MULTIPLYING OUT & FACTORING

Size: px
Start display at page:

Download "MULTIPLYING OUT & FACTORING"

Transcription

1 igitl ircuit Engineering MULTIPLYING OUT & FTORING I IGITL SIGN Except for fctoring st istributive X + X = X( + ) 2nd istributive (X + )(X + ) = X + (X + )(X + )(X + ) = X + Swp (X + )(X + ) = X + X The Most ommon Stupid Errors Using Not Using X. Y = XY YX + X = X X + = X Generl emorgn Y + XY = X + Y F(, b,... z,+,.,,0) F(, b,... z,.,+,0,) rleton University 2009 dig4fctoringh.fm p. 26 Revised; Jnury 29, 2009 Slide i Multiplying Out nd Fctoring Sum-of-Products, Product-of-Sums Multiplying Out Use Tke dul; Simplify & ; reverse dul; then use Use Krnugh Mp Fctoring Use 2 Tke the dul multiply out tke the dul bck Find F using emorgn, then use Krnugh mp rleton University dig4fctoringh.fm p. 27, Revised; Jnury 29, 2009 omment on Slide i

2 Stndrd Forms Sum-of-Products Product-of-Sums These re stndrd templtes or forms Every logicl expression cn be converted to either of these forms. Sum of Products Σ of Π, OR of Ns, S of P bc + cde + d + f + db Single vribles Ned together into terms. These terms re ORed together. Inversions re only over individul vribles. No brckets. Product of Sums. Π of Σ, N of ORs, P of S Single vribles ORed together into terms. These terms re Ned together. Inversions re only over individul vribles. rckets only round vribles in OR terms Questions Is b + cd + b(d+e) Σ of Π? Is ( +bc)(d+e) Π of Σ? (+b+c)(c+d+e)(+d)(f)(d+b) dig4fctoringh.fm p. 28 Revised; Jnury 29, 2009 Slide Stndrd Forms Terms Terms term in logicl expression is number of single letters Ned together. Exmples: bcd bcxq xy epending on context, it could lso represent number of single letters ORed together. Exmples: +b+c+y+q + Sum-of-Products These must be expressed lgebriclly s the OR of Ns. One ws not llowed to put in brckets, XORs, nor long inversion brs. Thus none of the exmples on the left below re true sum of products. NOT Σ of Π bc + b(+d) + de bc + bcd +bf (bc + bfg + bcd) c + bg d+bc bc These re Σ of Π fter finding n lgebric Σ of Π form, it cn be esily chnged to NN-NN logicfor implementtion. Product of Sums These must be expressed lgebriclly s the N of ORs. One must use single-letters ORed together into terms which re Ned together. There must be no XORs, nor long inversion brs. Thus none of the exmples on the left below re true product of sum. (+b+c)b(+d) (+b+c)(d+e)+d (e+b+c) (+b+c)(b+c+d)(+b) ( c+b)(g+h) +b+c These re Π of Σ NOT Π of Σ fter finding lgebric Π of Σ form, it cn be esily chnged to NOR-NOR logic for implementtion. rleton University dig4fctoringh.fm p. 29, Revised; Jnury 29, 2009 omment on Slide

3 Sum-of-Products Product-of-Sums Two Stndrd Forms Sum of Products (Σ of Π) bc + e + ce + bd +... OR of Ns c b e e c d b Σ of Π Σ of Π (NN-NN) n be implemented s NN-NN logic NN Product of Sum (Π of Σ) ul of Σ of Π (+b+c)(+e)(+c+e)(+b+d)(... N of ORs n be implemented s NOR-NOR logic c b e e c d b c b e e c d b Π of Σ NOR c b e e c d b Π of Σ (NOR-NOR) dig4fctoringh.fm p. 30 Revised; Jnury 29, 2009 Slide 2 Sum-of-Products Product-of-Sums Stndrd Forms oolen expression cn be represented in mny wys, see below. wy to see if two. There re mny templtes (forms) to define oolen Function. form tht cn define ll oolen functions is clled stndrd form. Some Stndrd Forms. b b The truth tble is very bsic form Σ of Π expression. b + b 0 0 Π of Σ expression. ( + b)( + b) (fctored form) 0 Krnugh mps form tht will be introduced shortly. Stndrd Forms inry decision digrms, which will be used lter to build rbitrry circuits with 2-input muxes. Why we sy Σ of Π for Sum of Product n Σ is used for repetitive ddition, s in x i = x 0 + x + x 2 + x, n i = 0 n Π is used for repetitive product, s in x i = x 0 x x 2 x n i = 0. specil type of stndrd form is the nonicl form. It is one which is unique. If two pprently different oolen expressions re reduced to cnonicl form, they re the sme if-nd-only-if their cnonicl forms re identicl. Truth tbles nd unlooped Krnugh mps re cnonicl stndrd forms. Σ of Π nd Π of Σ re only stndrd forms nonicl originlly ment ccording to the rule or lw, prticulrly the church lw. rleton University dig4fctoringh.fm p. 3, Revised; Jnury 29, 2009 omment on Slide 2

