# MATH 125 Midterm: Section 1.1 (Part 1)

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University of Saskatchewan

Mathematics

MATH 125

Scott

Fall

Description

x0 { The Real Number System
The symbol R denotes the set of all real numbers.
R is 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 2 R, there is a (unique)
b 2 R such 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 2 R, there is a (unique) b 2 R such that ab = 1.
(9) ▯ distributes over +: a(b + c) = ab + ac. 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 2 R, we set
( p
jaj = a if a ▯ 0 (i.e., jaj =a )
▯a if a < 0
Example:
If we think of R as an in▯nite straight line, then we can regard
jaj as giving the \distance from a to 0". We have:
Triangle inequality : ja + bj ▯ jaj + jbj. Chapter I { Vectors
x1 { The Geometry and Algebra of Vectors
We distinguish between two types of physical quantities:
▯ Scalars: Possess quality of magnitude only.
▯ Vectors: Possess qualities of magnitude and direction.
Example: In physics, \speed" is understood as a scalar quan-
tity, while \velocity" is a vector quantity.
Vectors in the physical sense may be perceived visually as
\
oating arrows" in Cartesian space.
Vectors in the Plane
Consider the familiar Cartesian (xy-) plane with origin O.
A vector in the plane is a directed line segment (or arrow)
corresponding to a displacement from a point A to

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