A Formulry for Mthemtis A olletion of the Formuls, Fts nd Figures often needed in mthemtis These re some of the pges of the first rough drft of ooklet whih hs now een pulished It is in hndier A5 size, ontins twie s muh mteril s this, nd uses seond olour (red) s help in piking out the slient points on eh pge. In ddition, there is set of work-sheets sed on the ooklet imed t enourging fmilirity in its use, nd developing some mthemtil ides. Full detils n e found t www.leveooks.om
Index to Contents Squre, Olong, Cirle, Setor, Prllelogrm, Trpezium, Tringle 3 Right-Angled Tringle 4 Generl Tringle 5 Cue, Cuoid, Polyhedrons 6 Sphere, Cylinder, Pyrmid, Cone 7 Seonds or Minutes (of Angle or Time) into Deiml Frtions 8 Degrees & Compss Points 9 Alger 0 Clulus Sttistis Vlues of n C r 3 Vlues of N n (n = to 5) 4 Ares under Curve of Norml Distriution 5 Symols nd Arevitions 6 The Greek Alphet 7 Frnk Tpson 004 [trolfb:] Formulry
Squre Cirle e d e = edge length d = digonl length P = perimeter length A = re d r r = rdius length d = dimeter length C = irumferene length A = re e P = 4 e P = 4 A P = d C = π r C = π d C = A π A = e A = d A = P 6 A = π r A = π d 4 A = C 4 π d = e d = A d = P 4 d = r d = π A d = C π e = A e = P 4 e = d r = d r = π A r = C π P = ( + ) = P d = + d = d = Olong, = edge lengths d = digonl length P = perimeter length A = re = d P A = = A = A r l s r l = π r s 80 Setor s = setor ngle (in degrees) l = length of r r = rdius of irle A = re of setor A = π r s 360 A = r l r = 80 l π s r = A l l = A r s = 80 l π r s = 360 A π r e p p e A = p e Prllelogrm e = edge lengths of two prllel edges p = perpendiulr distne etween them A = re Tringle = se length p = perpendiulr height A = re A = p, = edge lengths of two prllel edges p = perpendiulr distne etween them A = re Trpezium Frnk Tpson 004 [trolfb:3] Formulry 3 p A = p ( + )
Right-Angled Tringle B A C = 90 Tke re to mth given dt to the orret letters Given Use the formul from the pproprite ox elow to find A B = + tn A = tn B = = sin A = os B = = os A = sin B = A ê = sin A B = 90 A B = tn B = os B A = 90 B A = tn A = os A B = 90 A B = tn B = sin B A = 90 B A = sin A = os A B = 90 A B = os B = sin B A = 90 B Frnk Tpson 004 [trolfb:4] Formulry 4
Generl Tringle B The semi-perimeter is given y s = ( + + ) whih is more usully written s s = + + A C is the symol for re Are = sin C or sin B or sin A or ss ( )( s )( s ) Sine Rule = = sin A sin B sin C Cosine Rule = + os A or os A = ( + ) = + os B or os B = ( + ) = + os C or os C = ( + ) B C Tngent Rule tn = ot + A Hlf-ngle Formuls A sin = ( s )( s ) A os = ss ( ) A tn = ( s )( s ) ss ( ) Insried Cirle Cirumsried Cirle r rdius r = s = Are R Rdius R = 4 R = or or sin A sin B sin C Esried Cirles r r r The different rdii needed for the three possile esried irles re identified y the letters of the edge on whih eh irle is pled r r r r = s r = s r = s All the ove formuls re yli Tht is, the six vriles (,,, A, B, C) n e hnged round s long s the pttern of the formul is kept. This is est seen in the Cosine Rule where ll three possile vritions re given, nd the pttern is ler. Frnk Tpson 004 [trolfb:5] Formulry 5
