174 CHAPTER 5 ALGEBRA
SO b^2 + b + 1 lies strictly between two consecutive perfect squares. An impossibility!
These examples used very simple inequalities. It is essential that you master them,
as well as a few more sophisticated ideas.Fundamental Ideas
Let us begin with a brief review of the basic ideas, many of which we will present as a
series of statements that are problems (exercises?) for you to verify before moving on.Basic Arithmetic
The following are pretty simple, but you should ponder them carefully to make sure
that you really understand why they are true. Do pay attention to the signs of your
variables!5.5. 1 Addition. If x :2: y and a :2: b, then x + a :2: y + b.
5.5.2 Multiplication. If x :2: y and a :2: 0, then ax :2: ay. Conversely, if a < 0, then
ax:=:; ay.
5.5.3 Reciprocals. If x :2: y, then 1/ x:=:; l/y, provided that both x and y have the same
sign.5.5.4 Distance interpretation of absolute value. The set
{x:lx-al=b}
consists of all points x on the real number line^14 that lie within a distance b of the point
a.
Growth Rates of Functions
It is important to understand the hierarchy of growth rates for the most common func
tions. The best way to learn about this is to draw lots of graphs.5.5.5 Any quadratic function of x will dominate any linear function of x, provided that
x is "large enough." For example,(^0). 001 � > l00000x+ 20000000
is true for all x > 10^9.
5.5.6 By the same reasoning, JCl will "eventually dominate" l' provided that a> b > O.
5.5.7 Likewise, if a is any positive number, and b > 1, then lr eventually dominates
JCl. (In other words, exponential functions grow faster than polynomial functions.)5.5.8 Conversely, if a is any positive number, and b > 1, then JCl eventually dominates
logbx.(^14) you have probably noticed that we are restricting our attention to real numbers. That is because it is not
possible to define z > 0 in a meaningful way when z is complex. See Problem 2. 3. 16 on page 50.