MATH 125 Study Guide - Midterm Guide: Identity Element, Triangle Inequality
Course CodeMATH 125
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§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.
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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
−aif a < 0(i.e., |a|=√a2)
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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