Real Numbers Lecture PDF

Chapter P Prerequisites Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.  P.2 Real Numbers Stewart/Redlin/Watson,...

Chapter P Prerequisites Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.  P.2 Real Numbers Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Objectives Real Numbers Properties of Real Numbers Addition and Subtraction Multiplication and Division The Real Line Sets and Intervals Absolute Value and Distance Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.  Real Numbers Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Real Numbers (1 of 6) Let’s review the types of numbers that make up the real number system. We start with the natural numbers: 1, 2, 3, 4,... The integers consist of the natural numbers together with their negatives and 0:... , −3, −2, −1, 0, 1, 2, 3, 4,... Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Real Numbers (2 of 6) We construct the rational numbers by taking ratios of integers. Thus any rational number r can be expressed as m r = n where m and n are integers and n ≠ 0. Examples are 1 3 46 17 − 46 = 0.17 = 2 7 1 100 3 0 (We know that division by 0 is always ruled out, so expressions like and 0 0 are undefined.) There are also real numbers, such as 2, that cannot be expressed as a ratio of integers and are therefore called irrational numbers. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Real Numbers (3 of 6) It can be shown, with varying degrees of difficulty, that these numbers are also irrational: 3 3 5 3 2  2 The set of all real numbers is usually denoted by the symbol. When we use the word number without qualification, we will mean “real number.” Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Real Numbers (4 of 6) Figure 2 is a diagram of the types of real numbers. The real number system Figure 2 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Real Numbers (5 of 6) Every real number has a decimal representation. If the number is rational, then its corresponding decimal is repeating. For example, 1 2 = 0.5000... = 0.50 = 0.66666... = 0.6 2 3 157 9 = 0.3171717... = 0.317 = 1.285714285714... = 1.285714 495 7 (The bar indicates that the sequence of digits repeats forever.) If the number is irrational, the decimal representation is nonrepeating: 2 = 1.414213562373095...  = 3.141592653589793... Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Real Numbers (6 of 6) If we stop the decimal expansion of any number at a certain place, we get an approximation to the number. For instance, we can write   3.14159265 where the symbol ≈ is read “is approximately equal to.” The more decimal places we retain, the better our approximation. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Properties of Real Numbers Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Properties of Real Numbers (1 of 3) We all know that 2 + 3 = 3 + 2, and 5 + 7 = 7 + 5, and 513 + 87 = 87 + 513, and so on. In algebra we express all these (infinitely many) facts by writing a+b=b+a where a and b stand for any two numbers. In other words, “a + b = b + a” is a concise way of saying that “when we add two numbers, the order of addition doesn’t matter.” This fact is called the Commutative Property of addition. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Properties of Real Numbers (2 of 3) From our experience with numbers we know that the properties in the following box are also valid. PROPERTIES OF REAL NUMBERS Property Example Description Commutative Properties a+b=b+a 7+3=3+7 When we add two numbers, order doesn't matter. ab = ba 3·5=5·3 When we multiply two numbers, order doesn't matter. Associative Properties (a + b) + c = a + (b + c) (2 + 4) + 7 = 2 + (4 + 7) When we add three numbers, it doesn't matter which two we add first. (ab)c = a(bc) (3 · 7) · 5 = 3 · (7 · 5) When we multiply three numbers, it doesn't mailer which two we multiply first. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Properties of Real Numbers (3 of 3) PROPERTIES OF REAL NUMBERS Property Example Description Distributive Property a(b + c) = ab + ac 2 · (3 + 5) = 2 · 3 + 2 · 5 When we multiply a number by a sum of two numbers, we get the same result as we get if we multiply the number by (b + c)a = ab + ac (3 + 5) · 2 = 2 · 3 + 2 · 5 each of the terms and then add the results. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1 – Using the Distributive Property (1 of 2) (a) Distributive Property = 2x + 6 Simplify (b) Distributive Property = (ax + bx) + (ay + by) Distributive Property = ax + bx + ay + by Associative Property of Addition Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1 – Using the Distributive Property (2 of 2) In the last step we removed the parentheses because, according to the Associative Property, the order of addition doesn’t matter. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.  