Cue Cuoid e e = edge length d = digonl length S = surfe re V = volume S = 6 e V = e 3 d = e 3 e = S 6 e d e 3 e = V e = d 3 3 S = 6 V S = d V = S 3 6 V = d 3 3 9 V = d V = d = + + S = ( + + ) V =,, = edge lengths d = digonl length S = surfe re V = volume V = d = e S Exept for vlues whih re ext, ll others re given to 6 signifint figures. Regulr Polyhedrons Assoited with ny regulr onvex polyhedron re two prtiulr spheres. A irumsphere is the sphere drwn round the outside of regulr onvex polyhedron so s s to touh every vertex of tht polyhedron. An insphere is the sphere drwn round the inside of regulr onvex polyhedron so s s to touh every fe of tht polyhedron. If the edge length of the polyhedron is e then re of the surfe of the polyhedron is given y e A-ftor volume of the polyhedron is given y e 3 V-ftor rdius of the irumsphere is given y e C-ftor rdius of the insphere is given y e I-ftor The neessry ftors re to e found in the tle elow. The size of the dihedrl ngle (in degrees) etween fes is lso given No. of Dihedrl fes Nme A-ftor V-ftor C-ftor I-ftor Angle 4 tetrhedron.7305 0.785 0.637 0.044 70.5333 6 ue 6 0.86605 0.5 90 8 othedron 3.4640 0.47405 0.70707 0.40848 09.467 dodehedron 0.6458 7.663.406.35 6.565 0 ioshedron 8.6605.870 0.95057 0.75576 38.90 Frnk Tpson 004 [trolfb:6] Formulry 6
Sphere r = rdius d = dimeter C = irumferene A = re of surfe V = volume C = π r or π d A = 4 π r or π d V = 4 π r 3 3 or π d 3 6 d = r or A 3 6V or π π A r = d or or π 3 3V 4π h d V = π r h C = π r h or T = π r (r + h) r Cylinder r = rdius d = dimeter h = height C = urved re (without ends) T = totl re (with ends) V = volume or π d h 4 or π d h or C r V r s h h = 3 V s = h + h= s l s Pyrmid (Right squre-sed) = se edge length s = slnt edge length h = perpendiulr height l = slnt height V = volume V = h 3 = 3V h = ( s h ) = ( l h ) h Cone (Right irulr) r = rdius of se irle d = dimeter of se h = perpendiulr height l = slnt height C = urved re (without se) V = volume V = π r h 3 or π d h C = π r l d r = d l = r + h h = l r r = l h l r l r = θ 3V π h l h = 3V π r The setor needed to mke one hving se rdius of r nd slnt height of l n e ut from irle with rdius of l nd setor ngle of θ where θ = 360 r l l = h + 4 h = l 4 Frnk Tpson 004 [trolfb:7] Formulry 7
Deiml Prts of 60 The equivlent vlues of seonds or minutes (of time or ngle) & deiml frtion of minute, hour or degree 0.95 0 0.05 0.9 0. 0.85 55 0 5 0.5 0.8 50 Seonds or Minutes 0 0. 0.75 45 5 0.5 0.7 40 0 0.3 0.65 35 5 30 0.35 0.6 0.4 0.55 0.45 0.5 Deiml Frtion of Minute, Hour or Degree Time Angle 60 seonds = minute 60 seonds = minute 60 minutes = hour 60 minutes = degree Time is written in the form hh:mm:ss exmple :34:06 Angle is written in the form d m' s'' exmple 3 4' 56'' Frnk Tpson 004 [trolfb:8] Formulry 8
The equivlent vlues of degrees & the points of the ompss N Degrees & Points of the Compss NNW NNE NW 30 340 0 360 0 40 NE WNW 300 Degrees 60 ENE 80 80 W 70 90 E 60 00 WSW 40 0 ESE 0 40 SW 00 80 60 SE SSW SSE S Frnk Tpson 004 [trolfb:9] Formulry 9