4 Fctoring nd Multiplying Out Trnsforming Σ of Π Π of Σ Multiplying out Trnsforms Π of Σ Σ of Π ( + c)(b + + d) b + c + d Fctoring Trnsforms Σ of Χ Π of Σ b + c + d ( + c)(b + + d) Why Fctor? : Typiclly both forms hve bout the bout the sme size, but sometimes fctoring cn sve significnt logic. Σ of Π Π of Σ Σ of Π Π of Σ b + c + d + e = (b+c+d+e) 5 gtes, 8 letters 2 gtes, 5 letters cb + c b + cd + c d = ( + c)(b + d)( + c) 5 gtes, 2 letters 4 gtes, 6 letters c + e +bc + be = ( + b)(c + e) 5 gtes, 8 letters 3 gtes, 4 letters 2: Sometimes NOR-NOR logic my be desired (Fst fll time). dig4fctoringh.fm p. 32 Revised; Jnury 29, 2009 Slide 3 Fctoring nd Multiplying Out Σ of Π Π of Σ Σ of Π Π of Σ Why fctor? Usully the (Π of Σ) nd (Σ of Π) forms re bout the sme complexity, but chnging forms for prt of the circuit cn sometimes sve significnt logic. Logic minimiztion progrms will switch from one form to the other in different prts of lrge circuit. Π of Σ in NOR- NOR form, cn give fst results with flling signl => PROLEM Identify the following s: Σ of Π, Π of Σ, or neither.. (W + Y + Z)(X + Y + Z)(W + X)(W + Z)(X + Y + Z)(W + X + Z) 2. (b + c)(d + ce)( + b + c + d) 3. dc + (bc)+ cb 4. bc + c(+b) + dec +b 5. + b + cd. NMOS trnsistors re fster thn PMOS. NOR gte use prllel NMOS trnsistors to lower its output signl. ll other things being equl, this leds to slightly fster circuits. rleton University dig4fctoringh.fm p. 33, Revised; Jnury 29, 2009 omment on Slide 3

5 Multiplying Out; Methods Multiplying Out hnge Π of Σ Σ of Π ( )( )( ) ( ) + ( ) + ( ) st istributive Lw () x( + ) = x + x Methods of Multiplying Out I. Use the st distributive lw x( + b) = x + xb - This lwys works, but - it usully gives very very long result. II. Simplify (use 2, S2, Sw) OR tke dul (use, S, Sw) tke dul before using - Gives shorter nswers - Shorter steps to get nswer - Requires more thinking III. Use emorgn s Lw to get F Plot F on K-mp Then plot F - Esiest to do - Gives the simplest Σ of Π nswer. - Gets messy for 5 or more vribles. Exmple I ( + )( + + ) Use () = Use = 0 = Exmple II ( + )( + + ) = ( + )( + [ + ]) = [ + ] + = + + Exmple III F=( + )( + ) Use (em) F = + Use Mp F = + Use (Sw) (x+)(x+) = x + x Use () F F dig4fctoringh.fm p. 34 Revised; Jnury 29, 2009 Slide 4 Multiplying Out; Methods Σ of Π Π of Σ Three methods of multiplying out Using Using mny times is strightforwrd wy. is very esy to use; unfortuntely, when done utomticlly with no thinking, it cn get very long. If one uses simplifiction (S) X(X+)=X nd bsorption () X(X+)=X t every chnce, the work is shorter, but the finl expression still my be longer thn necessry. Hint: the simplify nd bsorb rules re esier to see in Σ of Π. Tke the dul so one uses X+X=X nd X+X=X+ which re esier to see. Then tke the dul bck to use. Use 2 (or equivlent) nd Swp before using The strnge distributed rule, 2, is the dul of. The equivlent of using 2 is to: tke the dul, use, then reverse the dul. Swp is the sme difficulty in both the dul nd in the originl form, so use it ny time. Following the hint bove, tke the dul nd use (S) nd () t the strt. While in the dul, use. Then reverse the dul nd continue using, remembering to wtch for convenient plces to use Swp, (S), nd (). Using Krnugh mps This is the esiest method for four or five vribles, it lwys gives the smllest nswer, it esily hndles don t cres, but gets very complex for over five input vribles. It is the method of choice for most smll problems. Three methods of fctoring Using 2 This is the strightforwrd wy, unfortuntely it uses the unfmilir 2 distributive lw which mkes the lgebr hrder for most people. Using dulity nd This is lgebriclly equivlent to the previous method. However using the more fmilir () mkes it esier for most people. Using Krnugh mps This is quite esy for four vribles, but more complex for over five input vribles. Esily hndles d s. rleton University dig4fctoringh.fm p. 35, Revised; Jnury 29, 2009 omment on Slide 4

