velo reeee 1

8/17/2019 Velo Reeee 1

1/16

V33d44cM@th33m@t44cs -M33thlldsPr33f@c33 ------------------------------------------------------------------------------------------------ 1

44. Why V33d44c M@th33m@t44cs? --------------------------------------------------------------------------

4444. V33d44c M@th33m@t44c@l Fllrmzzl@33 -----------------------------------------------------------51. 33k@dh44k33n@ Pzzrv33n@ --------------------------------------------------------------------------------

2. N44kh44l@m n@v@t@sc@r@m@m @s@t@h ------------------------------------------------------

3. zzrdhv@ - t44ry@"#hy@m ----------------------------------------------------------------------------------

4. P@r@v@rty@ $ll%@y33t --------------------------------------------------------------------------------------

5. &zzny@m &@my@ &@mzzcc@y33 -------------------------------------------------------------------------

'. @nzzrzz(y33 - &zzny@m@ny@t ----------------------------------------------------------------------------

7. &@nk@l@n@ - Vy@v@k@l@n@#hy@m -----------------------------------------------------------------

!. Pzzr@n@(zzr@n@#hy@m ------------------------------------------------------------------------------------

). *@l@n@ - +@l@n@#hy@m -------------------------------------------------------- -------------------------

1,. 33k@nyzzn33n@ Pzzrv33n@ ------------------------------------------------------------------------------

11. @nzzrzz(y33n@ ---------------------------------------------------------------------------------------------- 7

12. @dy@m@dy33n@nty@ - m@nty33n@ -----------------------------------------------------------------

13. $@v@dzzn@m @v@dzzn44krty@ V@r"@nc@ $ll%@y33t --------------------------------------

14. @nty@yllr @s@k33(44 ------------------------------------------------------------------------------------

15. @nty@yllr33v@ --------------------------------------------------------------------------------------------- )'

1'. ll(@n@ &th@(@n@#hy@m ---------------------------------------------------------------------------- 1

17. V44lllk@n@m ----------------------------------------------------------------------------------------------- 1,

1!. /zzn44t@ &@mzzcc@y@h 0 &@mzzcc@y@ /zzn44t@h -----------------------------------------

444444 V33d44c M@th33m@t44cs - @ #r4433f44n" --------------------------------------------------

1. 33rms @nd ll(33r@t44llns ---------------------------------------------------------------------------------

2. @dd44t44lln @nd &zz#tr@ct44lln --------------------------------------------------------------------------


2/16

3. Mzzlt44(l44c@t44lln ---------------------------------------------------------------------------------------- 13

4. 44v44s44lln ----------------------------------------------------------------------------------------------- 144

5. M44sc33ll@n33llzzs 44t33ms ------------------------------------------------------------------------------

44V *llnclzzs44lln --------------------------------------------------------------------------------------------- 15!


3/16

1

Pr33f@c33

Th33 S@nskr44t wllrd V33d@ 44s d33r44v33d frllm th33 rllllt V44d, m33@n44ng tll w44thllzzt l44m44t. Th33 wllrd V33d@ cllv33rs @ll V33d@-s@kh@s knllwn tll hzzm@V33d@ 44s @ r33plls44tllry llf @ll knllwl33dg33, f@thllml33ss, 33v33r r33v33@l44ng

d33lv33d d3333p33r.

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th33 pr44nc44pl33s cllnt@44n33d 44n th33 @phllr44sms @nd th3344r cllrllll@r4433V33d44c *@th33m@t44cs.

@ccllrd44ng tll h44m, th33r33 h@s '3333n cllns44d33r@'l33 l44t33r@tzzr33 lln *@

th33 V33d@-s@kh@s. zznfllrtzzn@t33ly mllst llf 44t h@s '3333n lllst tll hzzm@n44tynllw. Th44s 44s 33v44d33nt frllm th33 f@ct th@t wh44l33, 'y th33 t44m33 llf P@t@ c33ntzzr4433s @gll, 3 V33d@-s@kh@s w33r33 knllwn tll th33 V33d44c schlll@@'llzzt t33n V33d@-s@kh@s @r33 pr33s33ntly 44n th33 knllwl33dg33 llf th33 V33d

44n th33 cllzzntry.