Addition and Subtraction Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Addition and Subtraction (1 of 3) The number 0 is special for addition; it is called the additive identity because a + 0 = a for any real number a. Every real number a has a negative, −a, that satisfies a + (−a) = 0. Subtraction is the operation that undoes addition; to subtract a number from another, we simply add the negative of that number. By definition a − b = a + (−b) Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Addition and Subtraction (2 of 3) To combine real numbers involving negatives, we use the following properties. PROPERTIES OF NEGATIVES Property Example 1. (−1)a = −a (−1)5 = −5 2. −(−a) = a −(−5) = 5 3. (−a)b = a(−b) = −(ab) (−5)7 = 5(−7) = −(5 · 7) 4. (−a)(−b) = ab (−4)(−3) = 4 · 3 5. −(a + b) = −a − b −(3 + 5) = −3 − 5 6. −(a − b) = b − a −(5 − 8) = 8 − 5 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Addition and Subtraction (3 of 3) Property 6 states the intuitive fact that a − b and b − a are negatives of each other. Property 5 is often used with more than two terms: −(a + b + c) = −a − b − c Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 2 – Using Properties of Negatives Let x, y, and z be real numbers. (a) −(x + 2) = −x − 2 Property 5: −(a + b) = −a − b (b) −(x + y − z) = −x − y − (−z) Property 5: −(a + b) = −a − b = −x − y + z Property 2: −(−a) = a Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.  Multiplication and Division Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Multiplication and Division (1 of 4) The number 1 is special for multiplication; it is called the multiplicative identity because a · 1 = a for any real number a. 1  1 Every nonzero real number a has an inverse, , that satisfies a    = 1. a a Division is the operation that undoes multiplication; to divide by a number, we multiply by the inverse of that number. If b ≠ 0, then, by definition, 1 ab = a b Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Multiplication and Division (2 of 4)  1 a a We write a    as simply. We refer to as the quotient of a and b b b b or as the fraction a over b; a is the numerator and b is the denominator (or divisor). Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Multiplication and Division (3 of 4) To combine real numbers using the operation of division, we use the following properties. PROPERTIES OF FRACTIONS Property Example Description a c ac 2 /53) 2 (2 = * (5 5 / 7) = 10 = (2 * 5) / When multiplying fractions, multiply 1. (a/b) * =(c/d) = (ac)/(bd) b d bd 3 *7 7) 3= 7 (3 10 /21 21 numerators and denominators. 2. (aa / b)c divided a d by (c / d) (2 2 / 53) divided 2 7 14 by (5 / 7) = When dividing fractions, invert the divisor  =   =  = = (ab/ b)d* (db/ c)c (2 3 / 73) * 3(7 5/ 5)15 = 14 / 15 and multiply. a +/ c) 3. (aa / c)b + (b b = (a + b) / (2 7 + 2(7+ /75) =9 (2 + 7) / 2 / 5) When adding fractions with the same + = = c c+c = c 5=9 5/5 5 5 denominator, add the numerators. + bc  7/+7) 2 / 35) +2(3 3 =5 (2 29 When adding fractions with different 4. (aa / b)c + (c ad/ d) = ((a d) + (2+ = = * 7) + + = denominators, find a common denominator. (b d)) b /d (b d) bd 5 * 75) / 35 35 (3 = 29 / 3535 Then add the numerators. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Multiplication and Division (4 of 4) PROPERTIES OF FRACTIONS Property Example Description ac a 25 2 Cancel numbers that are common = = (a/b) 5. (ac/bc) = * 5) = (2/3) (2 * 5/3 bc b 35 3 factors in numerator and denominator. a c 2 6 = = ,(c/d), 6. IfIf (a/b) ad = ad then then bc = = =, (6/9), (2/3) so 2 so 9 =23* 96= 3 * 6 Cross-multiply. bc b d 3 9 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 3 – Using the LCD to Add Fractions 5 7 Evaluate: + 36 120 Solution: Factoring each denominator into prime factors gives 36 = 2 2 ×3 2 and 120 = 2 3 ×3 ×5 We find the least common denominator (LCD) by forming the product of all the prime factors that occur in these factorizations, using the highest power of each prime factor. Thus the LCD is 23  3 2  5 = 360. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 3 – Solution So 5 7 5  10 73 + = + Use common denominator 36 120 36  10 1 20  3 50 21 71 Property 3: Adding fractions = + = 360 360 360 with the same denominator Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.  The Real Line Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Real Line (1 of 4) The real numbers can be represented by points on a line, as shown in Figure 4. The real line Figure 4 The positive direction (toward the right) is indicated by an arrow. We choose an arbitrary reference point O, called the origin, which corresponds to the real number 0. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Real Line (2 of 4) 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, and each negative number −x is represented by the point x units to the left of the origin. The number associated with the point P is called the coordinate of P, and the line is then called a coordinate line, or a real number line, or simply a real line. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Real Line (3 of 4) 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. The symbol a ≤ b (or b ≥ a) means that either a < b or a = b and is read “a is less than or equal to b.” Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The Real Line (4 of 4) For instance, the following are true inequalities (see Figure 5): Figure 5 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.  Sets and Intervals Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sets and Intervals (1 of 6) A set is a collection of objects, and these objects are called the elements of the set. If S is a set, the notation a  S means that a is an element of S, and b  S means that b is not an element of S. For example, if Z represents the set of integers, then −3  Z but   Z. 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} Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sets and Intervals (2 of 6) We could also write A in set-builder notation as A =  x | x is an integer and 0  x  7 which is read “A is the set of all x such that x is an integer and 0 < x < 7.” 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 in both). The intersection of S and T is the set S  T consisting of all elements that are in both S and T. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sets and Intervals (3 of 6) In other words, S  T is the common part of S and T. The empty set, denoted by Ø, is the set that contains no element. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 4 – Union and Intersection of Sets If S = {1, 2, 3, 4, 5}, T = {4, 5, 6, 7}, and V = {6, 7, 8}, find the sets S  T , S  T , and S  V. Solution: 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 element in common Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sets and Intervals (4 of 6) Certain sets of real numbers, called intervals, occur frequently in calculus and correspond geometrically to line segments. If a < b, then the open interval from a to b consists of all numbers between a and b and is denoted (a, b). The closed interval from a to b includes the endpoints and is denoted [a, b]. Using set-builder notation, we can write. ( a, b ) =  x | a  x  b a, b  =  x | a  x  b Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sets and Intervals (5 of 6) Note that parentheses ( ) in the interval notation and open circles on the graph in Figure 6 indicate that endpoints are excluded from the interval, The open interval (a, b) Figure 6 whereas square brackets [ ] and solid circles in Figure 7 indicate that the endpoints are included. The closed interval [a, b] Figure 7 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Sets and Intervals (6 of 6) Intervals may also include one endpoint but not the other, or they may extend infinitely far in one direction or both. The following table lists the possible types of intervals. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.  Example 6 – Finding Unions and Intersections of Intervals Graph each set. (a) (1, 3)  [2, 7] (b) (1, 3)  [2, 7] Solution: (a) 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) Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 6 – Solution (1 of 2) This set is illustrated in Figure 8. (1, 3)  [2, 7] = [2, 3) Figure 8 (b) The union of two intervals consists of the numbers that are in either one interval or the other (or both). Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 6 – Solution (2 of 2) Therefore (1, 3 )   2, 7  = { x 1  x  3 and 2  x  7} = { x 1  x  7} = (1, 7] This set is illustrated in Figure 9. (1, 3)  [2, 7] = (1, 7] Figure 9 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Absolute Value and Distance Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Absolute Value and Distance (1 of 5) The absolute value of a number a, denoted by a , is the distance from a to 0 on the real number line (see Figure 10). Figure 10 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. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Absolute Value and Distance (2 of 5) DEFINITION OF ABSOLUTE VALUE If a is a real number, then the absolute value of a is  a if a  0 a =  −a if a  0 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 7 – Evaluating Absolute Values of Numbers (a) 3 = 3 (b) −3 = − ( −3 ) =3 (c) 0 = 0 (d) 3 −  = − (3 −  ) =  − 3 (since 3    3 −   0) Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Absolute Value and Distance (3 of 5) When working with absolute values, we use the following properties. PROPERTIES OF ABSOLUTE VALUE Property Example Description a  0>= 0 −3 = 3  0 3) = 3 >= 0 The absolute value of a number is always 1. abs(a) abs(negative positive or zero. 5 = −=5 abs(negative 5) A number and its negative have the same a = −=aabs(negative a) 2. abs(a) abs(5) absolute value. ab = a= b −2  5 = −2 5 abs(negative 2 * 5) = abs(negative The absolute value of a product is the 3. abs(ab) abs(a)abs(b) 2) abs(5) product of the absolute values. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Absolute Value and Distance (4 of 5) PROPERTIES OF ABSOLUTE VALUE Property Example Description a a abs 12 12 The absolute value of a quotient is the = 4. abs (a/b) = abs (a) / abs = (12/ negative 3) = abs (12)/ abs b b −3 (negative 3) −3 quotient of the absolute values. 5. abs a +(ab+ b)  less a +than b or equal to −3 + 5  −3 + 5 abs (negative 3 + 5) less than or equal to abs Triangle Inequality abs (a) + abs (b) (negative 3) + abs (5) Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Absolute Value and Distance (5 of 5) DISTANCE BETWEEN 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 The distance from a to b is the same as the distance from b to a. Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 8 – Distance Between Points on the Real Line The distance between the numbers −8 and 2 is d (a, b ) = 2 − ( −8 ) = −10 = 10 We can check this calculation geometrically, as shown in Figure 13. Figure 13 Stewart/Redlin/Watson, Algebra and Trigonometry, 4th Edition. © 2016 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.