6 Multiply Out Using st istributive Lw Method I: Using Step : Tke ul, Simplify Step 2: Tke ul bck Step 3: Use Step 2: Simplify, Use Step 3: Simplify, Use... Finl Step: Simplify, Simplify, (K-Mp mt help) Exmple ( + X)( + X)( + X) dul = X + X + X = X(++) f = +X = [( + X) + X( + X)]( + X) = [ + X + X + X]( + X) = [ + X]( + X) = ( + X) + ( + X)X = ( + X + X + X = + X Use () Use () Use (S) x + x = x Use () Use () Use (S) x + cx = x st istributive Lw x( + ) = x + x Wys to Simplify (S) - X +X = X - X +X = X + () - Krnugh mp - onsensus If too mny inputs for mp dig4fctoringh.fm p. 36 Revised; Jnury 29, 2009 Slide 5 Multiply Out Using st istributive Lw Exmple of Multiplying Out Exmple of Multiplying Out Simplifiction, bsorption 2 (or equivlent) nd Swp work well when their re repeted letters in the expression. However if ll the letters re unique, cn only use ( + )( + )(F + G + H) = ( )(F + G + H) = = ( ) F + ( ) G + ( ) H F + F + F + F + G + G + G + G + H + H + H + H ll letters re different, no simplifiction possible Use () rewrite Use () With ll the letters different, there is no wy to simplify. The expressions get long rpidly. Using () lwys works, it is esy on the brin, but hrd on the pencil. lso the simplifictions must be done by other mens. rleton University dig4fctoringh.fm p. 37, Revised; Jnury 29, 2009 omment on Slide 5

7 Multiply Out Using st istributive Lw Method I: Using Step : Simplify, Use Step 2: Simplify, Use Step 3: Simplify, Use... Finl Step: Simplify Exmple: ( + )( + F + )( + )( + F + ) How to Simplify. 2. X +X = X X +X = X + (S) () 3. Krnugh mp 4. oncensus if you hve to Use twice 0 0 = ( + F F + )( + F F + ) c +cx = c = (F + )(F F + ) bf + bfx = bf = FF + F + F + FF + F Use + F F + ) = F F + ) Simplify using 5-input mp but not done till next chpter NSWER, UT OUL E SIMPLER = F + F n simplify using consensus twice but tricky! F + SKIP THIS TWLE = F = F + ( + + ) ( + ) = F + + consensus consensus st istributive Lw () x( + ) = x + x onsensus F F b + bc + c F = b +c 5-vrible mp F Result is F + + dig4fctoringh.fm p. 38 Revised; Jnury 29, 2009 Slide 6 Multiply Out Using st istributive Lw Exmple of Multiplying Out Note on simplifying by consensus or 5-input mp (continued) The 5-input mp is not covered until the next section. Using the 5-input mp voids the use of consensus in the bove exmple. onsensus is tricky to see. Exmple: Multiply Out using Method I. F is not pure Π of Σ Multiply Out F = [( + )] [ ( + ) + ( + )] ( + + ) Use () nd xx=0 = { [ ( + ) + ( + )] + [ ( + ) + ( + )]} ( + + ) xx = 0 xx = 0 Use () = { [( + )] + [ ( + )]} ( + + ) = = {( + ) + ( + )} ( + + ) = { } ( + + ) Use () nd xx=0 ollect terms; use xy + x = x xx = 0 { } + +{ } = { } Plot mp xy + x = x = Mp shows there re no more simplifictions Use () Use () This psudo mtrix helps rrnge the messy multipliction rleton University dig4fctoringh.fm p. 39, Revised; Jnury 29, 2009 omment on Slide 6

8 Multiply Out; Use 2 (or equ) nd Swp, efore Method II Using 2 nd Swp Step : Simplify (esier to spot in dul) Step 2: Use 2 nd/or Swp, ( & mybe Swp in dul) Step 3: Use, Simplify nd mybe Swp.... Repet Steps 2 nd 3 Exmple: F = ( + + )( + )( + + )( + )( + + ) heck for simplifictions Rerrnge to use (2) nd Swp (x)(+x) = x = ( + + )( + )( + + )( + )( + + ) = ( + + )( + )( + + )( + ) = ( + + )( + + )( + )( + ) 2 Sw F = ( + + ) ( + ) F =( + + ) ( + ) n go no further with 2 nd Swp Must use Go to next slide ommuttive lws 2nd istributive Lw (2) (X+c)(X+d) = X + cd Swp (+X)(b+X)=X+bX Simplifiction (S) Simplifiction(dul) X+X = X (X)(X+) = X Tke dul to use esier forms of (S), () nd F d = ++++ heck for simplifictions F dul = ++++ = +++ Rerrnge to use () nd Swp x+xy = x (S) = +++ Sw = (+)+(+)(+) rcket LL product terms F dul = { (+)}+[(+) (+)] Tke dul bck F = {++( )} [( )+( )] rop extr brkets F = { + + } [ + ] dig4fctoringh.fm p. 40 Revised; Jnury 29, 2009 Slide 7 Multiply Out; Use 2 (or equ) nd Swp, efore Multiplying Out Using 2 nd Swp Multiplying Out Using 2 nd Swp before omments: For the initil step Simplify here mens using the dul rules X(X+)=X nd X(X+)=X However the rules X+X=X nd X+X=X re much esier to pply. pplying 2 here mens (X+)(X+b)=X+b However => X+Xb= X(+b) is esier thn using 2 Swp is self dul nd is eqully obnoxious in both forms. Insted of simplifying s bove, tke the dul s suggested in the brckets, nd s is on the right side of the slide. Tking the dul of Product-of-Sums is esy. fter tht one cn use the esy simplify rules X+X=X nd X+X=X. One cn lso use the simple insted of 2 One cn use swp here or in the next step, it mkes little difference. The next slide multiplies out product terms using => F dul =( + + ) ( + ) This is esier to do in the originl form. Thus tke the dul bck before going on to slide 2. This dul is error prone. The trick is to be sure you put RKETS ROUN LL THE N TERMS! If you think you cn write this on the sme line s converting +, I hve some lnd to sell you!. Using swp while in the dul form introduces more brckets. More brckets increse the error tendncy when tking the reverse dul. Thus there is slight dvntge in using Swp lter. rleton University dig4fctoringh.fm p. 4, Revised; Jnury 29, 2009 omment on Slide 7