Th33 Szztr@s @pply tll @nd cllv33r @lmllst 33v33ry 'r@nch llf *@th33m@t44cs. Th@pply 33v33n tll cllmpl33 prll'l33ms 44nvlllv44ng @ l@rg33 nzzm'33r llf m@th33m

llp33r@t44llns. @ppl44c@t44lln llf th33 Szztr@s s@v33s @ lllt llf t44m33 @nd 33ffllrth33 prll'l33ms, cllmp@r33d tll th33 fllrm@l m33thllds pr33s33ntly 44n vllgzz33. Thth33 slllzzt44llns @pp33@r l44k33 m@g44c, th33 @ppl44c@t44lln llf th33 Szztr@s 4

lllg44c@l @nd r@t44lln@l. Th33 cllmpzzt@t44lln m@d33 lln th33 cllmpzzt33rs fllllllwsw@y, th33 pr44nc44pl33s zznd33rly44ng th33 Szztr@s. Th33 Szztr@s prllv44d33 nll

m33thllds llf c@lczzl@t44lln, 'zzt @lsll w@ys llf th44nk44ng fllr th3344r @ppl44c@t4

Th44s 'llllk lln V33d44c *@th33m@t44cs s3333ks tll pr33s33nt @n 44nt33gr@t33d l33@rn44ng *@th33m@t44cs w44th k3333nn33ss llf ll's33rv@t44lln @nd 44n)zz44

@vll44d44ng th33 mllnlltllny llf @cc33pt44ng th33llr4433s @nd wllrk44ng frllm th33mm33ch@n44c@lly. Th33 33pl@n@t44llns llff33r33d m@k33 th33 prllc33ss33s cl33@l33@rn33rs. Th33 lllg44c@l prllllf llf th33 Szztr@s 44s d33t@44l33d 44n @lg33'r@, 33l44m44n@t33s th33 m44scllnc33pt44lln th@t th33 Szztr@s @r33 @ +zzggl33ry.

@ppl44c@t44lln llf th33 Szztr@s 44mprllv33s th33 cllmpzzt@t44lln@l sk44lls llf th33@ w44d33 @r33@ llf prll'l33ms, 33nszzr44ng 'llth sp3333d @nd @cczzr@cy, str44cr@t44lln@l @nd lllg44c@l r33@slln44ng. Th33 knllwl33dg33 llf szzch m33thllds 33n@

t33@ch33rs tll '33 mllr33 r33sllzzrc33fzzl tll mllzzld th33 stzzd33nts @nd 44mprllv33t@l33nt @nd cr33@t44v44ty. @ppl44c@t44lln llf th33 Szztr@s tll sp33c44f44c prll'l3r@t44lln@l th44nk44ng, wh44ch, 44n th33 prllc33ss, h33lps 44mprllv33 44ntzz44t44'llttllm - l44n33 llf th33 m@st33ry llf th33 m@th33m@t44c@l g33n44zzs33s llf th33pr33s33nt szzch @s @ry@'h@tt@, Bh@sk@r@ch@ry@, Sr44n44v@s@ /@m@nzz+@


4/16

2

Th44s 'llllk m@k33s zzs33 llf th33 Szztr@s @nd Szz'-Szztr@s st@t33d @'llv33 fllrpr33s33nt@t44lln llf th3344r @ppl44c@t44lln fllr l33@rn44ng *@th33m@t44cs @t tschlllll l33v33l 44n @ w@y d44ff33r33nt frllm wh@t 44s t@zzght @t pr33s33nt, 'zzt33m'lldy44ng th33 pr44nc44pl33s llf @lg33'r@ fllr 33mp44r44c@l @cczzr@cy. Th33

th33 pr33s33nt@t44lln 44s th33 @lg33'r@44c prllllf fllr 33v33ry 33lzzc44d@t44lln llth33 Szz'-Szztr@ cllnc33rn33d.

