Lecture 1 : Real Numbers, Absolute Value. Real Numbers We can represent the real numbers by the set of points on a line. The origin corresponds to the number 0. Numbers to the right of 0 are positive or > 0 and numbers to the left of 0 are negative or < 0. The set of Real numbers is denoted by R and contains all of the following number types: • The natural numbers (or counting numbers) denoted by N = f1; 2; 3; 4;::: g. • The integers (or whole numbers) denoted by Z = {· · · − 3; −2; −1; 0; 1; 2; 3;::: g. a • The rational numbers (or fractions) denoted by Q = the set of all numbers of the form b where a and b are integers. • The irrational numbers. These are numbers which do notp fit any of the above descriptions, in particular, they are not rational. Numbers such as π and 2 fit this description. Letting A ⊂ B denote that the set A is a subset of B, we see that N ⊂ Z ⊂ Q ⊂ R. 1 Intervals An important property of the real numbers is that they are totally ordered, so we can compare any two real numbers, a and b, and make a statement of the form a ≤ b or b ≤ a, with strict inequality if a 6= b. Given any two points on the real line, a and b, we call the7 set of points between a and b an interval. When discussing these intervals, it is important to indicate7 6 whether we are including one or both endpoints. 5 We also have the concept of intervals that extend to the6 ends of the number line +1 and −∞. We list the notations for each type of interval and their graphs below.4 (The statement fxja ≤ x ≤ bg should 5 be read as \the set of all x such that a is less than or equal3 to x which is less than or equal to b" (the symbol j denotes "such that")). We can represent intervals4 7 in three ways: 2 6 Notation Set Description Graphical3 1 Representation 7 9.5 5 – 2 2 2 4 6 8 (a; b) fxja < x < bg 9 a b 8.5 64– 1 1 8 – 2 7.5 53 9.5 7 9 – 2 – 3 2 4 6 8 6.5 2 48.5 a b 6 [a; b] fxja ≤ x ≤ bg 8 – 1 – 4 5.5 1 37.5 5 7 – 5 – 2 4.5 6.5 – 2 2 2 4 6 8 4 6 (a; b] fxja < x ≤ bg – 6 a b – 33.5 – 5.51 1 5 3 6.5 – 7 4.5 2.54.5– 2 – 4 4 2 – 2 4 2 4 6 8 3.5 a b 6 [a; b) fxja ≤ x < bg 1.5 – 3 – 5 –3.513 1 2.5 0.5 – 43 – 22 5.5 – 6 4.5 2.5 [a; 1) fxja ≤ x < 1g – 1 1.5 1 a 2 3 4 5 6 7 8 – 0.5 – 5 4 1 2 – 1– 3 – 7 5 0.5 3.5 – 1.51.5– 6 – 1 3 1 2 3 4 5 6 7 8 – 2– 4 a (a; 1) fxja < x < 1g – 0.51 – 2.5 – 7 2.5 4.53 – 1 0.5 – 3– 5 – 1.5 2 – 2 4 – 1 1.5 1 2 3 4 5 6 –– 62.5 b 2.5 (−∞; b] fxj − 1 < x ≤ bg – 0.5 – 3 1 – 1 3.5 – 7 0.5 – 1.5 2 – 1 1 2 3 4 5 6 9 (−∞; b) fxj − 1 < x < bg – 2 – 0.5 b 8.5 3 – 2.5 – 1 1.58 – 3 – 1.5 7.5 2.5(−∞; 1) fxj − 1 < x < 1g entire number line. – 2 7 – 3.5 – 2.5 Example Sketch6.5 1 the following intervals – 4 2 6 – 3 – 4.5 5.5 (1; 3); (1; 3]; [1; 1);– 3.5 (−∞; 3): 1.5 0.55 – 4 4.5 – 4.5 4 1 4.5 3.5 (1, 3) – 1 1 4 2 3 3 0.5 3.5 – 2.50.5 3 2 2.5 1.5 – 1 1 2 3 4 5 6 7 8 (1, 3] 1– 1 2 0.5– 0.5 1.5 1 [1; 1) – 1 1 2 3 4 5 6 7 –– 0.51.5 – 1 0.5 – 1 (−∞; 3) – 1.54 – 3 – 2 – 1 1 2 3 – 2 – 0.5 ––2 1.5 – 2.5 – 1 – 2.5– 3 – 2 – 1.5 2 – 2 – 2.5 – 2.5 – 3 – 3 – 3 – 3.5 – 4 – 3.5 – 4.5 – 4 – 4.5 – 5 – 5.5 Example Graph the intervals [1; 7), (0; 1), (−∞; −1], [−1; 3]. 3 Absolute Value The absolute value of a real number a is denoted by jaj and it is the distance from a to the origin 0 on the number line. The absolute value is always positive. We can give a formula for the absolute value of the number, which depends on whether a is positive or negative. Because of this we have to make two statements to describe the formula. Definition If a is a real number, the absolute value of a is a if a ≥ 0 jaj = −a if a < 0 Example Evaluate j2j, j − 10j, j5 − 9j, j9 − 5j. Algebraic properties of the Absolute Value 1. jaj ≥ 0 for all real numbers a. 2. jaj = j − aj for all real numbers a. 3. jabj = jajjbj, the absolute value of the product of two numbers is the product of the absolute values. 4. a = jaj , the absolute value of the quotient of two numbers is the quotient of the absolute values. b jbj Example Evaluate 7−10 and jj − 2j − j − 4jj. 21 Distance Between Two Points on The Real Line. If a and b are real numbers, then the distance between the points a and b on the real line is d(a; b) = jb − aj: Example Find the distance between the numbers −2 and 4. 4.