Page 13 - Schaum's Outline of Theory and Problems of Advanced Calculus

4 NUMBERS [CHAP. 1

These and extensions to any real numbers are possible so long as division by zero is excluded. In
0
particular, by using 2, with p ¼ q and p ¼ 0, respectively, we are lead to the definitions a ¼ 1,
q
a q ¼ 1=a .
p
p
If a ¼ N, where p is a positive integer, we call a a pth root of N written p ffiffiffiffi
N. There may be more
2
2
than one real pth root of N. For example, since 2 ¼ 4 and ð 2Þ ¼ 4, there are two real square roots of
ffiffiffiffi ffiffiffi
p p
4, namely 2 and 2. For square roots it is customary to define N as positive, thus 4 ¼ 2 and then
p ffiffiffi
4 ¼ 2.
p=q p ffiffiffiffiffi p
If p and q are positive integers, we define a ¼ q a .
LOGARITHMS
p
If a ¼ N, p is called the logarithm of N to the base a, written p ¼ log N.If a and N are positive
a
and a 6¼ 1, there is only one real value for p. The following rules hold:
M
1. log MN ¼ log M þ log N 2. log a N ¼ log M log N
a
a
a
a
a
r
3. log M ¼ r log M
a
a
In practice, two bases are used, base a ¼ 10, and the natural base a ¼ e ¼ 2:71828 ... . The logarithmic
systems associated with these bases are called common and natural, respectively. The common loga-
rithm system is signified by log N, i.e., the subscript 10 is not used. For natural logarithms the usual
notation is ln N.
Common logarithms (base 10) traditionally have been used for computation. Their application
replaces multiplication with addition and powers with multiplication. In the age of calculators and
computers, this process is outmoded; however, common logarithms remain useful in theory and
application. For example, the Richter scale used to measure the intensity of earthquakes is a logarith-
mic scale. Natural logarithms were introduced to simplify formulas in calculus, and they remain
effective for this purpose.
AXIOMATIC FOUNDATIONS OF THE REAL NUMBER SYSTEM
The number system can be built up logically, starting from a basic set of axioms or ‘‘self-evident’’
truths, usually taken from experience, such as statements 1–9, Page 2.
If we assume as given the natural numbers and the operations of addition and multiplication
(although it is possible to start even further back with the concept of sets), we find that statements 1
through 6, Page 2, with R as the set of natural numbers, hold, while 7 through 9 do not hold.
Taking 7 and 8 as additional requirements, we introduce the numbers 1; 2; 3; .. . and 0. Then
by taking 9 we introduce the rational numbers.
Operations with these newly obtained numbers can be defined by adopting axioms 1 through 6,
where R is now the set of integers. These lead to proofs of statements such as ð 2Þð 3Þ¼ 6, ð 4Þ¼ 4,
ð0Þð5Þ¼ 0, and so on, which are usually taken for granted in elementary mathematics.
We can also introduce the concept of order or inequality for integers, and from these inequalities for
rational numbers. For example, if a, b, c, d are positive integers, we define a=b > c=d if and only if
ad > bc, with similar extensions to negative integers.
Once we have the set of rational numbers and the rules of inequality concerning them, we can order
them geometrically as points on the real axis, as already indicated. We can then show that there are
ffiffiffi
p
points on the line which do not represent rational numbers (such as 2, , etc.). These irrational
numbers can be defined in various ways, one of which uses the idea of Dedekind cuts (see Problem 1.34).
From this we can show that the usual rules of algebra apply to irrational numbers and that no further
real numbers are possible.