Sr44 S@thy@ S@44 V33d@ Pr@t44sht@n


5/16

3

44. 0hy V33d44c *@th33m@t44cs1

*@ny 44nd44@n S33cllnd@ry Schlllll stzzd33nts cllns44d33r *@th33m@t44cs @ v3

szz'+33ct. Sllm33 stzzd33nts 33ncllzznt33r d44ff44czzlty w44th '@s44c @r44thm33llp33r@t44llns. Sllm33 stzzd33nts f3333l 44t d44ff44czzlt tll m@n44pzzl@t33 sym'll33)zz@t44llns. 44n llth33r wllrds, @'str@ct @nd lllg44c@l r33@slln44ng 44s th3344

*@ny szzch d44ff44czzlt4433s 44n l33@rn44ng *@th33m@t44cs 33nt33r 44ntll @ 'y @n 33p33r4433nc33d t33@ch33r llf *@th33m@t44cs. Vlllzzm33s h@v33 '3333d44@gnlls44s llf 2l33@rn44ng d44ff44czzlt4433s2 r33l@t33d tll *@th33m@t44cs @nt33chn44)zz33s. 33@rn44ng *@th33m@t44cs 44s @n zznpl33@s@nt 33p33r443stzzd33nts m@44nly '33c@zzs33 44t 44nvlllv33s m33nt@l 3333rc44s33.

llf l@t33, @ f33w t33@ch33rs @nd schlll@rs h@v33 r33v44v33d 44nt33r33st 44n V3*@th33m@t44cs wh44ch w@s d33v33lllp33d, @s @ syst33m d33r44v33d frllm V33

'y Sw@m44 Bh@r@t44 Kr44shn@ T44rth@+44 44n th33 33@rly d33c@d33s llf th33 r. 5@r44nd33r Pzzr44 llf th33 /llllrk33 zzn44v33rs44ty pr33p@r33d t33@ch44ng mlln V33d44c *@th33m@t44cs dzzr44ng "!# - !". @ f33w llf h44s llp44n44llns @r33h33r33zznd33r6

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33l33m33nt llf chll44c33 @nd fl3344'44l44ty @t 33@ch st@g33 k3333ps th33 m44tll d33v33lllp cl@r44ty llf thllzzght @nd 44ntzz44t44lln, @nd th33r33'y @ hlll44st44cllf th33 hzzm@n 'r@44n @zztllm@t44c@lly t@k33s pl@c33.

v% V33d44c *@th33m@t44cs w44th 44ts sp33c44@l f33@tzzr33s h@s th33 44n'zz4slllv33 th33 psychlllllg44c@l prll'l33m llf *@th33m@t44cs - @n4433ty.

&.T.7lllv33r (llndlln, ""% s@ys th@t th33 33p33r4433nc33 llf t33@ch44ng V33d4*@th33m@t44cs2 m33thllds tll ch44ldr33n h@s shllwn th@t @ h44gh d33gr3333 llf

m@th33m@t44c@l @'44l44ty c@n '33 @tt@44n33d frllm @n 33@rly st@g33 wh4433n+lly33d fllr 44ts llwn m33r44ts.


6/16

4

@.P. 544chlll@s ("!4% pzzts th33 V33d44c *@th33m@t44cs syst33m @s 2lln33 llf t

d33l44ghtfzzl ch@pt33rs llf th33 $th c33ntzzry m@th33m@t44c@l h44stllry2.

Prllf. /.8. 7zzpt@ (""4% s@ys 2th33 syst33m h@s gr33@t 33dzzc@t44lln@l v@lzz33th33 Szztr@s cllnt@44n t33chn44)zz33s fllr p33rfllrm44ng sllm33 33l33m33nt@ry m

llp33r@t44llns 44n s44mpl33 w@ys, @nd r33szzlts @r33 ll't@44n33d )zz44ckly2.

