pg017 [R] G1 5-36058 / HCG / Cannon & Elich KK 10-13-1995 QC1 1.3 Real Number Properties; Complex Numbers 17 1.3 REAL NUMBER PROPERTIES; COMPLEX NUMBERS . one of the central themes of science @is# the mysterious power of OP mathematics to prepare the ground for physical discoveries which could not –2 –1 0 1 2 3 have been foreseen by the mathematicians who gave the concepts birth. Freeman Dyson (a) Integers on a number line. Real Number Line 1 2 – –2 2 3 3 p One of the great ideas in the history of mathematics is that the set of real numbers can be associated with the set of points on a line. We assume a one-to-one corre- –2 –1 0 1 2 3 spondence that associates each real number with exactly one point 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 , cbis less than c. c 5 b 1 d, for some positive The mathematics course at San Diego High School number d was standard for that time: c . bcis greater than bb,c plane geometry in the b # cbis less than or b , c or b 5 c tenth grade, advanced equal to c. algebra in the eleventh, c $ bcis greater than b # c and trigonometry 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 , c: Name Notation Number-line diagram Open interval ~b, c! 5 $x _ b , x , c% Closed interval @b, c# 5 $x _ b # x # c% Half-open interval @b, c! 5 $x _ b # x , c% Half-open interval ~b, c# 5 $x _ b , x # c% Infinite intervals ~b, `! 5 $x _ x . b% @b, `! 5 $x _ x $ b% ~2`, b! 5 $x _ x , b% ~2`, b# 5 $x _ x # b% In these definitions, the symbol ` (infinity) does not represent a number, and we never use a closed bracket to indicate that ` 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 Distance We have no difficulty in finding the absolute value of specific numbers, as in _2 _ 5 2, _0 _ 5 0, _21.375 _ 5 1.375. There are always two numbers having the same nonzero absolute value, as 2p p p _2 _ 5 _22 _ 5 2, and 5 5 . U 3 U U 3 U 3 Following the pattern of the above examples, the absolute value of a positive number is itself; the absolute value of a negative number is its opposite. In working with an expression like _2x 2 3 _, the quantity 2x 2 3isneither positive nor negative until we give a value to x. All we can say is that _ 2x 2 3 _ is either 2x 2 3 or its opposite, 2~2x 2 3!. Thus _ 2x 2 3 _ 5 2x 2 3forallthex-valuesthatmake2x23 positive 9 ~2, 5 ,5p,...!, _2x23_ 522x13forallthex-valuesthatmake2x23negative p ~1, 3 ,0,...!. Looking at a number line, the numbers satisfying _ x _ 5 3are3and23, 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 2 5 _ 5 2, and _7 2 9 _ 5 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 $ 0 _ u _ 5 H2u if u , 0 If a and b are any real numbers, then the distance between a and b is given by _ a 2 b _ 5 _ b 2 a _ . It follows that _a _ 5 _a 2 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 _p 2 Ï10 _ is either p 2 Ï10 or Ï10 2 p, depending on which is positive. We do not need to use the calculator function 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 ABS~p 2 Ï10! < 0.020685. This is true, but not helpful in deciding whether _p 2 Ï10 _ is equal to p 2 Ï10 or Ï10 2 p. If, however, we use the calculator to learn that p 2 Ï10 is negative, about 20.020685, then we immediately know that _p 2 Ï10 _ 5 Ï10 2 p. pg019 [R] G1 5-36058 / HCG / Cannon & Elich ges 11-30-1995 QC2 1.3 Real Number Properties; Complex Numbers 19 cEXAMPLE 1 Absolute value (a) If t 5 1 2 Ï3, show both t and 2t on a number line and express _t _ in exact form without using absolute values. (b) Find all numbers x such that _ 2x 2 3 _ 5 1. Solution (a) Since t is negative ~t < 2.0732!, _t _ is the opposite of t: _t _ 5 _1 2 Ï3 _ 52~12Ï3!5Ï321. t =1– 3 –t Both t and 2t areshownonthenumberlineinFigure6. (b) x –1 0 1 The two numbers whose absolute value is 1 are 1 and 21. Thus, if _ 2 2 3 _ 5 1, we have either Distance is 3–1. 2x 2 3 5 1or2x23521. FIGURE 6 Solving each, we have x 5 2orx51. b Some useful properties of absolute values Suppose x and y are any real numbers. 1. _ x _ $ 0 2. _ x _ 5 _ 2x _ 3. _x · y _ 5 _ x _ · _ y _ 4. Ïx 2 5 _ x _ x _ x _ 5. 5 if y 5/ 0 6. _ x 1 y _ # _ x _ 1 _ y _ UyU _ y _ cEXAMPLE 2 Absolute value arithmetic Let x 523, y 5 2, and z 5 1 2 Ï3. Evaluate the expressions. Are the values in each pair equal or not? (a) Ïx 2, x (b) _ x · y _ , _ x _ · _ y _ (c) _ y 1 z _ , _ y _ 1 _ z _ (d) _ x 1 z _ , _ x _ 1 _ z _ Strategy: Identify each Solution number as positive or nega- 2 2 tive before applying a (a) Ïx 5 Ï~23! 5 Ï9 5 3; x 523. definition of absolute value. (b) _ x · y _ 5 _~23! ·2_ 5_26_ 56; _ x _ · _ y _ 5 _23 _ · _2 _ 5 3·256. (c) _ y 1 z _ 5 _ 2 1 ~1 2 Ï3!_ 5 _3 2 Ï3_ 5 3 2 Ï3 < 1.27. _ y _ 1 _ z _ 5 _2 _ 1 _1 2 Ï3 _ 5 2 1 ~Ï3 2 1! 5 1 1 Ï3 < 2.73. (d) _ x 1 z _ 5 _ 23 1 ~1 2 Ï3!_ 5 _22 2 Ï3_ 5 2 1 Ï3 < 3.73. _ x _ 1 _ z _ 5 _23 _ 1 _1 2 Ï3 _ 5 3 1 ~Ï3 2 1! 5 2 1 Ï3. The pairs in parts (b) and (d) are equal; those in (a) and (c) are not. b 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 polynomial equations, 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 square roots. Weinclude a picture of the complex plane 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 Ï2and2Ï2. 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 1 Ï215 and terms of the length of the side of Boethius (left), using written 5 2 Ï215.