Page 11 - Schaum's Outline of Theory and Problems of Advanced Calculus

2 NUMBERS [CHAP. 1

DECIMAL REPRESENTATION OF REAL NUMBERS
Any real number can be expressed in decimal form, e.g., 17=10 ¼ 1:7, 9=100 ¼ 0:09,
1=6 ¼ 0:16666 ... . In the case of a rational number the decimal exapnsion either terminates, or if it
does not terminate, one or a group of digits in the expansion will ultimately repeat, as for example, in
ffiffiffi
1 ¼ 0:142857 142857 142 ... . In the case of an irrational number such as 2 ¼ 1:41423 ... or
p
7
¼ 3:14159 ... no such repetition can occur. We can always consider a decimal expansion as unending,
e.g., 1.375 is the same as 1.37500000 . . . or 1.3749999 . . . . To indicate recurring decimals we some-
_
_ _ _ _ _ _
1
5
4
1
8
2
times place dots over the repeating cycle of digits, e.g., ¼ 0:1428577, 19 ¼ 3:166.
7 6
The decimal system uses the ten digits 0; 1; 2; ... ; 9. (These symbols were the gift of the Hindus.
They were in use in India by 600 A.D. and then in ensuing centuries were transmitted to the western world
by Arab traders.) It is possible to design number systems with fewer or more digits, e.g. the binary
system uses only two digits 0 and 1 (see Problems 32 and 33).
GEOMETRIC REPRESENTATION OF REAL NUMBERS
The geometric representation of real numbers as points on a line called the real axis,asin the figure
below, is also well known to the student. For each real number there corresponds one and only one
point on the line and conversely, i.e., there is a one-to-one (see Fig. 1-1) correspondence between the set of
real numbers and the set of points on the line. Because of this we often use point and number
interchangeably.
_ p _ 4 1 √2 e p
3 2
_ 5 _ 4 _ 3 _ 2 _ 1 0 1 2 3 4 5
Fig. 1-1

(The interchangeability of point and number is by no means self-evident; in fact, axioms supporting
the relation of geometry and numbers are necessary. The Cantor–Dedekind Theorem is fundamental.)
The set of real numbers to the right of 0 is called the set of positive numbers; the set to the left of 0 is
the set of negative numbers, while 0 itself is neither positive nor negative.
(Both the horizontal position of the line and the placement of positive and negative numbers to the
right and left, respectively, are conventions.)
Between any two rational numbers (or irrational numbers) on the line there are infinitely many
rational (and irrational) numbers. This leads us to call the set of rational (or irrational) numbers an
everywhere dense set.

OPERATIONS WITH REAL NUMBERS
If a, b, c belong to the set R of real numbers, then:

1. a þ b and ab belong to R Closure law
2. a þ b ¼ b þ a Commutative law of addition
3. a þðb þ cÞ¼ða þ bÞþ c Associative law of addition
4. ab ¼ ba Commutative law of multiplication
5. aðbcÞ¼ ðabÞc Associative law of multiplication
6. aðb þ cÞ¼ ab þ ac Distributive law
7. a þ 0 ¼ 0 þ a ¼ a,1 a ¼ a 1 ¼ a
0is called the identity with respect to addition,1 is called the identity with respect to multi-
plication.