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pg017 [R] G1 5-36058 / HCG / Cannon & Elich KK 10-13-1995 QC1

1.3 Real Properties; Complex 17

1.3 PROPERTIES; COMPLEX NUMBERS . . . one of the central themes of science ͓is͔ the mysterious power of OP to prepare the ground for physical discoveries which could not Ð2 Ð1 0 1 2 3 have been foreseen by the who gave the concepts birth. Freeman Dyson (a) on a number .

Real 1 2 Ð Ð2 2 3 3 ␲ One of the great ideas in the is that the of real numbers can be associated with the set of points on a line. We assume a one-to-one corre- Ð2 Ð1 0 1 2 3 spondence that associates each real number with exactly one on a line, and every point on the line corresponds to exactly one real number. (b) Real numbers on a number line. Wefrequently identify a number with its point, and vice versa, speaking of “the FIGURE 5 point 2” rather than “the point that corresponds to 2.” Figure 5 shows a few numbers and the corresponding points on a number line.

Order Relations and Intervals The number line also represents the ordering of the real numbers. We assume that the ideas of less than and greater than, and the following notation are familiar: Order relations for real numbers Notation Terminology Meaning b Ͻ cbis less than c. c ϭ b ϩ d, for some positive The mathematics course at San Diego High School number d was standard for that : c Ͼ bcis greater than bbϽc geometry in the b Յ cbis less than or b Ͻ c or b ϭ c tenth grade, advanced equal to c. in the eleventh, c Ն bcis greater than b Յ c and and solid geometry in the twelfth. or equal to b. . . . After plane geometry, I was the only girl still We also need notation for sets of all numbers between two given numbers, or taking mathematics. all numbers less than or greater than a given number. Such sets are called Julia Robinson intervals. Definition: intervals Suppose b and c are real numbers and b Ͻ c: Name Notation Number-line diagram Open ͑b, c͒ ϭ ͕x ͉ b Ͻ x Ͻ c͖ Closed interval ͓b, c͔ ϭ ͕x ͉ b Յ x Յ c͖ Half-open interval ͓b, c͒ ϭ ͕x ͉ b Յ x Ͻ c͖ Half-open interval ͑b, c͔ ϭ ͕x ͉ b Ͻ x Յ c͖ Infinite intervals ͑b, ϱ͒ ϭ ͕x ͉ x Ͼ b͖ ͓b, ϱ͒ ϭ ͕x ͉ x Ն b͖ ͑Ϫϱ, b͒ ϭ ͕x ͉ x Ͻ b͖ ͑Ϫϱ, b͔ ϭ ͕x ͉ x Յ b͖

In these definitions, the symbol ϱ (infinity) does not represent a number, and we never use a closed to indicate that ϱ is included in an infinite interval. pg018 [V] G2 5-36058 / HCG / Cannon & Elich cr 11-10-95 QC2

18 Chapter 1 Basic Concepts: Review and Preview

Absolute Value and We have no difficulty in finding the of specific numbers, as in ͉2 ͉ ϭ 2, ͉0 ͉ ϭ 0, ͉Ϫ1.375 ͉ ϭ 1.375. There are always two numbers having the same nonzero absolute value, as Ϫ␲ ␲ ␲ ͉2 ͉ ϭ ͉Ϫ2 ͉ ϭ 2, and ϭ ϭ . ͯ 3 ͯ ͯ 3 ͯ 3 Following the pattern of the above examples, the absolute value of a positive number is itself; the absolute value of a is its opposite. In working with an expression like ͉2x Ϫ 3 ͉, the 2x Ϫ 3isneither positive nor negative until we give a value to x. All we can say is that ͉ 2x Ϫ 3 ͉ is either 2x Ϫ 3 or its opposite, Ϫ͑2x Ϫ 3͒. Thus ͉ 2x Ϫ 3 ͉ ϭ 2x Ϫ 3forallthex-valuesthatmake2xϪ3 positive 9 ͑2, 5 ,5␲,...͒, ͉2xϪ3͉ ϭϪ2xϩ3forallthex-valuesthatmake2xϪ3negative ␲ ͑1, 3 ,0,...͒. Looking at a number line, the numbers satisfying ͉ x ͉ ϭ 3are3andϪ3, the two numbers whose distance from 0 is 3. More generally, the numbers located two units from 7 are the numbers 5 and 9, and ͉7 Ϫ 5 ͉ ϭ 2, and ͉7 Ϫ 9 ͉ ϭ 2. These examples lead to two ways of looking at absolute values, both of which are useful, so we include them in our definition. Definition: absolute value For any expression u,theabsolute value of u, denoted by ͉u ͉, is given by u if u Ն 0 ͉ u ͉ ϭ ͭϪu if u Ͻ 0 If a and b are any real numbers, then the distance between a and b is given by ͉ a Ϫ b ͉ ϭ ͉ b Ϫ a ͉ . It follows that ͉a ͉ ϭ ͉a Ϫ 0 ͉ is the distance between a and 0.