Prllf. &.5. K@pzzr s@ys 2V33d44c *@th33m@t44cs c@n '33 zzs33d tll r33mllv33 m@phll'44@, @nd c@n '33 t@zzght tll (schlllll% ch44ldr33n @s 33nr44chm33nt m@t33rw44th llth33r h44gh sp3333d m33thllds2.

r. *44ch@33l 03344nl33ss, 8h@44rm@n llf th33 33p@rtm33nt llf *@th33m@t4*.44.zz, 44llw@ s@ys thzzs6 2V33d44c *@th33m@t44cs 44s 33@s4433r tll l33@rn, l33ss prlln33 tll 33rrllr th@n cllnv33nt44lln@l m33thllds. 9zzrth33rmllr33, th33 t33ch

llf V33d44c *@th33m@t44cs nllt llnly 33n@'l33 th33 stzzd33nts tll slllv33 sp33c44f4m@th33m@t44c@l prll'l33ms: th33y @lsll d33v33lllp cr33@t44v44ty, lllg44c@l th4444ntzz44t44lln.2

K3333p44ng th33 @'llv33 ll's33rv@t44llns 44n v4433w, l33t zzs 33nt33r V33d44c *g44v33n 'y Sr44 Bh@r@t44 Kr44shn@ T44rth@+44 (!!4 - "#$%, S@nk@r@ch@ry7llv@rdh@n@ *@th, Pzzr44. 33nt33r44ng 44ntll th33 m33thllds @nd prllc33dzzr33sr33@l44z33 th33 44mpllrt@nc33 @nd @ppl44c@'44l44ty llf th33 d44ff33r33nt fllrm

m33thllds.


7/16

5

4444. V33d44c *@th33m@t44c@l 9llrmzzl@33

0h@t w33 c@ll V33448 *@T33*@T448S 44s @ m@th33m@t44c@l 33l@'llr@t44S44t3333n

S44mpl33 *@th33m@t44c@l fllrmzzl@33 frllm th33V33d@s2 @s 'rllzzght llzzt 'ySr44

Bh@r@t44 Kr44shn@ T44rth@+44. 44n th33 t33t @zzthllr33d 'y th33 Sw@m44+44, nllwh33r

th33 l44st llf th33 *@th33m@t44c@l fllrmzzl@33 (Szztr@s% '3333n g44v33n. Bzzt tht33t h@s cllmp44l33d th33 l44st llf th33 fllrmzzl@33 frllm str@y r33f33r33nc33s 44nl44st sll cllmp44l33d cllnt@44ns S44t3333n Szztr@s @nd Th44rt3333n Szz' - Szztr@h33r33zznd33r.

S44;T33335 SzzT/@S


8/16

'

T44/T33335 SzzB < SzzT/@S

44n th33 t33t, th33 wllrds Szztr@, @phllr44sm, fllrmzzl@ @r33 zzs33d synllnymllzzs@lsll th33 wllrds zzp@-szztr@, Szz'-szztr@, Szz'-fllrmzzl@, cllrllll@ry zzs33d.

5llw w33 sh@ll h@v33 th33 l44t33r@l m33@n44ng, cllnt33tzz@l m33@n44ng, prllc3m33thllds llf @ppl44c@t44lln @lllng w44th 33@mpl33s fllr th33 Szztr@s. 33pl@n@tm33thllds, fzzrth33r shllrt-czzts, @lg33'r@44c prllllf, 33tc fllllllw. 0h@t fllllllws r33l@t@ s44ngl33 fllrmzzl@ llr @ grllzzp llf fllrmzzl@33 r33l@t33d tll th33 m33thllds llf V33d

*@th33m@t44cs.


9/16

7

. 33k@dh44k33n@ Pzzrv33n@

Th33 Szztr@ (fllrmzzl@% 33k=dh44k33n@ P>rv33n@ m33@ns6 ?By lln33 mllr33 th@n

pr33v44llzzs lln33.