Alger Qudrti Equtions If x + x + = 0 then ± x = 4 If 4 > 0 there re two, rel, different roots. If 4 = 0 there is only one root. If 4 < 0 the roots re omplex. Indies m n m n = m + n = m n ( m ) n = m n n m = m n n n = n = n 0 = ( ) n = n n ( ) n = n n Expnsions & Ftoristions ( + ) = + + ( ) = + ( + ) 3 = 3 + 3 + 3 + 3 ( + ) 3 = 3 + 3 + 3 ( + ) ( ) 3 = 3 3 + 3 3 ( ) 3 = 3 3 3 ( ) = ( + )( ) 3 + 3 = ( + )( + ) 3 3 = ( )( + + ) 4 4 = ( + )( 3 + 3 ) 4 4 = ( )( 3 + + + 3 ) n + n is divisile y ( + ) when n is odd ut y ( ) never n n is divisile y ( + ) when n is even nd y ( ) lwys Logrithms If N = x then log N = x nd N = log N log ( ) log ( ) log n log n log N = log + log = log log = n log = n log = log N log log e N =.306 log 0 N log = 0 Arithmeti Progressions The generl form of n AP is, + d, + d, + 3d, + 4d, + (n )d where = the first term d = the ommon differene n = the numer of terms the lst term is l = + (n l)d the totl sum of n terms is S n = n( + ) or n[ + (n )d] Geometi Progressions The generl form of GP is, r, r, r 3, r 4, r 5, r n where = the first term r = the ommon rtio or multiplier n = the numer of terms the totl sum of n terms is S n = ( r n ) ( r) if r < or S n = (r n ) (r ) if r > if n is infinity nd r < then S = ( r) The geometri men of two numers nd = Sums of Powers of Nturl Numers The first n nturl numers re,, 3, 4, 5, 6, 7, n Their sum when eh hs een rised to the power r is Σn r = r + r + 3 r + 4 r + 5 r + 6 r + + n r For ny given vlue of r there is formul for Σn r The first six re (r = ) Σn = n(n + ) (r = ) Σn = n(n + )(n + ) 6 (r = 3) Σn 3 = n (n + ) 4 or (Σn) (r = 4) Σn 4 = n(n + )(n + )(3n + 3n ) 30 (r = 5) Σn 5 = n (n + ) (n + n ) (r = 6) Σn 6 = n(n + )(n + )(3n 4 + 6n 3 3n + ) 4 Comintions Given n different ojets nd required to hoose r t time, this formul gives the numer of wys in whih it n e done, negleting the order in whih they re hosen. n n! C r = (n r)! r! Given the importne of these numers in the Binomil Theorem elow, they re lso known s the Binomil Coeffiients. (see Tle of Vlues t the k) Binomil Theorem ( + ) n = n + n C n + n C n + n C 3 n 3 3 + + n C r n r r + + n Frnk Tpson 004 [trolfb:0] Formulry 0
Clulus funtion (st) derivtive integrl d dy f(x) or y = f(x) f (x) or f(x) or dx dx f(x)dx or ydx x n nx n n x + n + e x e x e x e x e x e x x log e x x x log e x x log e x x log e x x log e x + x tn x - x tnh x x - oth x + x sinh x x - osh x sin x os x os x os x sin x sin x tn x se x log e se x sin x x os x x tn x + x onstnts of integrtion hve not een shown Given tht u nd v re oth funtions of x Produt rule dy du dv if y = u v then = v + u dx dx dx Quotient rule dy du dv if y = u v then = v u v dx dx dx Chin rule dy dy du if y is funtion of u then = dx du dx Frnk Tpson 004 [trolfb:] Formulry
In sttistis, when the dt ontent is numeril, it is usul to use the symol x to represent the generl se, nd individul piees of dt s x x x 3 x 4 x 5 x 6 x 7 x n Another ommonly used symol is Σ (Greek sigm) whih mens find the sum of. So formul ontining Σx would men dd up ll the x-numers, nd Σx would men squre ll the x-numers nd dd up ll those vlues. The numer of piees of dt is given y n. If the dt is grouped, then f is used to refer to the frequeny of the dt in eh group nd tht would require hnge to some of the formuls given here. Arithmeti Men Generlly this is referred to simply s the men. Symol is x This my e found y Adding up the vlues of ll the dt Dividing y the numer of piees of dt x Expressed s formul it is x = n Rnge is the solute vlue of the differene etween the gretest nd