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LECTURE 3: INDUCTION AND AN INTRODUCTION TO THE
REAL NUMBERS
1. N and induction
Definition (3.1). We let N denote the set of natural numbers; that is,
N : t1, 2, 3, . . . u.
The principle of mathematical induction (3.2). Suppose P pxq is a property
that may or may not hold for some x P N. If
i) P p1q is true and
ii) whenever P pk q is true it follows P pk
1q is true,
then P pxq is true for all x P N.
Lemma (3.3). For any k
P N,
1
2
k
kpk 2 1q .
The principle of complete induction (3.4). Suppose P pxq is a property that
may or may not hold for some x P N. If
i) P p1q is true and
ii) whenever P p1q, . . . , P pk q is true it follows P pk
then P pxq is true for all x P N.
1q is true,
Lemma (3.5). Induction (3.2) and strong induction (3.4) are equivalent.
Lemma (3.6). Every n
2.
P N can be written as a sum of distinct integer powers of
Exercise. Read p22-p23 for alternative formulations of induction.
Closely related to mathematical induction are recursive definitions.
P N define n! : 1.2. . . . .n.
Example (3.7, Factorials). For n
recursively by:
1!
n!
:
1;
:
n.pn 1q!
This can be defined
for n ¥ 2.
Example (3.8, Σ-notation). Given numbers a1 , . . . , ak we write
¸a
k
j
a1 ak .
j 1
For example, by Lemma 3.4,
¸ j kpk
1q
k
j 1
1
2
.
2
LECTURE 3: INDUCTION AND AN INTRODUCTION TO THE REAL NUMBERS
This notation can be defined recursively by
¸a
1
j
¸a
:
a1 ;
¸ a
j 1
k
j
:
k 1
j 1
Question. What is
?
j
ak
for k
¥ 2.
j 1
2. R: the set of real numbers
2?
Tentative answer: The number a such that a2
a number exists? How do we know if it is unique?
2. But how do we know such
Theorem (3.9). There exists no number a P Q such that a2
?
2.
To work with objects such as 2 in a systematic way we need to consider a
much larger system of numbers - the real numbers R.
To begin with we’ll work with a heuristic idea of R as the set of all numbers which
can be represented by an infinite decimal expansion. It therefore corresponds to
our intuitive picture of a continuous number line. We will assume in R things like
?
2 are all defined.
Defining R takes some work. It turns out that R is ‘essentially’ the unique set of
numbers which satisfies (P1) - (P12) together with an additional axiom (P13) which
ensures there are ‘no holes’ or ‘gaps’ in our number line. This will be discussed in
later lectures.
Definition (3.10). If x P R but x does not belong to Q, then we say x is irrational.
Hence Theorem 3.9 states that
?
2 is irrational.
Jonathan Hickman, Department of mathematics, University of Chicago, 5734 S. University Avenue, Eckhart hall Room 414, Chicago, Illinois, 60637.
E-mail address: [email protected]