Calculators and Absolute Value In finding the absolute value of any particular number, we shouldn’t have to rely on a calculator, but a calculator can be helpful nonetheless. We know, for example, that ͉␲ Ϫ ͙10 ͉ is either ␲ Ϫ ͙10 or ͙10 Ϫ ␲, depending on which is positive. We do not need to use the calculator 5ABS7; in fact, if we were to try (see the Technology Tip in Section 1.5 for suggestions about how to enter 5ABS7), we would find only that

ABS͑␲ Ϫ ͙10͒ Ϸ 0.020685. This is true, but not helpful in deciding whether ͉␲ Ϫ ͙10 ͉ is equal to ␲ Ϫ ͙10 or ͙10 Ϫ ␲. If, however, we use the calculator to learn that ␲ Ϫ ͙10 is negative, about Ϫ0.020685, then we immediately know that ͉␲ Ϫ ͙10 ͉ ϭ ͙10 Ϫ ␲. pg019 [R] G1 5-36058 / HCG / Cannon & Elich ges 11-30-1995 QC2

1.3 Real Number Properties; Complex Numbers 19

᭤EXAMPLE 1 Absolute value (a) If t ϭ 1 Ϫ ͙3, show both t and Ϫt on a number line and express ͉t ͉ in exact form without using absolute values. (b) Find all numbers x such that ͉ 2x Ϫ 3 ͉ ϭ 1. Solution (a) Since t is negative ͑t Ϸ Ϫ.0732͒, ͉t ͉ is the opposite of t: ͉t ͉ ϭ ͉1 Ϫ ͙3 ͉ ϭϪ͑1Ϫ͙3͒ϭ͙3Ϫ1. t =1Ð 3 Ðt Both t and Ϫt areshownonthenumberlineinFigure6. (b) x Ð1 0 1 The two numbers whose absolute value is 1 are 1 and Ϫ1. Thus, if ͉ 2 Ϫ 3 ͉ ϭ 1, we have either Distance is 3Ð1. 2x Ϫ 3 ϭ 1or2xϪ3ϭϪ1. FIGURE 6 Solving each, we have x ϭ 2orxϭ1. ᭣

Some useful properties of absolute values Suppose x and y are any real numbers. 1. ͉ x ͉ Ն 0 2. ͉ x ͉ ϭ ͉ Ϫx ͉ 3. ͉x · y ͉ ϭ ͉ x ͉ · ͉ y ͉ 4. ͙x 2 ϭ ͉ x ͉ x ͉ x ͉ 5. ϭ if y ϭր 0 6. ͉ x ϩ y ͉ Յ ͉ x ͉ ϩ ͉ y ͉ ͯyͯ ͉ y ͉