9 Multiply Out Using 2 nd Swp, efore Exmple from lst slide (continued) Originl Problem Multiply Out: F = ( + + )( + )( + + )( + )( + + ) st istributive Lw () x( + ) = x + x On lst slide, using 2 (or equ) nd Swp obtined ( + + ) ( + ) Use () Use () x +xy = x = ( + + ) + ( + + ) ollect terms = + + = + + = heck mp for further simplifictions re there ny? dig4fctoringh.fm p. 42 Revised; Jnury 29, 2009 Slide 8 Multiply Out Using 2 nd Swp, efore Exmple With Method II: Multiply out Exmple With Method II: Multiply out using 2 (or equ) nd Sw before using F = ( + )( + + )( + )( + + )( + + ) F d = ( ) + ( ) + ( ) + ( ) + ( ) Tke dul; F d = ( ) + ( ) + ( ) + ( ) + ( ) 0 use +X = F d = ( ) + ( ) + ( ) + ( ) F d = ( + ) + ( + ) F d = [ + ( + )][ + ( + )] F d = [ + + { }] [ + + { }] (dul) 2 =F = [( {+}] + [ ( {+}] Fctor out nd then using Use Swp rcket Ned terms (redy to tke dul) Reverse dul F = Use () This reduces to the sme function s the exmple on Slide 6 The Krnugh mp there shows there re no further simplifictions for the Σ of Π form F = PROLEM Multiply out. Remember to check for obvious simplifictions before strting. (W + Y + Z)(X + Y + Z)(W + X)(W + Z)(X + Y + Z)(W + X + Z) Hint: Tke the dul, simplify the dul nd use if possible, then tke the reverse dul, before you multiply out PROLEM Multiply out lgebriclly. The best nswer hs 8 letters. The hint from the lst problem my be useful. ( + )( + + )( + + )( + + )( + + ) rleton University dig4fctoringh.fm p. 43, Revised; Jnury 29, 2009 omment on Slide 8

10 Multiply Out Using emorgn nd K-Mps Method III: Using K-Mps Step : Find F using Generl emorgn Step 2: Plot F on K-Mp Step 3: Plot F using the 0 Squres Step 4: Get expression for F from mp Exmple: () Find the inverse using generl emorgn F = ( + + )( + )( + + )( + ) F dul = () + ( ) + () + () F = (2) Plot F on K-mp (3) Plot the F on K-mp; Put s where F ws 0 s. (4) ircle F on K-mp F = + + Mp of function, F Mp of function, F Mp of inverse, F dig4fctoringh.fm p. 44 Revised; Jnury 29, 2009 Slide 9 Multiply Out Using emorgn nd K-Mps The Esy Wy to Multiply Out The Esy Wy to Multiply Out F is esy to find using emorgn s lw. F is esy to find from K-mp of F. The bove re the two essentil fcts used for multiplying out using K-mp. When is the K-mp the best method? For most problems done by hnd is by fr the esiest wy to fctor. Six or more vribles will give problem too big for K-mp. Using lgebr my be esier if ll the vribles, or lmost ll, re different.see omment on Slide 5. lgebr my be esier for converting smll prtil expression inside long expression PROLEM: Multiply out using emorgn s Lw nd Krnugh mp, to get two terms of 2 letters nd one of 3 letters. ( + + )( + )( + + )( + )( + + ) 4-5. PROLEM (Solution to lter fctoring problem,nd n erlier lgebric problem) Multiply out to get n expression with eight letters. Use Krnugh mp. ( + )( + + )( + + )( + + )( + + ) 4-6. PROLEM (Remember Simplify, simplify, simplify!) multiply out problem using,,,,e nd F s vribles. ( + )( + + )(E + + )( + + )(E + + )( + F) rleton University dig4fctoringh.fm p. 45, Revised; Jnury 29, 2009 omment on Slide 9

11 Multiply Out: Using emorgn nd K-Mps Exmple Using Krnugh Mp Steps: () Given F = (Π of Σ expression) F = ( + )( + + )( + + )( + + ) (2) Invert F using emorgn s lw to get F s Σ of Π F = (3) Plot it on mp. (4) Mke mp for F, It hs where F hd 0 (5) ircle the F mp (6) Write out the eqution for F F = Mp of F Mp of F Mp of F dig4fctoringh.fm p. 46 Revised; Jnury 29, 2009 Slide 0 Multiply Out: Using emorgn nd K-Mps Multiply Out With d s Multiply Out With d s If some input combintions re never used, these become don t cre outputs. o the norml steps up through finding F () Mke the mp for F (2) Mke the mp for F from tht of F. Then identify the d s on the mp of F. Finlly circle the mp normlly to find the minimum Σ of Π expression PROLEM: Find the minimum Σ of Π expression for F using the don t cres to best dvntge. F = (W+X+Y)(W+X+Z)(X+Y+Z) The input combintions W XYZ, nd WXYZ never hppen, so these mp squres re d s. Z YZ WX d 0 X W 0 d Y Helpful mp 4-8. PROLEM: Find the minimum Σ of Π expression for G using the don t cres to best dvntge. G = ( + )( + + )( + + )( + + ) The input combintions, nd never hppen. rleton University dig4fctoringh.fm p. 47, Revised; Jnury 29, 2009 omment on Slide 0

