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:
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 .

It is possible to use Maple to verify these laws (associativity, commutativity, etc.). Here a a few examples:
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);
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);
z[1] = Z1;
conjugate(z[1]) = conjugate(Z1); ` `;
Z2 := 7 + 5*I:
z[2] = Z2;
conjugate(z2) = conjugate(Z2);
The following proposition is very simple; we leave the task of proving it to the reader.
Denote , where and are real numbers. Then we have:
Let us consider the two cases separately:
In other words, we have:
The property in Proposition 1.13 (ii) justifies the following definition:
For , where and are real numbers, we have:
The proof is an easy consequence of Proposition 1.16.
Here the proof is an easy consequence of Corollary 1.18 and Proposition 1.16.
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.
Noah DanaPicard 20071224