Elementary Algebra and Geometry

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1 1 Elemetary Algera ad Geometry 1.1 Fudametal Properties (Real Numers) a + = + a Commutative Law for Additio (a + ) + c = a + ( + c) Associative Law for Additio a + 0 = 0 + a Idetity Law for Additio a + ( a) = ( a) + a = 0 Iverse Law for Additio a(c) = (a)c Associative Law for Multiplicatio æ 1ö 1 aç a 1 a 0 è a ø = æ è ç ö =, ¹ Iverse Law for a ø Multiplicatio (a)(1) = (1)(a) = a Idetity Law for Multiplicatio 1

2 Itegrals ad Mathematical Formulas a = a Commutative Law for Multiplicatio a( + c) = a + ac Distriutive Law Divisio y zero is ot defied. 1. Expoets For itegers m ad, aa = a m + m a / a = a m -m ( a ) m ( a) = a m = a m m m ( a/ ) = a / m m m 1. Fractioal Expoets a p/ q / q p = ( a 1 ) where a 1/q is the positive qth root of a if a > 0 ad the egative qth root of a if a is egative ad q is odd. Accordigly, the five rules of expoets give aove (for itegers) are also valid if m ad are fractios, provided a ad are positive.

3 Elemetary Algera ad Geometry 1.4 Irratioal Expoets If a expoet is irratioal, e.g.,, the quatity, such as a, is the limit of the sequece a 1.4, a 1.41, a 1.414,. Operatios with Zero m 0 0 = 0; a = Logarithms If x, y, ad are positive ad 1, log ( xy) = log x+ log y log ( x/ y) = log x-log y p log x = plog x log ( 1/ x) =-log x log = 1 log x log 1= 0 Note: = x. Chage of Base (a 1) log x = log xlog a a

4 4 Itegrals ad Mathematical Formulas 1.6 Factorials The factorial of a positive iteger is the product of all the positive itegers less tha or equal to the iteger ad is deoted! Thus,! = 1 ¼. Factorial 0 is defied 0! = 1. Stirlig s Approximatio lim ( / e) p =! (See also Sectio 9..) 1.7 Biomial Theorem For positive iteger, ( - 1) ( x+ y) = x + x y+ x! -1 - ( -1)( -) + x! + xy y. y - y +

5 Elemetary Algera ad Geometry Factors ad Expasio ( a+ ) = a + a+ ( a- ) = a - a+ ( a+ ) = a + a + a + ( a- ) = a -a + a - ( a - ) = ( a- )( a+ ) ( a - ) = ( a- )( a + a+ ) ( a + ) = ( a+ )( a - a+ ) 1.9 Progressio A arithmetic progressio is a sequece i which the differece etwee ay term ad the precedig term is a costat (d): aa, + da, + d,, a+ ( -1) d. If the last term is deoted l[= a + ( 1)d], the the sum is s = a+ l ( ).

6 6 Itegrals ad Mathematical Formulas A geometric progressio is a sequece i which the ratio of ay term to the precedig terms is a costat r. Thus, for terms, aar,, ar,, ar 1 - The sum is a ar S = - 1- r 1.10 Complex Numers A complex umer is a ordered pair of real umers (a, ). Equality: (a, ) = (c, d) if ad oly if a = c ad = d Additio: (a, ) + (c, d) = (a + c, + d) Multiplicatio: (a, )(c, d) = (ac d, ad + c) The first elemet of (a, ) is called the real part; the secod, the imagiary part. A alterate otatio for (a, ) is a + i, where i = ( 1, 0), ad i(0, 1) or 0 + 1i is writte for this complex umer as a coveiece. With this uderstadig, i ehaves as a umer, i.e., ( i)(4 + i) = 8 1i + i i = 11 10i. The cojugate of a + i is a i, ad the product of a complex umer ad its cojugate is a +.