᭤EXAMPLE 2 Absolute value Let x ϭϪ3, y ϭ 2, and z ϭ 1 Ϫ ͙3. Evaluate the expressions. Are the values in each pair equal or not? (a) ͙x 2, x (b) ͉ x · y ͉ , ͉ x ͉ · ͉ y ͉ (c) ͉ y ϩ z ͉ , ͉ y ͉ ϩ ͉ z ͉ (d) ͉ x ϩ z ͉ , ͉ x ͉ ϩ ͉ z ͉ Strategy: Identify each Solution number as positive or nega- 2 2 tive before applying a (a) ͙x ϭ ͙͑Ϫ3͒ ϭ ͙9 ϭ 3; x ϭϪ3. definition of absolute value. (b) ͉ x · y ͉ ϭ ͉͑Ϫ3͒ ·2͉ ϭ͉Ϫ6͉ ϭ6; ͉ x ͉ · ͉ y ͉ ϭ ͉Ϫ3 ͉ · ͉2 ͉ ϭ 3·2ϭ6. (c) ͉ y ϩ z ͉ ϭ ͉ 2 ϩ ͑1 Ϫ ͙3͉͒ ϭ ͉3 Ϫ ͙3͉ ϭ 3 Ϫ ͙3 Ϸ 1.27. ͉ y ͉ ϩ ͉ z ͉ ϭ ͉2 ͉ ϩ ͉1 Ϫ ͙3 ͉ ϭ 2 ϩ ͙͑3 Ϫ 1͒ ϭ 1 ϩ ͙3 Ϸ 2.73. (d) ͉ x ϩ z ͉ ϭ ͉ Ϫ3 ϩ ͑1 Ϫ ͙3͉͒ ϭ ͉Ϫ2 Ϫ ͙3͉ ϭ 2 ϩ ͙3 Ϸ 3.73. ͉ x ͉ ϩ ͉ z ͉ ϭ ͉Ϫ3 ͉ ϩ ͉1 Ϫ ͙3 ͉ ϭ 3 ϩ ͙͑3 Ϫ 1͒ ϭ 2 ϩ ͙3. The pairs in parts (b) and (d) are equal; those in (a) and (c) are not. ᭣

Complex Numbers Although most of our work deals exclusively with real numbers, sometimes we must expand to a larger set, the set of complex numbers. We need complex numbers mostly in two settings: for the solutions of , and for some trigonometric applications (Chapter 7). For the time being, all the informa- tion we need is simple complex-number arithmetic and how to take roots. Weinclude a picture of the and some properties of complex numbers for reference. pg020 [V] G6 5-36058 / HCG / Cannon & Elich jb 11-21-95 mp1

20 Chapter 1 Basic Concepts: Review and Preview

HISTORICAL NOTE GROWTH OF THE NUMBER SYSTEM The ancient Greeks believed irrationals have negatives, so two numbers expressed the essence of numbers have the square 2, namely the whole world. Numbers to the ͙2andϪ͙2. Pythagorean philosophers meant Complex numbers have a whole numbers and their history somewhat shorter than that ratios—what we would call the of irrational numbers. Cardan made positive rational numbers. It was the first public use of complex extremely distressing to some when numbers in 1545 when he showed they discovered that something as how to find two numbers with a simple as the diagonal of a square sumof10andaproductof40, cannot be expressed rationally in giving the result as 5 ϩ ͙Ϫ15 and terms of the length of the side of Boethius (left), using written 5 Ϫ ͙Ϫ15. Although he observed the square. (ca. 550 Arabic numerals, triumphs that the equals 52 Ϫ ͑Ϫ15͒ B.C.) is said to have sacrificed 100 over Pythagoras and his or 40, he considered such oxen in honor of the discovery of abacus in a mathematical expressions no more real than irrational numbers. Nonetheless, contest. The goddess negative numbers describing lengths irrational numbers were called Arithmetica presides over the of line segments. In 1777 Euler first competition. alogos in Greek, carrying the double used the symbol i to denote ͙Ϫ1. meaning that such numbers were not ratios and As with ͙2, people first considered only one root also that they were not to be spoken. of Ϫ1, but then recognized that i and Ϫi both Hundreds of years passed before satisfy the x 2 ϩ 1 ϭ 0. mathematicians became comfortable with the use We have come gradually to recognize that our of numbers like ͙2. Even we are reluctant to number system is much larger and richer than accept new numbers, as our language reflects. We childhood experience conceives. We extend our equate rational with reasonable, and dislike numbers to accommodate irrational or negative concepts. and and then to solve simple equations Not until the Middle Ages did mathematicians such as x 2 ϭ 2andx2 ϭϪ1. Other extensions become secure with and negative are possible, and we hope to be open to accepting numbers. Also at that time, they recognized that whatever is useful for solving new problems.

