This

**preview**shows pages 1-2. to view the full**6 pages of the document.**§0 – The Real Number System

The symbol Rdenotes the set of all real numbers.

Ris equipped with the familiar operations of addition (de-

noted +) and multiplication (denoted ·, though we usually

just write ab instead of a·b).

Main algebraic properties of the real number system:

(1) + is commutative:a+b=b+a.

(2) + is associative:a+ (b+c)=(a+b) + c.

(3) + has a neutral element (i.e., 0):a+ 0 = a.

(4) + has inverses: For every a∈R, there is a (unique)

b∈Rsuch that a+b= 0.

(5) ·is commutative:ab =ba.

(6) ·is associative:a(bc)=(ab)c.

(7) ·has a neutral element (i.e, 1):a·1 = a.

(8) ·has inverses for non-zero elements: For any

non-zero a∈R, there is a (unique) b∈Rsuch that ab = 1.

(9) ·distributes over +: a(b+c) = ab +ac.

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

A more subtle feature of the real number system is the no-

tion of positivity, and the existence of the absolute value

operation: Recall that, for any a∈R, we set

|a|=(aif a≥0

−aif a < 0(i.e., |a|=√a2)

Example:

If we think of Ras an inﬁnite straight line, then we can regard

|a|as giving the “distance from ato 0”. We have:

Triangle inequality:|a+b| ≤ |a|+|b|.

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