lest vlues of the dt. Expressed s formul it is rnge = x mx x min Root Men Squre Vlue is given y x n Stndrd Devition This my e found y Squring the vlues of ll the dt Adding them ll up Dividing y how mny there re Sutrting the squre of the men vlue Tking the squre root. Symol is σ Expressed s formul it is σ = Vrine is the squre of the Stndrd Devition = σ x χ (hi-squred) Test For ny prtiulr piee of dt, if O is its Oserved frequeny nd E is its Expeted frequeny then ( O E) χ = E whih is the summtion rried out over ll the groups of the dt x n Sttistis Correltion Coeffiient More preisely it is Person s produt moment orreltion oeffiient Symol is r When the dt is in the form of ordered pirs of numers suh s (x, y) nd there re n suh pirs, then the mount of orreltion etween them n e determined y A. Multiplying the mthing x nd y vlues together, dding them ll up nd multiplying the totl y n B. Adding up ll x-vlues; dding up ll y-vlues; nd multiplying the two results together. C. Sutrting the result of B from A (It might e negtive) D. Squring ll x-vlues, dding them up, multiplying the totl y n. Repeting for y-vlues. E. Adding together ll x-vlues, nd squring the totl. Repeting for y-vlues. F. Sutrting the x-result in E from tht in D nd repeting tht for y-result. G. Multiplying the two nswers from F together nd tking the squre root. Then r = result from C result from G Expressed s formul it is r = n xy x y ( ) ( ) n x x n y y Stright Line Formul When the dt is in the form of ordered pirs of numers suh s (x, y) nd there is good degree of orreltion etween them (s determined ove) then it is possile, s well s useful, to drw stright line whih n serve s the sis of further lultions. The eqution for this line will e of the form y = mx + The neessry vlues of m nd n e found from nd ( ) n xy x y m = n x x y m n = Rnk Order Correltion Coeffiient More preisely it is Spermn s rnk order orreltion oeffiient Symol is ρ When two sets of dt hve een rnked in order y some riteri or other, this oeffiient is used to determine how losely the two lists gree (or differ). Given tht there re n items listed, it is found y Finding the differene in vlue (y their list order) of eh orresponding pir of rnkings. Squring ll the differenes. Adding the squred vlues together nd multiplying y 6 Dividing the previous result y (n 3 n) Sutrting tht from = ρ x Frnk Tpson 004 [trolfb:] Formulry
n n! C r = (n r)! r! Vlues of n C r n r 3 4 5 6 7 8 9 0 n 3 3 3 4 6 4 4 5 0 0 5 5 6 5 0 5 6 6 7 35 35 7 7 8 8 56 70 56 8 8 8 9 36 84 6 6 84 36 9 9 0 45 0 0 5 0 0 45 0 0 55 65 330 46 46 330 65 55 66 0 495 79 94 79 495 0 66 3 78 86 75 87 76 76 87 75 86 78 3 4 9 364 00 00 3 003 3 43 3 003 00 00 364 4 5 05 455 365 3 003 5 005 6 435 6 435 5 005 3 003 365 5 6 0 560 80 4 368 8 008 440 870 440 8 008 4 368 6 7 36 680 380 6 88 376 9 448 4 30 4 30 9 448 376 7 8 53 86 3 060 8 568 8 564 3 84 43 758 48 60 43 758 3 84 8 9 7 969 3 876 68 7 3 50 388 75 58 9 378 9 378 75 58 9 0 90 40 4 845 5 504 38 760 77 50 5 970 67 960 84 756 67 960 0 0 330 5 985 0 349 54 64 6 80 03 490 93 930 35 76 35 76 3 540 7 35 6 334 74 63 70 544 39 770 497 40 646 646 705 43 3 53 77 8 855 33 649 00 947 45 57 490 34 87 90 44 066 35 078 3 4 76 04 0 66 4 504 34 596 346 04 735 47 307 504 96 56 496 44 