We are familiar with the fact that whenever we take the square of a nonzero real number, we always get a positive number. There is no real number that satisfies the simple equation x 2 ϩ 1 ϭ 0. Accordingly, we extend the real number system by introducing a new number i, whose distinguishing is that its square equals Ϫ1, i 2 ϭϪ1, which is sufficient to define the set C of complex numbers. See the Historical Note, “Growth of the Number System.” Definition: the set of complex numbers C ϭ ͕c ϩ di ͉ c and d are real numbers, and i 2 ϭϪ1͖. The set of complex numbers is really an extension of the set of real numbers because for any real number a, a ϭ a ϩ 0i. This observation gives the conclusion: Every real number is also a . pg021 [R] G1 5-36058 / HCG / Cannon & Elich jn 11-10-95 qc3

1.3 Real Number Properties; Complex Numbers 21

The , Complex Numbers, and Principal Square Roots The roots of a may or may not be real numbers. For solving such an equation, we rely on another familiar tool from introductory algebra, the quadratic formula. Quadratic formula The roots of the equation ax2 ϩ bx ϩ c ϭ 0, where a ϭր 0, are given by Ϫb Ϯ ͙b 2 Ϫ 4ac x ϭ 2a The expression b 2 Ϫ 4ac is called the of the equation and determines the nature of the roots. Ϫb If b 2 Ϫ 4ac ϭ 0, there is only one root, given by x ϭ . 2a If b 2 Ϫ 4ac Ͼ 0, the quadratic formula gives two real roots. If b 2 Ϫ 4ac Ͻ 0, there are two nonreal complex roots.

To apply the quadratic formula in the case with a negative discriminant, we need to extend the idea of square roots from our definition in Section 1.2 to the following. Definition: principal Suppose p is a positive real number. Then the principal square roots of p and Ϫp are given by: ͙p is the nonnegative number whose square is p, as ͙4 ϭ 2. ͙Ϫp is the complex number ͙pi,as͙Ϫ9ϭ3i.

For example, if x 2 Ϫ 4x ϩ 5 ϭ 0, the discriminant is negative and the 4 Ϯ ͙Ϫ4 4 Ϯ 2i quadratic formula gives the roots as x ϭ ϭ ϭ 2 Ϯ i.The 2 2 nonreal complex roots given by the quadratic formula always occur in what are y called conjugate pairs, in this case 2 ϩ i and 2 Ϫ i. In general, for the complex number z ϭ c ϩ di,theconjugate of z, denoted by z, is given by z ϭ c Ϫ di.

c + di The Complex Plane d (c, d) Just as we identify each real number with a point on a number line, we identify each complex number c ϩ di with a point in the plane having coordinates ͑c, d͒. See Figure 7. In this correspondence, the x-axisistherealnumberlineandallreal x multiples of i are located on the y-axis. c The standard form for a complex number is c ϩ di, where c and d are real numbers. The number c is called the real part and d is the imaginary part. If the FIGURE 7 imaginary part is nonzero, then we call the complex number c ϩ di a nonreal- The complex number c ϩ di is complex number. identified with the point (c, d). Complex Number Arithmetic When are two complex numbers equal? How do we add, subtract, multiply, and divide complex numbers? Given complex numbers z and w in standard form, say z ϭ a ϩ bi and w ϭ c ϩ di, we treat these numbers as we would any algebraic pg022 [V] G2 5-36058 / HCG / Cannon & Elich cr 11-10-95 QC2

