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College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson. Prerequisites. P. The Real Number Line and Order. P.3. The Real Line. Introduction. The real numbers can be represented by points on a line, as shown.
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College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson
Introduction • The real numbers can be represented by points on a line, as shown. • The positive direction (toward the right) is indicated by an arrow.
Origin • We choose an arbitrary reference point O, called the origin, which corresponds to the real number 0.
Origin • Given any convenient unit of measurement, • Each positive number x is represented by the point on the line a distance of x units to the right of the origin. • Each negative number –x is represented by the point x units to the left of the origin.
Coordinate • The number associated with the point P is called: • The coordinate of P
Real Line • Then, the line is called any of the following: • Coordinate line • Real number line • Real line
Real Line • Often, we identify the point with its coordinate and think of a number as being a point on the real line.
Order of Numbers • The real numbers are ordered. • We say that a is less than b,and write a <b if b –a is a positive number. • Geometrically, this means that a lies to the left of b on the number line. • Equivalently, we can say that b is greater than a, and write b >a.
Symbol a ≤ b • The symbol a≤ b (or b ≥ a): • Means that either a <b or a =b. • Is read “a is less than or equal to b.”
Inequalities • For instance, these are true inequalities:
E.g. 1—Graphing Inequalities • On the real line, graph all the numbers x that satisfy the inequality: • x < 3 • x≥ –2
Example (a) E.g. 1—Graphing Inequalities • We must graph the real numbers that are smaller than 3. • Those that lie to the left of 3 on the real line.
Example (a) E.g. 1—Graphing Inequalities • The graph is shown here. • Note that the number 3 does not satisfy the inequality. • So, it is indicated with an open dot on the real line.
Example (b) E.g. 1—Graphing Inequalities • We must graph the real numbers that are greater than or equal to–2. • Those that lie to the right of –2 on the real line, including the number–2.
Example (b) E.g. 1—Graphing Inequalities • The graph is shown here. • Note that the number –2 does satisfy the inequality. • So, it is indicated with a solid dot on the real line.
Sets & Elements • A set is a collection of objects. • These objects are called the elements of the set.
Sets • If S is a set, the notation a S means that a is an element of S. • b S means that b is not an element of S. • For example, if Z represents the set of integers, then –3 Z but πZ.
Braces • Some sets can be described by listing their elements within braces. • For instance, the set A that consists of all positive integers less than 7 can be written as: A = {1, 2, 3, 4, 5, 6}
Set-Builder Notation • We could also write A in set-builder notation as: • A = {x | x is an integer and 0 < x < 7} • This is read: “A is the set of all x such that x is an integer and 0 < x < 7.”
Union of Sets • If S and T are sets, then their union S T is: • The set that consists of all elements that are in S or T (or both).
Intersection of Sets • The intersection of S and T is the set S T consisting of all elements that are in both S and T. • That is, S T is the common part of S and T.
Empty Set • The empty set, denoted by Ø, is: • The set that contains no element.
E.g. 2—Union & Intersection of Sets • If S = {1, 2, 3, 4, 5} T = {4, 5, 6, 7} V = {6, 7, 8} • find the sets S T, S T, S V
E.g. 2—Union & Intersection of Sets • S T = {1, 2, 3, 4, 5, 6, 7}(All elements in S or T) • S T = {4, 5}(Elements common to both S and T) • S V = Ø(S and V have no elements in common)
Intervals • Certain sets of real numbers occur frequently in calculus and correspond geometrically to line segments. • These are called intervals.
Open Interval • If a <b, the open interval from a to b consists of all numbers between a and b. • It isdenoted (a, b).
Closed Interval • The closed interval from a to b includes the endpoints. • It is denoted [a, b].
Open & Closed Intervals • Using set-builder notation, we can write: (a, b) = {x | a < x < b} [a, b] = {x | a≤ x ≤ b}
Open Intervals • Note that parentheses ( ) in the interval notation and open circles on the graph in this figure indicate that: • Endpoints are excluded from the interval.
Closed Intervals • Note that square brackets and solid circles in this figure indicate that: • Endpoints are included.
Intervals • Intervals may also include one endpoint but not the other. • They may also extend infinitely far in one direction or both.
Types of Intervals • The table lists the possible types of intervals.
E.g. 3—Graphing Intervals • Express each interval in terms of inequalities, and then graph the interval. • [–1, 2) • [1.5, 4] • (–3, ∞)
Example (a) E.g. 3—Graphing Intervals • [–1, 2) = {x | –1 ≤ x < 2}
Example (b) E.g. 3—Graphing Intervals • [1.5, 4] = {x | 1.5 ≤ x ≤ 4}
Example (c) E.g. 3—Graphing Intervals • (–3, ∞) = {x | –3 < x}
E.g. 4—Finding Unions & Intersections of Intervals • Graph each set. • (a) (1, 3) [2, 7] • (b) (1, 3) [2, 7]
Example (a) E.g. 4—Intersection of Intervals • The intersection of two intervals consists of the numbers that are in both intervals. • Therefore, (1, 3) [2, 7] = {x | 1 < x < 3 and 2 ≤ x ≤ 7} = {x | 2 ≤ x < 3} = [2, 3)
Example (a) E.g. 4—Intersection of Intervals • This set is illustrated here.
Example (b) E.g. 4—Union of Intervals • The union of two intervals consists of the numbers that are in either one of the intervals. • Therefore, (1, 3) [2, 7] = {x | 1 < x < 3 or 2 ≤ x ≤ 7} = {x | 1 < x ≤ 7} =(1, 7]
Example (b) E.g. 4—Union of Intervals • This set is illustrated here.
Absolute Value • The absolute value of a number a, denoted by |a|, is: • The distance from a to 0 on the real number line.
Distance • Distance is always positive or zero. • So, we have: • |a| ≥ 0 for every number a • Remembering that –a is positive when a is negative, we have the following definition.
Absolute Value—Definition • If a is a real number, the absolute value of a is:
