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MATH 125 Midterm: Section 1.1 (Part 1)

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University of Saskatchewan
MATH 125

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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