22 Chapter 1 Basic Concepts: Review and Preview

expressions, combining like terms in the usual fashion, with one exception. In 2 z we replace i by Ϫ1 wherever it occurs. For division, w ͑w ϭր 0͒,we multiply numerator and denominator by w as follows: z a ϩ bi a ϩ bi c Ϫ di ϭ ϭ · w c ϩ di c ϩ di c Ϫ di ac Ϫ adi ϩ bci Ϫ bdi 2 ͑ac ϩ bd͒ ϩ ͑bc Ϫ ad͒i ϭ ϭ c 2 Ϫ d 2i 2 c 2 ϩ d 2 Definitions: complex number arithmetic Suppose z ϭ a ϩ bi, w ϭ c ϩ di, where a, b, c,anddare real numbers. : z ϭ w if and only if a ϭ c and b ϭ d. : z ϩ w ϭ ͑a ϩ c͒ ϩ ͑b ϩ d͒i Subtraction: z Ϫ w ϭ ͑a Ϫ c͒ ϩ ͑b Ϫ d͒i Multiplication: z · w ϭ ͑ac Ϫ bd͒ ϩ ͑ad ϩ bc͒i z ac ϩ bd bc Ϫ ad Division: ϭ ϩ i, w c 2 ϩ d 2 c 2 ϩ d 2 where c 2 ϩ d 2 ϭր 0.

y ᭤EXAMPLE 3 Complex number arithmetic If z ϭ 2 Ϫ i and w ϭϪ1ϩi, zw write each expression as a complex number in standard form and locate each on a diagram of the complex plane: Ðz w z 1 (a) z ϩ w (b) z and Ϫz (c) zw (d) z+w w x Solution 1 w z (a) z ϩ w ϭ ͑2 Ϫ i͒ ϩ ͑Ϫ1 ϩ i͒ ϭ 1 ϩ 0i ϭ 1. (b) z ϭ 2 ϩ i,andϪzϭϪ2ϩi. (c) zw ϭ ͑2 Ϫ i͒͑Ϫ1 ϩ i͒ ϭϪ2ϩ2iϩiϪi2 ϭϪ2ϩ3iϪ͑Ϫ1͒ FIGURE 8 ϭϪ1ϩ3i. Points in the complex plane. 1 1w– 1͑Ϫ1 Ϫ i͒ Ϫ1 Ϫ i Ϫ1 Ϫ i 1 1 (d) ϭ ϭ ϭ ϭ ϭϪ Ϫ i. w ww– ͑Ϫ1 ϩ i͒͑Ϫ1 Ϫ i͒ 1 Ϫ i2 1 Ϫ ͑Ϫ1͒ 2 2 y a + bi The points are shown in Figure 8. ᭣

ͦ a + bi ͦ Absolute Value of a Complex Number = a2 + b2 We define the absolute value of a real number x as the distance between the point b x and the 0. In a similar manner, we define the absolute value of a ϩ bi as the distance in the plane between ͑a, b͒ and the origin ͑0, 0͒. The diagram in Figure 9 and the give a distance of ͙a2 ϩ b2. Definition: absolute value of a complex number a x Suppose z is the complex number a ϩ bi.Theabsolute value of z, denoted by ͉ z ͉ ,is͙a2 ϩb2, and we write ͉ z ͉ ϭ ͙a2 ϩ b 2. FIGURE 9 The absolute value of a Many of the properties of absolute values of complex numbers are the same as complex number is the distance to the origin in the those for real numbers. From the definition of distance in the next section, we can complex plane. also observe that ͉ z Ϫ w ͉ is the distance between the complex numbers z and w. pg023 [R] G1 5-36058 / HCG / Cannon & Elich madan 10-12-95 QC

1.3 Real Number Properties; Complex Numbers 23

Properties of absolute value of a complex number If z and w are any complex numbers, then: 1. ͉ z ͉ Ն 0 2. ͉ z ͉ ϭ ͉ Ϫz ͉ ϭ ͉ z ͉ 3. ͉ z · w ͉ ϭ ͉ z ͉ · ͉ w ͉ 4. ͙z · z ϭ ͉ z ͉ z ͉ z ͉ 5. ϭ if w ϭր 0 6. ͉ z ϩ w ͉ Յ ͉ z ͉ ϩ ͉ w ͉ ͯwͯ ͉ w ͉