7 Elemetary Algera ad Geometry 7 Thus, quotiets are computed y multiplyig umerator ad deomiator y the cojugate of the deomiator, as illustrated elow: + i ( 4- i)( + i) i 7 4 = = = + i 4+ i ( 4- i)( 4+ i) Polar Form The complex umer x + iy may e represeted y a plae vector with compoets x ad y: ( ) x+ iy = r cosq+ i siq (see Figure 1.1). The, give two complex umers z 1 = r 1 (cosθ 1 + i siθ 1 ) ad z = r (cosθ + i siθ ), the product ad quotiet are: Product: z 1 z = r 1 r [cos(θ 1 + θ ) + i si(θ 1 + θ )] Quotiet: z/ z = ( r/ r )[ cos( q - q ) + i si( q -q )] ( ) Powers: z = é ë r cosq+ i siq ù û = r éë cos q+ isi qù û ( ) 1/ Roots: z = é ë r cosq+ i si q ù û 1/ é q+ k. 60 q+ k. 60 ù = r + i ë ê cos si û ú, k = 01,,, ¼, - 1 1/

8 8 Itegrals ad Mathematical Formulas y P (x, y) r θ x 0 FIGURE 1.1 Polar form of complex umer. 1.1 Permutatios A permutatio is a ordered arragemet (sequece) of all or part of a set of ojects. The umer of permutatios of ojects take r at a time is pr (, ) = ( -1)( -) ( - r+ 1)! = ( - r)! A permutatio of positive itegers is eve or odd if the total umer of iversios is a eve

9 Elemetary Algera ad Geometry 9 iteger or a odd iteger, respectively. Iversios are couted relative to each iteger j i the permutatio y coutig the umer of itegers that follow j ad are less tha j. These are summed to give the total umer of iversios. For example, the permutatio 41 has four iversios: three relative to 4 ad oe relative to. This permutatio is therefore eve. 1.1 Comiatios A comiatio is a selectio of oe or more ojects from amog a set of ojects regardless of order. The umer of comiatios of differet ojects take r at a time is Pr (, )! C( r, ) = = r! r!( - r)! 1.14 Algeraic Equatios Quadratic If ax + x + c = 0, ad a 0, the roots are ac x = - ± - 4 a

10 10 Itegrals ad Mathematical Formulas Cuic To solve x + x + cx + d = 0, let x = y /. The the reduced cuic is otaied: y + py+ q= 0 where p = c (1/) ad q = d (1/)c + (/7). Solutios of the origial cuic are the i terms of the reduced cuic roots y 1, y, y : x = y -( 1 / ) x = y -( 1 / ) 1 1 x = y -( 1 / ) The three roots of the reduced cuic are where y = ( A) + ( B) 1 1/ 1/ y = W( A) + W ( B) 1/ 1/ y = W ( A) + W( B) 1/ 1/ 1 1 A=- q+ ( 17 / ) p + q, B=- q- ( 17 / ) p + q, 4 i W = - 1+ W = - 1 -i,.

11 Elemetary Algera ad Geometry 11 Whe (1/7)p + (1/4)p is egative, A is complex; i this case, A should e expressed i trigoometric form: A = r(cos θ + i si θ), where θ is a first or secod quadrat agle, as q is egative or positive. The three roots of the reduced cuic are y y 1 y 1/ r cos ( q/ ) 1/ æ q r ç è 10 ö cos ø 1/ æ q ö r cosç + 40 è ø = ( ) = ( ) + = ( ) 1.15 Geometry Figures 1. through 1.1 are a collectio of commo geometric figures. Area (A), volume (V), ad other measurale features are idicated. h FIGURE 1. Rectagle. A = h.

12 1 Itegrals ad Mathematical Formulas h FIGURE 1. Parallelogram. A = h. h FIGURE 1.4 Triagle. A= 1 h. a h FIGURE Trapezoid. A= a+ h ( ).

13 Elemetary Algera ad Geometry 1 R S θ FIGURE 1.6 Circle. A = πr ; circumferece = πr; arc legth S = Rθ (θ i radias). R θ FIGURE Sector of circle. Asector = R q; Asegmet = R ( q-siq). R θ FIGURE 1.8 Regular polygo of sides. A = p R 4 = p ct ; csc.

14 14 Itegrals ad Mathematical Formulas h R FIGURE 1.9 Right circular cylider. V = πr h; lateral surface area = πrh. h A FIGURE 1.10 Cylider (or prism) with parallel ases. V = Ah.

15 Elemetary Algera ad Geometry 15 l h R FIGURE 1.11 Right circular coe. V = 1 p R h; lateral surface area = prl = pr R + h. R FIGURE 1.1 Sphere. V = 4 p R ; surface area = 4πR.

16 16 Itegrals ad Mathematical Formulas 1.16 Pythagorea Theorem For ay right triagle with perpedicular sides a ad, the hypoteuse c is related y the formula c = a + This famous result is cetral to may geometric relatios, e.g., see Sectio 4..

Appendix F: Complex Numbers

Appendix F: Complex Numbers Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write