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 = {1, 2, 3, 4,... }.

• The integers (or whole numbers) denoted by Z = {· · · − 3, −2, −1, 0, 1, 2, 3,... }.

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 not√ 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 +∞ and −∞. We list the notations for each type of interval and their graphs below.4 (The statement {x|a ≤ x ≤ b} 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 | 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) {x|a < x < b} 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] {x|a ≤ x ≤ b} 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] {x|a < x ≤ b} – 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) {x|a ≤ x < b} 1.5 – 3 – 5 –3.513 1

2.5 0.5 – 43 – 22 5.5 – 6 4.5 2.5 [a, ∞) {x|a ≤ x < ∞} – 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, ∞) {x|a < x < ∞} – 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] {x| − ∞ < x ≤ b} – 0.5 – 3 1

– 1 3.5 – 7 0.5 – 1.5 2 – 1 1 2 3 4 5 6 9 (−∞, b) {x| − ∞ < x < b} – 2 – 0.5 b

8.5 3 – 2.5 – 1 1.58 – 3 – 1.5 7.5 2.5(−∞, ∞) {x| − ∞ < x < ∞} 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, ∞),– 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 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, 3].

3 Absolute Value The absolute value of a real number a is denoted by |a| 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 |a| = −a if a < 0

Example Evaluate |2|, | − 10|, |5 − 9|, |9 − 5|.

Algebraic properties of the Absolute Value

1. |a| ≥ 0 for all real numbers a.

2. |a| = | − a| for all real numbers a.

3. |ab| = |a||b|, the absolute value of the product of two numbers is the product of the absolute values.

4. a = |a| , the absolute value of the quotient of two numbers is the quotient of the absolute values. b |b|

Example Evaluate 7−10 and || − 2| − | − 4||. 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) = |b − a|.

Example Find the distance between the numbers −2 and 4.

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