᭤EXAMPLE 4 Absolute values of complex numbers Suppose z ϭ 2 Ϫ i and w ϭϪ1ϩi(the complex numbers of Example 3). Verify that the properties of absolute values hold for each pair. (a) ͉ z ͉ ϩ ͉ w ͉ , ͉ z ϩ w ͉ (b) ͉ z ͉ , ͉ Ϫz ͉ (c) ͉ zw ͉, ͉z ͉ · ͉w ͉ Solution (a) ͉ z ͉ ϩ ͉ w ͉ ϭ ͉ 2 Ϫ i ͉ ϩ ͉ Ϫ1 ϩ i ͉ ϭ ͙22 ϩ ͑Ϫ1͒2 ϩ ͙͑Ϫ1͒2 ϩ 12 ϭ ͙5 ϩ ͙2. ͉ z ϩ w ͉ ϭ ͉ 1 ϩ 0i ͉ ϭ ͙12 ϩ 02 ϭ 1. Since 1 Ͻ ͙5 ϩ ͙2, ͉ z ϩ w ͉ Ͻ ͉ z ͉ ϩ ͉ w ͉ (Property 6). (b) ͉ z ͉ ϭ ͉ 2 ϩ i ͉ ϭ ͙22 ϩ 12 ϭ ͙5; ͉ Ϫz ͉ ϭ ͉ Ϫ2 ϩ i ͉ ϭ ͙͑Ϫ2͒2 ϩ 12 ϭ ͙5. Thus ͉ z ͉ ϭ ͉ Ϫz ͉ (Property 2). (c) zw ϭϪ1ϩ3i,andso͉zw ͉ ϭ ͉Ϫ1 ϩ 3i͉ ϭ ͙10. ͉ z ͉ · ͉ w ͉ ϭ ͙5·͙2ϭ͙10. Therefore ͉zw ͉ ϭ ͉z ͉ · ͉w ͉ (Property 3). ᭣

Ordering of the Complex Numbers Real numbers are ordered by the less than , so that if b is any real number, then exactly one of the following is true: b ϭ 0orbϽ0orbϾ0. Since R is a of C, any ordering of the complex numbers should be consistent with the ordering of R. Consider the nonzero complex number i.Ifwecouldextend the ordering of R to C, then we would have to have i Ͻ 0oriϾ0. If i Ͼ 0, then we can multiply both sides by i and get i · i Ͼ i ·0 or Ϫ1Ͼ0, which is not a true statement in R. That leaves only the possibility that i is negative, i Ͻ 0. If we multiply both sides by a negative number, we must reverse the direc- tion of the , and we get the same contradiction: i · i Ͼ i ·0 or Ϫ1Ͼ0. We are forced to conclude that there is no consistent way to order the set of complex numbers using the less than relation. Thus, for example, we cannot say that one of the two numbers 3 Ϫ 4i and Ϫ1 ϩ 2i is less than the other. pg024 [V] G2 5-36058 / HCG / Cannon & Elich clb 11-9-95 QC1

