Section 13: NUMBER SYSTEMS (up to R)
all have a well ordering (Trichotomy Law) and a metric (“absolute value”)

1. Natural Numbers N = {1, 2, 3, 4, …}. Oldest and most respectable. Loved by all.
   [Operations: addition and multiplication.]

2. Integer numbers Z={…, -2, -1, 0, 1, 2, …}.
   Historical resistance to the number 0. Negative numbers even worse.
   [Operations: Addition, subtraction, multiplication. Properties given at beginning of term]

3. Rational Numbers Q
   
   

We define the rationals to be the set Q={ | }

Are representations of rational numbers by fractions unique? Yes/No

Example:

Def: a/b=c/d ó ??

Def: Let <a/b> be the rational number represented by the fraction a/b.

Let’s make it a unique representation.

Def: The fraction a/b is “in lowest terms” iff ??

Operations: sum, multiplication (can derive subtraction and division)

Sum: a/b+c/d=??=

Product: (a/b)(c/d)=??=

Question: Can you see any possible problem with these definitions of operations?

4. Real Numbers

R= completion of Q under Cauchy sequences (later course). Intuitively: filling in all the gaps between the rational numbers, to get a continuum.

Real numbers are represented by their (infinite) decimal expansions: if a is in R then:

where n is an integer, and .

That is,

Notation: A bar or dots above some of the decimal digits:

signifies that that portion of the decimal gets repeated infinitely many times from there on. (recurring portion)

WARNING: Is the correspondence “real nbs <-> infinite decimals” one-to-one? (i.e. are decimal representation unique?) Yes/No

Example:

Proof:

How the other number systems get represented inside R:

1) The integers correspond to decimal expansions such that ??

2) The rationals correspond to what sort of decimal expansions ?

(no proof. See exercise 5 in Problem set IV, page 225)

Converse is also true. Example: Write as a fraction.

3) Hence the irrationals correspond to what sort of decimals?