4 5 300 300 650 53 30 77 00 480 700 08 575 04 975 3 68 760 4 457 400 5 6 35 600 4 950 65 780 30 30 657 800 56 75 3 4 550 5 3 735 7 76 60 6 7 35 95 7 550 80 730 96 00 888 030 0 075 4 686 85 8 436 85 3 037 895 7 8 378 3 76 0 475 98 80 376 740 84 040 3 08 05 6 906 900 3 3 0 474 80 8 9 406 3 654 3 75 8 755 475 00 560 780 4 9 45 0 05 005 0 030 00 34 597 90 9 30 435 4 060 7 405 4 506 593 775 035 800 5 85 95 4 307 50 30 045 05 54 67 300 30 3 465 4 495 3 465 69 9 736 8 69 575 7 888 75 0 60 075 44 35 65 84 67 35 3 3 496 4 960 35 960 0 376 906 9 3 365 856 0 58 300 8 048 800 64 5 40 9 04 480 3 33 58 5 456 40 90 37 336 07 568 4 7 048 3 884 56 38 567 00 9 56 040 93 536 70 33 34 56 5 984 46 376 78 56 344 904 5 379 66 8 56 04 5 45 56 3 8 40 86 097 760 34 35 595 6 545 5 360 34 63 63 60 6 74 50 3 535 80 70 607 460 83 579 396 47 5 900 35 36 630 7 40 58 905 376 99 947 79 8 347 680 30 60 340 94 43 80 54 86 856 600 805 96 36 37 666 7 770 66 045 435 897 34 784 0 95 47 38 608 00 4 403 60 348 330 36 854 99 5 37 38 703 8 436 73 85 50 94 760 68 60 56 48 903 49 63 0 640 47 733 756 03 3 88 38 39 74 9 39 8 5 575 757 3 6 63 5 380 937 6 53 748 95 3 635 745 396 676 056 044 39 40 780 9 880 9 390 658 008 3 838 380 8 643 560 76 904 685 73 438 880 847 660 58 3 80 440 40 Frnk Tpson 004 [trolfb:3] Formulry 3
Powers of N N N N 3 N 4 N 5 N 6 4 8 6 3 64 3 9 7 8 43 79 4 6 64 56 04 4 096 5 5 5 65 3 5 5 65 6 36 6 96 7 776 46 656 7 49 343 40 6 807 7 649 8 64 5 4 096 3 768 6 44 9 8 79 6 56 59 049 53 44 0 00 000 0 000 00 000 000 000 33 4 64 6 05 77 56 44 78 0 736 48 83 985 984 3 69 97 8 56 37 93 4 86 809 4 96 744 38 46 537 84 7 59 536 5 5 3 375 50 65 759 375 390 65 6 56 4 096 65 536 048 576 6 777 6 7 89 4 93 83 5 49 857 4 37 569 8 34 5 83 04 976 889 568 34 0 4 9 36 6 859 30 3 476 099 47 045 88 0 400 8 000 60 000 3 00 000 64 000 000 44 9 6 94 48 4 084 0 85 766 484 0 648 34 56 5 53 63 3 379 904 3 59 67 79 84 6 436 343 48 035 889 4 576 3 84 33 776 7 96 64 9 0 976 5 65 5 65 390 65 9 765 65 44 40 65 6 676 7 576 456 976 88 376 308 95 776 7 79 9 683 53 44 4 348 907 387 40 489 8 784 95 64 656 7 0 368 48 890 304 9 84 4 389 707 8 0 5 49 594 83 3 30 900 7 000 80 000 4 300 000 79 000 000 3 96 9 79 93 5 8 69 5 887 503 68 3 04 3 768 048 576 33 554 43 073 74 84 33 089 35 937 85 9 39 35 393 9 467 969 34 56 39 304 336 336 45 435 44 544 804 46 35 5 4 875 500 65 5 5 875 838 65 65 36 96 46 656 679 66 60 466 76 76 78 336 37 369 50 653 874 6 69 343 957 565 76 409 38 444 54 87 085 36 79 35 68 3 00 936 384 39 5 59 39 33 44 90 4 99 3 58 743 76 40 600 64 000 560 000 0 400 000 4 096 000 000 4 68 68 9 85 76 5 856 0 4 750 04 4 4 764 74 088 3 696 30 69 3 5 489 03 744 43 849 79 507 3 48 80 47 008 443 6 3 363 049 44 936 85 84 3 748 096 64 96 4 7 56 33 856 45 05 9 5 4 00 65 84 58 5 8 303 765 65 46 6 97 336 4 477 456 05 96 976 9 474 96 896 47 09 03 83 4 879 68 9 345 007 0 779 5 39 48 304 0 59 5 308 46 54 803 968 30 590 464 49 40 7 649 5 764 80 8 475 49 3 84 87 0 50 500 5 000 6 50 000 3 500 000 5 65 000 000 Frnk Tpson 004 [trolfb:4] Formulry 4