24 Chapter 1 Basic Concepts: Review and Preview

EXERCISES 1.3 Check Your Understanding 10. (a) Ϫ4 Ϫ6 Exercises 1–5 True or False. Give reasons. (b) Ϫ␲ Ϫ͙10 5 1. If both b and d are negative real numbers, then 11. (a) 0.45 ͙Ϫb ͙Ϫd ϭ ͙bd. 11 2. ͉ 3 Ϫ ͙10 ͉ ϭ ͙10 Ϫ 3. (b) 1 ϩ ͙2 2.9 47 3. 1 Ϫ ͙10 Ͼ 1 Ϫ ␲ 12. (a) 16 3 4. In the complex plane, 3 ϩ 4i is farther from the origin 7 than 5i. (b) 0.63 11 5. If 0 Ͻ b Ͻ 1andϪ1ϽcϽ0, then 13. (a) ͉ 1 Ϫ ͙3 ͉ ͙3 Ϫ 1 Ϫ1 Ͻ b Ϫ c Ͻ 1. (b) ͉ Ϫ5 ͉ 4 Exercises 6–10 Fill in the blank so that the resulting Exercises 14–17 Ordering Numbers Order the set of statement is true. three numbers from smallest to largest. Express the result using the symbol Ͻ as, for instance, y Ͻ z Ͻ x 6. Thesmallestintegerthatisgreaterthan3Ϫ͙10 is , . . 14. x ϭ 5, y ϭϪ7, z ϭϪ3 3 Ϫ ␲ 16 5 7 7. The greatest less than is . 15. x ϭ , y ϭ , z ϭ 4 23 12 15 8. The largest integer in the set ͕x ͉ Ϫ͙3 Ͻ x Ͻ ͙148͖ 16. x ϭ 1 Ϫ ͙3, y ϭ ͙3 Ϫ 1, z ϭϪ1 is . 7 6 1 17. x ϭ 1 Ϫ , y ϭ 1 Ϫ , z ϭ 1 Ϫ 9. The number of integers between ͙29 ϩ 1and8␲is ͯ 5 ͯ ͯ 5 ͯ ͯ 5 ͯ . Exercises 18–19 True or False. 10. The number of prime numbers between ͙17 Ϫ 2and ͙83 ϩ 8is . 1 18. (a) ␲ 2 Ͻ 10 (b) Ͼ 2.28 ͙2 Ϫ 1 Develop Mastery 6 Exercises 1–6 Number Line Show the set on a number 19. (a) 1.33 Ͻ 1.3 (b) 0.54 Ͼ 11 line. 1. ͕x ͉ x ϾϪ2andxϽ2͖ Exercises 20–25 Intervals on Number Line Show the intervals on a number line. 2. ͕x ͉ x ՆϪ1andxՅ1͖ 20. ͑Ϫ1, 4͒ 21. ͑Ϫϱ,2͒ 3. ͕x ͉ x Ͻ 1orxϾ3͖ 22. ͓Ϫ2, ϱ͒ 23. ͓1, 4͔ ʝ ͑0, 5͒ 4. ͕x ͉ x ՅϪ1͖ʜ͕x͉xϾ4͖ 24. ͑Ϫϱ,3͒ʜ͑3, 4͔ 25. ͓Ϫ3, 2͔ ʝ ͓2, 5͔ 5. ͕x ͉ x Ͼ 0͖ ʝ ͕x ͉ x Ͻ 3͖ 6. ͕x ͉ 1 Յ x Ͻ ͙7͖ Exercises 26–31 Verbal to Number Line SetSisde- scribed verbally. Show S on the number line. Exercises 7–9 Absolute Value Simplify. Express in ex- 26. Set S contains all negative numbers greater than Ϫ5. act form without using absolute values, and as a rounded off to four decimal places. 27. Set S contains all real numbers greater than Ϫ2 and less than 3. 1 3 9 7. (a) Ϫ (b) 3 Ϫ 28. Set S consists of all integers between Ϫ3 and 8. ͯ 4 2 ͯ ͯ 2 ͯ 29. Set S consists of all prime numbers between 0 and 16. 4 8. (a) 1 Ϫ (b) ͉ 3 Ϫ ͙17 ͉ 30. Set S consists of all real numbers between Ϫ͙3and ͯ 7 ͯ ͙5. 22 31. Set S consists of all less than 4. 9. (a) ͉␲ Ϫ 3 ͉ (b) ␲ Ϫ ͯ 7 ͯ 32. What is the largest integer that is (a) less than or equal to Ϫ5? (b) less than Ϫ5? Exercises 10–13 Enter one of the three symbols Ͻ, Ͼ, or ϭ in each blank to make the resulting statement 33. Whatisthelargestintegerthatislessthan1ϩ͙17? true. pg025 [R] G1 5-36058 / HCG / Cannon & Elich cr 11-10-95 QC2