44% S)zz@r33s llf nzzm'33rs 33nd44ng 44n 6

5llw w33 r33l@t33 th33 szztr@ tll th33 As)zz@r44ng llf nzzm'33rs 33nd44ng 44n . 8

33@mpl33 .

33r33 th33 nzzm'33r 44s . 033 h@v33 tll f44nd llzzt th33 s)zz@r33 llf th33 nzzmnzzm'33r , th33 l@st d44g44t 44s @nd th33 2pr33v44llzzs2 d44g44t 44s . 33ncth@n th33 pr33v44llzzs lln332, th@t 44s, CD3. Th33 Szztr@, 44n th44s cllnt33t, gprllc33dzzr332tll mzzlt44ply th33 pr33v44llzzs d44g44t 'y lln33 mllr33 th@n 44ts33l

44t '33cllm33s th33 ..S (l33ft h@nd s44d33% llf th33 r33szzlt, th@t 44s, ; 3 D #(r44ght h@nd s44d33% llf th33 r33szzlt 44s, th@t 44s, .

Thzzs D ; 3 E D #.

44n th33 s@m33 w@y,

352= 3 X (3+1) /25 = 3 X 4/ 25 = 1225;

652= 6 X 7 / 25 = 4225;

1052= 10 X 11/25 = 11025;

1352

= 13 X 14/25 = 18225;

@pply th33 fllrmzzl@ tll f44nd th33 sqzz@r33s llf th33 nzzmb33rs 15, 45, 85, 125, 175 @nd v33r44fy th

@lg33'r@44c prllllf6

@) Cllns44d33r (@x + b)2 F @

2. x

2+ 2@bx + b

2.

Th44s44d33nt44ty fllr x = 10 @nd b = 5 b33cllm33s

(10@ + 5)2= @

2. 10

2 + 2. 10@ . 5 + 5

2

= @2. 10

2+ @.10

2+ 5

2

= (@2+ @ ) . 10

2+5

2


10/16

!

= @ (@ + 1) . 102+ 25.

Cl33@rly 10@ + 5 r33pr33s33nts twll-d44g44t nzzmb33rs 15, 25, 35, -------,95 fllr th33 v@lzz33s @-----,9 r33sp33ct44v33ly. 44n szzch @ c@s33 th33 nzzmb33r (10@ + 5)

244s llf th33 fllrm whlls33 L.H.S 44s @ (@ + 1

@nd R.H.S 44s 25, th@t 44s, @ (@ + 1) / 25.

Thzzs @ny szzch twll d44g44t nzzmb33r g44v33s th33 r33szzlt 44n th33 s@m33 f@sh44lln.

33x@mpl33: 45 = (40 + 5)2, 44t 44s llf th33 fllrm (@x+b)

2 fllr @ = 4, x=10

@nd b = 5. g44v44ng th33 @nsw33r @ (@+1) / 25th@t 44s, 4 (4+1) / 25 + 4 X 5 / 25 = 2025.

'% @ny thr3333 d44g44t nzzm'33r 44s llf th33 fllrm @C'Cc fllr D$, @ G $, @, ', c H0.

Nllw (@x2+bx+ c)

2= @

2x

4+ b

2x

2+ c

2+ 2@bx

3 + 2bcx + 2c@x

2

= @2x

[email protected]

3+(b

2+ 2c@)x

2+2bc . x+ c

2.

Th44s 44d33nt44ty fllr D $, c D '33cllm33s (@ . $ C ' .$ C %

D @.$4C .@.'.$3 C ('C ..@%$C .'..$ C

D @.$4C .@.'.$3 C (' C $ @%$ C '.$C

D @.$4C @'.$3C '.$C @ . $3 C ' $C

D @.$4 C (@' C @%.$3 C ('C '%$ C

= [ @2.10

[email protected] + @.10 + b

2+ b] 10

2+ 5

2

D ($@ C '% ( $@C'C%.$ C


11/16

)

D P (PC% $C , wh33r33 P D $@C'.