12 Fctoring Fctoring, the ul of Multiplying Out hnge Σ of Π Π of Σ Exmple I ( ) + ( ) + ( ) ( )( )( ) Methods of Fctoring I. Use the 2nd distributive lw x + b = (x + )(x + b) - lwys works, but - very long nd slow. II. Tke the dul Use multiply-out method Tke the dul bck - hnges unfmilir 2 to fmilr - In theory the sme mount of work, but esier to grsp. III. Plot F on K-mp Plot F using the 0 squres Find F using emorgn - Esiest to do. - Gives the simplest Π of Σ nswer. - Very messy for 5 vribles or more. + + Use (2) = (+)(+) + Use (2) = [(+)(+) + ][(+)(+) + ] Use (2) = [(+) + ][(+) + ][(+) + ][(+) + ] Use (2) = [ + ][++ ][++ ][+ + ] nd += = [ + ][+ + ] Use x(x+y) = x Exmple II F = + + F = ( + )+ F = {( + )}+ F UL = {+ )} F UL = + (ul) 2 = F = (+)( + + ) Exmple III F = + Plot on Mp Get F from Mp F = + Use (em) F=( + )( + ) Use () Tke ul Use () Tke ul ck F F dig4fctoringh.fm p. 48 Revised; Jnury 29, 2009 Slide Fctoring Four methods of fctoring Four methods of fctoring I) Using 2 Using 2 mny times is the brute force wy. Unfortuntely students find 2 hrd to use, nd the expnsion my get very long. It helps to use simplifiction X (X+y) = X nd bsorption X (X+y) = Xy t every chnce, but these rules re lso more difficult thn their dul rules. If ll the letters re different, then ll one cn use is 2. II) Using dulity nd This in theory, is just s difficult s the previous method, but the more fmilir rules mkes it seem esier. II) Using bit, then use dulity nd, s in II) bove cn do some fctoring nd often helps t the strt. Exmple I bove, using before 2, cn be done in two lines. ++ =(+) + (using first) ={+}{+(+)} (using 2) III) Using Krnugh mps This is the esiest method for four or five inputs, it lwys gives the smllest nswer, it esily hndles don t cres, but it gets very messy for over five inputs. It is the method of choice for most smll problems. Three methods of multiplying out (compre with fctoring) Using Using mny times is the strightforwrd wy, is very esy to use. Unfortuntely the result cn get very long. Using simplifiction (X + Xy =X) nd bsorption (X + Xy = X y) frequently will help. Using 2 (or equ) nd Swp before using Use 2 or the equivlent (tke the dul nd use ) nd the Swp rule cn do initil consolidtion before using. In most cses one must use for finl clenup. Using Krnugh mps This is quite esy for four vribles, but more complex for over five input vribles. Esily hndles d s rleton University dig4fctoringh.fm p. 49, Revised; Jnury 29, 2009 omment on Slide

13 Fctoring Using 2 Method III: Using 2 Step : Simplify, Use 2 Repet: Step until done... Unless the problem is very simple the other methods will be esier. 2nd istributive Lw (2) X + cd = (X+c)(X+d) Exmple + X Use (2) ( + X) ( + X) Use (2) gin ( + X)( + X)( + X) Get extended (2) + X = ( + X)( + X)( + X) Exmple + Use (2) gin ( + ) ( + ) Use (2) gin, twice ( + ) ( + )( + ) ( + ) dig4fctoringh.fm p. 50 Revised; Jnury 29, 2009 Slide 2 Fctoring Using 2 Fctoring Using 2 Fctoring Using 2 The expression proven on the slide is: The extended (2) + X = ( + X)( + X)( + X) The dul is the extended () ( + + )X = X + X + X Exmple + ( + ) ( + ) (2) (2) (2) ( + ) ( + )( + ) ( + ) ( + )( + ) ( + ) ( + = ) If we hd strted by using the Swp rule, we would hve simpler nswer in one step PROLEM Fctor + + rleton University dig4fctoringh.fm p. 5, Revised; Jnury 29, 2009 omment on Slide 2

