The Complex Numbers

Part (g) follows immediately from this.
Part (h) follows directly from Part (g) and the other parts of the theorem. In fact,

On taking square roots, we conclude , which is Part (h).
The inequalities in the following theorem are used extensively - particularly in the next section. The proofs are very simple and are left as an exercise (Exercise 1.1.12).
Theorem 1.1.8. If iy, then .
Inversion and Division. Recall that the inverse of a non-zero complex number is

Stating this in the last form makes the identity obvious.
This also suggests the right way to do complex division problems in general: to express as a complex number in standard form (as a real number plus times a real number), simply multiply both numerator and denominator by . That is,

The number is then easily put in standard form and the problem is finished by dividing by the real number .
Example 1.1.9. Express in the standard form .
Solution:

Example 1.1.10. Express and in standard form.
Solution:

2. Exercise Set 1.1

1. 
   Express and in the standard form .
   
2. 
   Express in the standard form .
   
3. 
   Express and in the standard form .
   
4. 
   Express in the standard form .
   
5. 
   Find a square root for .
   
6. 
   If , express in standard form.
   
7. 
   By direct substitution, prove that the two solutions to the quadratic equation given in Example 1.1.3 really do satisfy equation (1.1.1).
   
8. 
   Prove Part (e) of Theorem 1.1.4.
   
9. 
   Prove Parts (a), (b) and (c) of Theorem 1.1.7.
   
10. 
    Prove Parts (d), (e) and (f) of Theorem 1.1.7.
    
11. 
    For and prove the following:
    

(a) and ;
(b) and .

12. 
    Prove Theorem 1.1.8 - that is, prove that if , then
    

13. 
    Graph the set of points which satisfy the equation .
    
14. 
    Graph the set of points which satisfy the equation . Hint: If , rewrite this equation as an equation in and .
    
15. 
    Prove that if is a non-zero complex number, then and .
    
16. 
    If is any non-zero complex number, prove that has modulus one.
    
17. 
    Prove that every complex number of modulus 1 has the form for some angle .
    
18. 
    Prove that every line or circle in is the solution set of an equation of the form
    

where and are real numbers and is a complex number. Conversely, show that every equation of this form has a line, circle, point, or the empty set as its solution set.