Page 17 - Schaum's Outline of Theory and Problems of Advanced Calculus

8 NUMBERS [CHAP. 1

from which it follows that there are in general n different values of z 1=n . Later (Chap. 11) we will show
i
that e ¼ cos þ i sin where e ¼ 2:71828 ... . This is called Euler’s formula.

MATHEMATICAL INDUCTION
The principle of mathematical induction is an important property of the positive integers. It is
especially useful in proving statements involving all positive integers when it is known for example that
the statements are valid for n ¼ 1; 2; 3 but it is suspected or conjectured that they hold for all positive
integers. The method of proof consists of the following steps:

1. Prove the statement for n ¼ 1 (or some other positive integer).
2. Assume the statement true for n ¼ k; where k is any positive integer.
3. From the assumption in 2 prove that the statement must be true for n ¼ k þ 1. This is part of
the proof establishing the induction and may be difficult or impossible.
4. Since the statement is true for n ¼ 1 [from step 1] it must [from step 3] be true for n ¼ 1 þ 1 ¼ 2
and from this for n ¼ 2 þ 1 ¼ 3, and so on, and so must be true for all positive integers. (This
assumption, which provides the link for the truth of a statement for a finite number of cases to
the truth of that statement for the infinite set, is called ‘‘The Axiom of Mathematical Induc-
tion.’’)

Solved Problems

OPERATIONS WITH NUMBERS
1
3
2
1.1. If x ¼ 4, y ¼ 15, z ¼ 3, p ¼ , q ¼ , and r ¼ , evaluate (a) x þðy þ zÞ, (b) ðx þ yÞþ z,
3 6 4
(c) pðqrÞ,(d) ðpqÞr,(e) xðp þ qÞ
(a) x þðy þ zÞ¼ 4 þ½15 þð 3ފ ¼ 4 þ 12 ¼ 16
(b) ðx þ yÞþ z ¼ð4 þ 15Þþð 3Þ¼ 19 3 ¼ 16
The fact that (a) and (b)are equal illustrates the associative law of addition.
(c) 2 1 3 2 3 2 1 2 1
3 6 4 3 24 3 8 24 ¼ 12
pðqrÞ¼ fð Þð Þg ¼ ð Þð Þ¼ð Þð Þ¼
2 1 3 2 3 1 3 3 1
3 6 4 18 4 9 4 36 12
(d) ðpqÞr ¼fð Þð Þgð Þ¼ð Þð Þ¼ð Þð Þ¼ ¼
The fact that (c) and (d)are equal illustrates the associative law of multiplication.
(e) 2 1 4 1 3 12 ¼ 2
3 6 6 6 6 6
xðp þ qÞ¼ 4ð Þ¼ 4ð Þ¼ 4ð Þ¼
1
2
8
6
2
4
8
Another method: xðp þ qÞ¼ xp þ xq ¼ð4Þð Þþð4Þð Þ¼ ¼ ¼ ¼ 2using the distributive
3 6 3 6 3 3 3
law.
1.2. Explain why we do not consider (a) 0 (b) 1 as numbers.
0 0
(a)If we define a=b as that number (if it exists) such that bx ¼ a,then0=0isthat number x such that
0x ¼ 0. However, this is true for all numbers. Since there is no unique number which 0/0 can
represent, we consider it undefined.
(b)As in (a), if we define 1/0 as that number x (if it exists) such that 0x ¼ 1, we conclude that there is no
such number.
Because of these facts we must look upon division by zero as meaningless.