14 Fctoring Using ulity Method II: Fctoring Using ulity Step. Simplify nd use if possible ut for F dul Step 2. Tke the dul; get fctored, or semi fctored form Step 3. Multiply out the dul to get sum-of products. The right one Step 4. Tke the dul bck to get the fctored form. Fctoring Using ulity The expression to fctor is Σ of Π F = + + () Use F = ( + ) + (2) Tke its dul to get Π of Σ. F = [ ( + { })] + { } rcket LL N terms F UL = [ + ( { + })] { + } (3) Multiply out to get Σ of Π. See box F UL = (4) Tke the dul bck F = ( + )( + )( + )( + ) Get the desired Π of Σ. + + = F = ( + )( + )( + )( + ) Multiply Out etils F UL =[ + { + }] { + } () = [ + + ] { + } = [ + + ] () +[ + + ] bdx+bd=bd F UL = Multiplying out is bsed on (). Esier for people, thn fctoring bsed on (2). lgebr of one is the dul of the lgebr of the other dig4fctoringh.fm p. 52 Revised; Jnury 29, 2009 Slide 3 Fctoring Using ulity hnging Fctoring into Multiplying Out hnging Fctoring into Multiplying Out Fctoring is onverted to Multiplying Out, its ul Problem We tke fctoring problem which is confusing, becuse fctoring is bsed on (2). This lw is not fmilir high-school type lgebric lw nd is hrder to work with. In the dul spce, the dul expression is lredy fctored. The problem is trnsformed into multiplying out, which is bsed on the first distributive lw (). () is more fmilir, nd hence multiplying out is usully esier thn fctoring. Multiplying out in the dul spce does not give the nswer. One tke the dul to get the nswer. This will then be the fctored form of the originl expression PROLEM Show lgebriclly tht F = tkes only 8 letters or 2 gte inputs in fctored form. rleton University dig4fctoringh.fm p. 53, Revised; Jnury 29, 2009 omment on Slide 3

15 Fctoring Using ulity Method II: Fctoring Using ulity Exmple: equl F = First use () F = ( + ) + ( + ) F =[ ({ } + { })] + [ ({ } + )] Tke dul F dul = [ + ({ + } { + })] [+ ({ + } )] Multiply out = [ + ( + )( + )] [ + ( + )] Sw = ( + )) + ( + )( + ) Sw = ( + )) + ( + ) = First use () twice Put brkets round ll the N terms Minus 25% if you don t differentite between F nd F dul Rerrnge to use Swp Use Swp (+stuff)(+junk)= junk+ stuff Use Swp (+)(+)= + Use () F dul = heck on mp Tke dul bck OK on mp F = ( + + )( + + )( + + )( + + ) Fctored form -> F = ( + + )( + + )( + + )( + + ) mp of F dul dig4fctoringh.fm p. 54 Revised; Jnury 29, 2009 Slide 4 Fctoring Using ulity Method II: Fctoring In the ul Spce Method II: Fctoring In the ul Spce Exmple; Fctor F = F = ( +) + ( + ) F = ( +) + ( + ) F = [+ ( + )] [+( + )] F = [+{ ( + )}] [+({ } + { })] Put brckets round ll Ns redy to tke dul Tke the dul F d = [ {+( )}] + [ ({+} {+})] Remove extr brckets F d = {+( )} + {+} {+} Use () In Generl F d = + + {+} {+} Simplify, use F Use Swp d = + + [ + ] nd mybe Swp fter tking the Use () F d = dul F d = Mp shows no more simplifictions Tke the dul bck F = ( + ) ( + + )( + + )( + + ) 4-. PROLEM Fctor EF + E + E + EF Use () Use () Use Swp In Generl Simplify, use nd mybe Swp before tking the dul rleton University dig4fctoringh.fm p. 55, Revised; Jnury 29, 2009 omment on Slide 4

16 Fctoring Krnugh Mps nd emorgn Method III: Fctoring Using K-Mp Step. Plot function F on K-mp Step 2. Plot F by interchnging 0 on the mp. Step 3. ircle the mp to get F. Step 4. Write out the expression for F. Step 5. Use emorgn to get bck F in fctored form. Exmple: Given F = ( Σ of Π expression) F = () Plot it on mp. (2) Mke mp for F, It hs where F hd 0 (3) ircle the F mp (4) Write out the eqution for F F = (5) Invert F using emorgn s lw to get F s Π of Σ F = ( + )( + + )( + + )( + + ) Mp of F Mp of F Mp of F dig4fctoringh.fm p. 56 Revised; Jnury 29, 2009 Slide 5 Fctoring Krnugh Mps nd emorgn Method III: Fctoring Using Krnugh Method III: Fctoring Using Krnugh Mps This method is probbly the esiest, nd lest error prone, for up to four vribles. Five vribles is t lest twice the work of four. bove 5 it gets very messy. It is very esy to incorporte don t cres with this method PROLEM Fctor EF + E + E + EF Using Krnugh mp nd compre your nswer with the previous problem if you did it PROLEM Fctor Use Krnugh mp nd obtin the minimum Π of Σ expression PROLEM Show, using Krnugh mp, tht F = tkes only 8 letters or 2 gte inputs in fctored form. ompre with Problem 4-0. rleton University dig4fctoringh.fm p. 57, Revised; Jnury 29, 2009 omment on Slide 5

