2.13. THE COMPLEX NUMBERS 31

19. Suppose f (x) = −5x2 + 8x− 7. Find f (R) . In particular, find the largest value off (x) and the value of x at which it occurs. Can you conjecture and prove a resultabout y = ax2 +bx+ c in terms of the sign of a based on these last two problems?

20. Show that if it is assumed R is complete, then the Archimedean property can beproved. Hint: Suppose completeness and let a > 0. If there exists x ∈ R such thatna≤ x for all n ∈N, then x/a is an upper bound for N. Let l be the least upper boundand argue there exists n ∈ N∩ [l−1/4, l] . Now what about n+1?

21. Suppose you numbers ak for each k a positive integer and that a1 ≤ a2 ≤ ·· · . Let Abe the set of these numbers just described. Also suppose there exists an upper boundL such that each ak ≤ L. Then there exists N such that if n ≥ N, then (supA− ε <an ≤ supA].

22. If A⊆ B for A ̸= /0 and A,B are sets of real numbers, show that inf(A)≥ inf(B) andsup(A)≤ sup(B).

2.13 The Complex NumbersJust as a real number should be considered as a point on the line, a complex number isconsidered a point in the plane which can be identified in the usual way using the Cartesiancoordinates of the point. Thus (a,b) identifies a point whose x coordinate is a and whosey coordinate is b. In dealing with complex numbers, such a point is written as a+ ib. Forexample, in the following picture, I have graphed the point 3+ 2i. You see it correspondsto the point in the plane whose coordinates are (3,2) .

3+2i

Multiplication and addition are defined in the most obvious way subject to the conven-tion that i2 =−1. Thus,

(a+ ib)+(c+ id) = (a+ c)+ i(b+d)

and(a+ ib)(c+ id) = ac+ iad + ibc+ i2bd = (ac−bd)+ i(bc+ad) .

Every non zero complex number, a+ ib, with a2 + b2 ̸= 0, has a unique multiplicativeinverse.

1a+ ib

=a− ib

a2 +b2 =a

a2 +b2 − ib

a2 +b2 .

You should prove the following theorem.

Theorem 2.13.1 The complex numbers with multiplication and addition defined asabove form a field satisfying all the field axioms listed on Page 9.

The field of complex numbers is denoted as C. An important construction regardingcomplex numbers is the complex conjugate denoted by a horizontal line above the number.It is defined as follows.

a+ ib≡ a− ib.