4 CHAPTER 1. SOME PREREQUISITE TOPICS
The theorem will be proved if this last expression is less than1√
2n+3. This happens if
and only if (1√
2n+3
)2
=1
2n+3>
2n+1
(2n+2)2
which occurs if and only if (2n+2)2 > (2n+3)(2n+1) and this is clearly true which maybe seen from expanding both sides. This proves the inequality.
Lets review the process just used. If S is the set of integers at least as large as 1 for whichthe formula holds, the first step was to show 1 ∈ S and then that whenever n ∈ S, it followsn+ 1 ∈ S. Therefore, by the principle of mathematical induction, S contains [1,∞)∩Z,all positive integers. In doing an inductive proof of this sort, the set S is normally notmentioned. One just verifies the steps above. First show the thing is true for some a ∈ Zand then verify that whenever it is true for m it follows it is also true for m+1. When thishas been done, the theorem has been proved for all m≥ a.
1.3 The Complex NumbersRecall that a real number is a point on the real number line. Just as a real number should beconsidered as a point on the line, a complex number is considered a point in the plane whichcan be identified in the usual way using the Cartesian coordinates of the point. Thus (a,b)identifies a point whose x coordinate is a and whose y coordinate is b. In dealing with com-plex numbers, such a point is written as a+ ib. For example, in the following picture, I havegraphed the point 3+2i. You see it corresponds to the point in the plane whose coordinatesare (3,2) .
•3+2iMultiplication and addition are defined in the most obvious way sub-ject to the convention 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 ΜΈ= 0, has a unique multiplicativeinverse.
1a+ ib
=a− ib
a2 +b2 =a
a2 +b2 − ib
a2 +b2 .
You should prove the following theorem.
Theorem 1.3.1 The complex numbers with multiplication and addition defined as aboveform a field satisfying all the field axioms. These are the following list of properties.
1. x+ y = y+ x, (commutative law for addition)
2. x+0 = x, (additive identity).
3. For each x ∈ R, there exists −x ∈ R such that x+(−x) = 0, (existence of additiveinverse).