Ares under Curve of Norml Distriution 3 34 345 3456 3456 34567 345678 345678 3456789 3456789 3456789 3456789 3456789 3456789 3456789 3456789 3456789 3456789 3456789 3456789 0 z The tle gives the frtion of the totl re under the urve for the shded re shown, whih lies etween the middle ordinte (the men) nd the ordinte t z for vlues of z from 0.00 to 3.99 (All vlues rounded to 4 deiml ples) z 0 3 4 5 6 7 8 9 0.0 0.0000 0.0040 0.0080 0.00 0.060 0.099 0.039 0.079 0.039 0.0359 0. 0.0398 0.0438 0.0478 0.057 0.0557 0.0596 0.0636 0.0675 0.074 0.0754 0. 0.0793 0.083 0.087 0.090 0.0948 0.0987 0.06 0.064 0.03 0.4 0.3 0.79 0.7 0.55 0.93 0.33 0.368 0.406 0.443 0.480 0.57 0.4 0.554 0.59 0.68 0.664 0.700 0.736 0.77 0.808 0.844 0.879 0.5 0.95 0.950 0.985 0.09 0.054 0.088 0.3 0.57 0.90 0.4 0.6 0.58 0.9 0.34 0.357 0.389 0.4 0.454 0.486 0.58 0.549 0.7 0.580 0.6 0.64 0.673 0.704 0.734 0.764 0.794 0.83 0.85 0.8 0.88 0.90 0.939 0.967 0.996 0.303 0.305 0.3078 0.306 0.333 0.9 0.359 0.386 0.3 0.338 0.364 0.389 0.335 0.3340 0.3365 0.3389.0 0.343 0.3438 0.346 0.3485 0.3508 0.353 0.3554 0.3577 0.3599 0.36. 0.3643 0.3665 0.3686 0.3708 0.379 0.3749 0.3770 0.3790 0.380 0.3830. 0.3849 0.3869 0.3888 0.3907 0.395 0.3944 0.396 0.3980 0.3997 0.405.3 0.403 0.4049 0.4066 0.408 0.4099 0.45 0.43 0.447 0.46 0.477.4 0.49 0.407 0.4 0.436 0.45 0.465 0.479 0.49 0.4306 0.439.5 0.433 0.4345 0.4357 0.4370 0.438 0.4394 0.4406 0.448 0.449 0.444.6 0.445 0.4463 0.4474 0.4484 0.4495 0.4505 0.455 0.455 0.4535 0.4545.7 0.4554 0.4564 0.4573 0.458 0.459 0.4599 0.4608 0.466 0.465 0.4633.8 0.464 0.4649 0.4656 0.4664 0.467 0.4678 0.4686 0.4693 0.4699 0.4706.9 0.473 0.479 0.476 0.473 0.4738 0.4744 0.4750 0.4756 0.476 0.4767.0 0.477 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.48 0.487. 0.48 0.486 0.4830 0.4834 0.4838 0.484 0.4846 0.4850 0.4854 0.4857. 0.486 0.4864 0.4868 0.487 0.4875 0.4878 0.488 0.4884 0.4887 0.4890.3 0.4893 0.4896 0.4898 0.490 0.4904 0.4906 0.4909 0.49 0.493 0.496.4 0.498 0.490 0.49 0.495 0.497 0.499 0.493 0.493 0.4934 0.4936.5 0.4938 0.4940 0.494 0.4943 0.4945 0.4946 0.4948 0.4949 0.495 0.495.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.496 0.496 0.4963 0.4964.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.497 0.497 0.4973 0.4974.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.498.9 0.498 0.498 0.498 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986 3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990 3. 0.4990 0.499 0.499 0.499 0.499 0.499 0.499 0.499 0.4993 0.4993 3. 0.4993 0.4993 0.4994 0.4994 0.4994 0.4994 0.4994 0.4995 0.4995 0.4995 3.3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4996 0.4996 0.4997 3.4 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4998 3.5 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 3.6 0.4998 0.4998 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 3.7 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 3.8 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 3.9 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 Frnk Tpson 004 [trolfb:5] Formulry 5