17 Fctoring Using K-Mps nd emorgn Fctoring Using Krnugh Mps Steps: Given F = ( Σ of Π expression) F = Not the minimum but it doesn t mtter () Plot it on mp. (2) Mke mp for F, It hs where F hd 0 (3) ircle the F mp (4) Write out the eqution for F F = + + (5) Invert F using emorgn s lw to get F s Π of Σ F = ( + )( + + )( + ) Mp of F Mp of F Mp of F dig4fctoringh.fm p. 58 Revised; Jnury 29, 2009 Slide 6 Fctoring Using K-Mps nd emorgn Method III: Fctoring Using Krnugh Exmple: Fctoring 5-Vrible Expression without using mp! Method II: uses initilly, then dulity nd swp before the finl clenup. Tke dul Multiply Out Fctor F = E Use F = ( + + E) + ( +) F = [ ( +{ } +{ E})] +[ ({ } +)] F dul = [ + ( { + } { + E})] [ + ({ + } )] F dul = [ { + } ] + [ { + } { + E}] F dul = [ { + } ] + [{ + } { + E}] [{ E} + { }] Use ut brckets round LL N terms + Use Swp Redy to use Swp Use Swp F dul = [ { + } ] + [{ E} + { }] F dul = + + E + Use () twice Tke dul bck get (dul) 2 = F = (++)(++)(+++E)(+++) rleton University dig4fctoringh.fm p. 59, Revised; Jnury 29, 2009 omment on Slide 6

18 Wrnings on t sy (+b+c)(e+b)(+d) is lredy Π of Σ lwys simplify. Look for x+xz = x, x + xy = x + Y, xy + xy = Y before nd fter ech step. Pick the best method: ) for 5 vribles or under use generl emorgn nd mp. ) for 6 vribles or more, use lgebr or find computer progrm. For multiplying out lgebriclly Tke the dul. Simplify nd use Tke the reverse dul. Look for complemented letters, use swp, nd simplify. Use for wht is left. For fctoring lgebriclly EFORE you tke the dul Simplify nd use Tke the dul use Look possible Swp, simplify Use nd simplify Tke the reverse dul Tking the dul The hrd prt is getting the brckets round the LL the N terms. Try: (+ + E ) + ( +) dig4fctoringh.fm p. 60 Revised; Jnury 29, 2009 Slide 7 Originl the slide bove sid, on t sy (+b+c)b(+d) is lredy Π of Σ." ctully it ws Π of Σ, nd the slide embrrsed the lecturer PROLEM Explin why the (+b+c)b(+d) is Π of Σ. Hint: tke the dul. Method III: Fctoring Using Krnugh 4-6. PROLEM ON ULS Tke the dul of F = ( + + E ) + ( + ) To check your nswer look in the omment on Slide 6 I ll bet you cn t get it right the first time without looking. rleton University dig4fctoringh.fm p. 6, Revised; Jnury 29, 2009 omment on Slide 7

19 dig4fctoringh.fm p. 62 Revised; Jnury 29, 2009 Slide 8 Method III: Fctoring Using Krnugh rleton University dig4fctoringh.fm p. 63, Revised; Jnury 29, 2009 omment on Slide 8

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 4721 Money and Banking Problem Set 2 Answer Key Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

Helicopter Theme and Variations

Helicopter Theme and Variations Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

More information

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

How To Network A Smll Business

How To Network A Smll Business Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

Integration by Substitution

Integration by Substitution Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00

Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00 COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

Lecture 5. Inner Product

Lecture 5. Inner Product Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

1.2 The Integers and Rational Numbers

1.2 The Integers and Rational Numbers .2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl

More information

Lec 2: Gates and Logic

Lec 2: Gates and Logic Lec 2: Gtes nd Logic Kvit Bl CS 34, Fll 28 Computer Science Cornell University Announcements Clss newsgroup creted Posted on we-pge Use it for prtner finding First ssignment is to find prtners Due this

More information

Solving BAMO Problems

Solving BAMO Problems Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

MODULE 3. 0, y = 0 for all y

MODULE 3. 0, y = 0 for all y Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)

More information

Boolean Algebra Part 1

Boolean Algebra Part 1 Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

FUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation

FUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does

More information

Words Symbols Diagram. abcde. a + b + c + d + e

Words Symbols Diagram. abcde. a + b + c + d + e Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

More information

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

More information

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required

More information

PROGRAMOWANIE STRUKTUR CYFROWYCH

PROGRAMOWANIE STRUKTUR CYFROWYCH PROGRAMOWANIE STRUKTUR CYFROWYCH FPGA r inż. Igncy Pryk, UJK Kielce Mteriły źrółowe:. Slies to ccompny the textbook Digitl Design, First Eition, by Frnk Vhi, John Wiley n Sons Publishers, 7, http://www.vhi.com.

More information

19. The Fermat-Euler Prime Number Theorem

19. The Fermat-Euler Prime Number Theorem 19. The Fermt-Euler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout

More information

6.2 Volumes of Revolution: The Disk Method

6.2 Volumes of Revolution: The Disk Method mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

More information

Learning Outcomes. Computer Systems - Architecture Lecture 4 - Boolean Logic. What is Logic? Boolean Logic 10/28/2010

Learning Outcomes. Computer Systems - Architecture Lecture 4 - Boolean Logic. What is Logic? Boolean Logic 10/28/2010 /28/2 Lerning Outcomes At the end of this lecture you should: Computer Systems - Architecture Lecture 4 - Boolen Logic Eddie Edwrds eedwrds@doc.ic.c.uk http://www.doc.ic.c.uk/~eedwrds/compsys (Hevily sed

More information

Repeated multiplication is represented using exponential notation, for example:

Repeated multiplication is represented using exponential notation, for example: Appedix A: The Lws of Expoets Expoets re short-hd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

