider the two cases separately:

• If , then and (a negative real number).
• If , then itself is a real number, and the square of a real number is a real number.

Definition 1.1.11 (Inverse of a complex)   If , then it has a complex inverse . Let , where and are real numbers; then we have:

Proof.

In other words, we have:

Example 1.1.12

Proposition 1.1.13   For any , we have:
(i)
.
(ii)
.

Proof.

(i)
If , with real and , we have:

(ii)
If , with real and , we have:

The property in Proposition 1.13 (ii) justifies the following definition:

Definition 1.1.14 (Absolute value)   .

For , where and are real numbers, we have:

Example 1.1.15

Proposition 1.1.16   For any , we have:
(i)
.
(ii)
.

Proof. We denote and , where are real numbers.
(i)
, thus on the one hand, we have:

 (1.1)

On the other hand, we have:

 (1.2)

We develop the right-hand sides of Equation 1 and of Equation 2 and the required equality follows.
(ii)
We can leave it to the reader. Section 2 explains why it is possible here to use results from analytic geometry.

Example 1.1.17   If and ,
• .
• .
• and .
Then we have:

Corollary 1.1.18   For any , we have:

The proof is an easy consequence of Proposition 1.16.

Corollary 1.1.19   For any and any we have:

Here the proof is an easy consequence of Corollary 1.18 and Proposition 1.16.

Remark 1.1.20   For any , we have:

It follows that the best way to compute a quotient is to multiply both the numerator and the denominator by the conjugate of the denominator, and then simplify.

Example 1.1.21

Noah Dana-Picard 2007-12-24