1.4 Rectangular Coordinates, Technology, and Graphs 25

348 34. What is the smallest integer that is greater than 37 ? 51. z ϩ 3w 52. zw Ϫ 4 53. z · w 35. What is the smallest even integer that is greater than z Ϫ w– 54. ͉ z ϩ w͉ 55. 12 ϩ ͙5? w 36. What is the largest that is less than 23 ? 0.23 Exercises 56–59 Complex Plane For the given z and w, Exercises 37–50 Complex Number Arithmetic Perform show in the complex plane: the indicated operations. Express the result as a complex (a) z (b) w (c) z (d) z ϩ w (e) z·w number in standard form. 56. z ϭ 2 Ϫ 2i; w ϭ 3 ϩ 4i 37. ͑5 ϩ 2i͒ ϩ ͑3 Ϫ 6i͒ 38. ͑3 Ϫ i͒ ϩ ͑Ϫ1 ϩ 5i͒ 57. z ϭϪ3ϩ2i;wϭϪ2Ϫi 39. ͑6 Ϫ i͒ Ϫ ͑3 Ϫ 4i͒ 40. 8 Ϫ ͑3 ϩ 5i͒ ϩ 2i 58. z ϭϪ1ϩ2i;wϭ3i 41. ͑2 ϩ i͒͑3 Ϫ i͒ 42. ͑Ϫ1 ϩ i͒͑2 ϩ 3i͒ 59. z ϭ 5 Ϫ i; w ϭϪ1ϩi 43. ͑1 ϩ 3i͒͑1 Ϫ 3i͒ 44. ͑7 Ϫ 2i͒͑7 ϩ 2i͒ 1 ͙3 2 3 4 5 6 1 ϩ 3i 1 ϩ i 60. (a) If z ϭ ϩ i; find z , z , z , z , z . 45. 46. 2 2 i 1 Ϫ i (b) Evaluate each of ͉ z ͉ , ͉ z 2 ͉ , ͉ z 3 ͉ ,...,͉z6͉. 2i 2 3 4 47. ͑1 ϩ ͙3i͒2 48. 61. From Exercise 60 draw a diagram showing z, z , z , z , ͑1 Ϫ i͒͑2 Ϫ i͒ z 5,andz6 in the complex plane. Note the distance from 49. (a) i 2 (b) i 6 (c) i 12 (d) i 18 each of these points to the origin. On what circle do these points lie? 50. (a) i 5 (b) i 9 (c) i 15 (d) i 21 Exercises 62–63 In Exercises 60–61, replace z with z ϭ Exercises 51–55 If z is 1 Ϫ iandwisϪ2ϩi,express in 1 ͑1 ϩ i͒. standard form. ͙2

1.4 RECTANGULAR COORDINATES, TECHNOLOGY, AND GRAPHS Creative people live in two worlds. One is the ordinary world which they share with others and in which they are not in any special way set apart from their fellow men. The other is private and it is in this world that the creative acts take place. It is a world with its own passions, elations and despairs, and it is here that, if one is as great as Einstein, one may even hear the voice of God. Mark Kac

Rectangular Coordinates y Few intellectual discoveries have had more far-reaching consequences than coordi- natizing the plane by Rene´ Descartes nearly 400 years ago. We speak of Cartesian 3 P(c, d) or rectangular coordinates in his honor. d A rectangular uses two perpendicular number lines in the 2 plane, which we call coordinate axes. The more common is a horizontal 1 x-axis andaverticaly-axis, but other names and orientations are some- useful. x Each point P in the plane is identified by an of real numbers c, d , Ð1 1234c ͑ ͒ Ð1 called the coordinates of P, where c and d are numbers on the respective axes as shown in Figure 10. Conversely, every pair of real numbers names a unique point FIGURE 10 on the plane.

A rectangular system of coordinates provides a one-to-one correspondence between the set of ordered pairs of real numbers and the points in the plane.