Review guide for the final exam in Math 233

Review guide for the final exam in Math 233 Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

More information

How To Set Up A Network For Your Business

How To Set Up A Network For Your Business Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer

More information

Basic Analysis of Autarky and Free Trade Models

Basic Analysis of Autarky and Free Trade Models Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

More information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

Data replication in mobile computing

Data replication in mobile computing Technicl Report, My 2010 Dt repliction in mobile computing Bchelor s Thesis in Electricl Engineering Rodrigo Christovm Pmplon HALMSTAD UNIVERSITY, IDE SCHOOL OF INFORMATION SCIENCE, COMPUTER AND ELECTRICAL

More information

Small Businesses Decisions to Offer Health Insurance to Employees

Small Businesses Decisions to Offer Health Insurance to Employees Smll Businesses Decisions to Offer Helth Insurnce to Employees Ctherine McLughlin nd Adm Swinurn, June 2014 Employer-sponsored helth insurnce (ESI) is the dominnt source of coverge for nonelderly dults

More information

Answer, Key Homework 10 David McIntyre 1

Answer, Key Homework 10 David McIntyre 1 Answer, Key Homework 10 Dvid McIntyre 1 This print-out should hve 22 questions, check tht it is complete. Multiple-choice questions my continue on the next column or pge: find ll choices efore mking your

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values) www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

More information

Small Business Cloud Services

Small Business Cloud Services Smll Business Cloud Services Summry. We re thick in the midst of historic se-chnge in computing. Like the emergence of personl computers, grphicl user interfces, nd mobile devices, the cloud is lredy profoundly

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Protocol Analysis. 17-654/17-764 Analysis of Software Artifacts Kevin Bierhoff

Protocol Analysis. 17-654/17-764 Analysis of Software Artifacts Kevin Bierhoff Protocol Anlysis 17-654/17-764 Anlysis of Softwre Artifcts Kevin Bierhoff Tke-Awys Protocols define temporl ordering of events Cn often be cptured with stte mchines Protocol nlysis needs to py ttention

More information

3 The Utility Maximization Problem

3 The Utility Maximization Problem 3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best

More information

Distributions. (corresponding to the cumulative distribution function for the discrete case).

Distributions. (corresponding to the cumulative distribution function for the discrete case). Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive

More information

Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials

Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

baby on the way, quit today

baby on the way, quit today for mums-to-be bby on the wy, quit tody WHAT YOU NEED TO KNOW bout smoking nd pregnncy uitting smoking is the best thing you cn do for your bby We know tht it cn be difficult to quit smoking. But we lso

More information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 8 and 9 1 Rectangular waveguides 1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

More information

Basically, logarithmic transformations ask, a number, to what power equals another number?

Basically, logarithmic transformations ask, a number, to what power equals another number? Wht i logrithm? To nwer thi, firt try to nwer the following: wht i x in thi eqution? 9 = 3 x wht i x in thi eqution? 8 = 2 x Biclly, logrithmic trnformtion k, number, to wht power equl nother number? In

More information

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting

More information

Chapter. Contents: A Constructing decimal numbers

Chapter. Contents: A Constructing decimal numbers Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives

More information

Pure C4. Revision Notes

Pure C4. Revision Notes Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd

More information

Version 001 Summer Review #03 tubman (IBII20142015) 1

Version 001 Summer Review #03 tubman (IBII20142015) 1 Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This print-out should he 35 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03

More information

Unleashing the Power of Cloud

Unleashing the Power of Cloud Unleshing the Power of Cloud A Joint White Pper by FusionLyer nd NetIQ Copyright 2015 FusionLyer, Inc. All rights reserved. No prt of this publiction my be reproduced, stored in retrievl system, or trnsmitted,

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

Unit 6: Exponents and Radicals

Unit 6: Exponents and Radicals Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

More information

Week 11 - Inductance

Week 11 - Inductance Week - Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n

More information

NQF Level: 2 US No: 7480

NQF Level: 2 US No: 7480 NQF Level: 2 US No: 7480 Assessment Guide Primry Agriculture Rtionl nd irrtionl numers nd numer systems Assessor:.......................................... Workplce / Compny:.................................

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

Brillouin Zones. Physics 3P41 Chris Wiebe

Brillouin Zones. Physics 3P41 Chris Wiebe Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction

More information

VoIP for the Small Business

VoIP for the Small Business Reducing your telecommunictions costs VoIP (Voice over Internet Protocol) offers low cost lterntive to expensive trditionl phone services nd is rpidly becoming the communictions system of choice for smll

More information

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is

More information

6 Energy Methods And The Energy of Waves MATH 22C

6 Energy Methods And The Energy of Waves MATH 22C 6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this

More information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

More information

4 Approximations. 4.1 Background. D. Levy

4 Approximations. 4.1 Background. D. Levy D. Levy 4 Approximtions 4.1 Bckground In this chpter we re interested in pproximtion problems. Generlly speking, strting from function f(x) we would like to find different function g(x) tht belongs to

More information

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right. Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

More information

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, cross-clssified

More information

PHY 140A: Solid State Physics. Solution to Homework #2

PHY 140A: Solid State Physics. Solution to Homework #2 PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.

More information

Rotating DC Motors Part II

Rotating DC Motors Part II Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors

More information