Symols nd Arevitions Mthemtis uses mny symols nd revitions to represent instrutions, or numers, in more onise form. Here, with rief note s to their mening, re the ones most ommonly used. see lso The Greek Alphet + dd or plus or positive minus or sutrt or negtive ~ find the solute differene of times or multiplied y * times or multiplied y divided y / divided y ± dd or sutrt plus or minus positive or negtive = equls or is equl to does not equl or is not equl to is pproximtely equl to is equivlent to or hs the sme vlue s is identilly equl to is ongruent to < is less thn! is less thn or equl to > is greter thn " is greter thn or equl to vries s or is proportionl to : proportion. deiml (or frtion) point, deiml mrker % per ent or out of hundred per mil or out of thousnd ( ) rkets or prentheses ngle rkets [ ] squre rkets { } urly rkets or res lso used to enlose set [x] the lrgest whole numer whih is not greter thn x x the solute vlue of x x ² x squred x 3 x ued x n x to the nth power x the squre root of x 3 x the ue root of x ngle! is prllel to! is not prllel to is perpendiulr to degrees minutes seonds " the set of nturl numers # the set of whole numers $ the set of rtionl numers % the set of rel numers & the set of omplex numers is memer of is not memer of is suset of is not suset of inludes union intersetion null or empty set implies is implied y implies nd is implied y therefore infinity n! ftoril n!n su-ftoril or derngements of n i squre root of e.788 π 3.459 f(x) funtion of x f (x) first derivtive of f(x) integrl or nti-derivtive & hexdeiml numer follows AP APR u d p g d h f l d l m m mod QED s f sq UT rithmeti progression nnul perentge rte ui (referring to units of volume) deiml ples gretest ommon denomintor highest ommon ftor lowest ommon denomintor lowest ommon multiple grdient of line modulus whih ws to e proved signifint figures squre (referring to units of re) Universl Time (Greenwih Men Time) Frnk Tpson 004 [trolfb:6] Formulry 6
The Greek Alphet The Greek lphet is rih soure of symols used in oth mthemtis nd siene, to the extent tht nerly every one of them (oth pitls nd lower se) is used in some wy or other. Some of them pper more thn one to represent different things. Below is the full lphet, nd the nmes of the vrious symols. The pitl form of the letter is given in the first olumn, followed y the lower se version nd its nme. Then some of the more ommonly seen menings of usge re given. Α α lph α β γ re often used to identify ngles in plne figures. Β β Γ γ et gmm δ delt is sometimes used to represent the re of plne figure. δ is used (in lulus) to show tht smll mount is onsidered. Ε ε Ζ ζ Η η epsilon zet et Θ θ thet θ is used to indite generl ngle Ι ι iot Κ κ kpp Λ λ lmd λ is used to represent slr in vetor work Μ µ mu µ is used (in the SI system) to represent the prefix miro µ is sometimes used to represent the rithmeti men Ν ν nu Ξ ξ xi ξ is sometimes used s the symol for the universl set Ο ο omiron Π π pi Π is used to show tht ontinued produt is needed π is used to represent the vlue of the irrtionl numer 3.459 π(n) mens the numer of primes less thn, or equl to n Ρ ρ rho Σ σ sigm Σ is used to show tht the sum of series is to e found σ is used to represent the stndrd devition of popultion Τ τ tu τ is used to represent the golden rtio.680 (see lso phi) Υ υ upsilon Φ φ phi Φ is sometimes used s the symol for the empty set φ is used to represent the golden rtio.680 (see lso tu) φ (n) mens the numer of positive integers less thn, nd reltively prime to, n Χ χ hi χ is used in sttistis in referene to the hi-squred test Ψ ψ Ω ω psi omeg Frnk Tpson 004 [trolfb:7] Formulry 7