33nc33 @ny thr3333 d44g44t nzzm'33r whlls33 l@st d44g44t 44s g44v33s th33 s

(@% fllr PD$@ C ', th33 Apr33v44llzzs llf .

33@mpl33 6 #D ( . $ C # . $ C % .

44t 44s llf th33 fllrm (@C'Cc%fllr @ D , ' D #, c D @nd D $. 44t g44v33s th33@nsw33r P(PC% E , wh33r33 P D $@ C ' D $ ; C # D #, th33 Apr33v44llzzs.

@nsw33r 44s # (#C% E D # ; I E D I.

@pply 33k@dh44k33n@ pzzrv33n@tll f44nd th33 s)zz@r33s llf th33 nzzm'33rs ", ,

3I, #3, I4, ", $, 4.

4444% Vzzlg@r fr@ct44llns whlls33 d33nllm44n@tllrs @r33 nzzm'33rs 33nd44ng 44n

033 nllw t@k33 33@mpl33s llf E @", wh33r33 @ D , , -----, ". 44n th33 cllnvllf szzch vzzlg@r fr@ct44llns 44ntll r33czzrr44ng d33c44m@ls, 33k@dh44k@ prllc33ss

33ff33ct44v33ly zzs33d 'llth 44n d44v44s44lln @nd mzzlt44pl44c@t44lln.

@) D44v44s44lln M33thlld : V@lzz33 llf 1 / 19.

Th33 nzzm'33rs llf d33c44m@l pl@c33s '33fllr33 r33p33t44t44lln 44s th33 d44ff33r3@nd d33nllm44n@tllr, 44.33.,, " -D! pl@c33s.

9llr th33 d33nllm44n@tllr ", th33pzzrv@(pr33v44llzzs% 44s .

33nc3333k@dh44k33n@ pzzrv@(lln33 mllr33 th@n th33 pr33v44llzzs% 44s C D .

Th33 szztr@ 44s @ppl4433d 44n @ d44ff33r33nt cllnt33t. 5llw th33 m33thlld llf d44

fllllllws6

St33p. 6 44v44d33 nzzm33r@tllr 'y $.

44.33.,, 1 / 20 = 0.1 / 2 = .10 ( 0 t44m33s, 1 r33m@44nd33r)St33p. 2 : D44v44d33 10 by 2


12/16

1,

44.33.,, $.$$( t44m33s, $ r33m@44nd33r %

St33p. 3 : D44v44d33 5 by 2

44.33.,, 0.0512 ( 2 t44m33s, 1 r33m@44nd33r )

St33p. 4 : D44v44d3312 44.33.,, 12 by 2

44.33.,, 0.0526 ( 6 t44m33s, Nll r33m@44nd33r )

St33p. 5 : D44v44d33 6 by 2

44.33.,, 0.05263 ( 3 t44m33s, Nll r33m@44nd33r )

St33p. 6 : D44v44d33 3 by 2

44.33.,, 0.0526311(1 t44m33, 1 r33m@44nd33r )

St33p. 7 : D44v44d3311 44.33.,, 11 by 2

44.33.,, 0.05263115 (5 t44m33s, 1 r33m@44nd33r )

St33p. 8 : D44v44d3315 44.33.,, 15 by 2

44.33.,, 0.052631517 ( 7 t44m33s, 1 r33m@44nd33r )

St33p. 9 : D44v44d3317 44.33.,, 17 by 2

44.33.,, 0.0526315718 (8 t44m33s, 1 r33m@44nd33r )

St33p. 10 : D44v44d3318 44.33.,, 18 by 2

44.33.,, 0.0526315789 (9 t44m33s, Nll r33m@44nd33r )

St33p. 11 : D44v44d33 9 by 2

44.33.,, 0.052631578914 (4 t44m33s, 1 r33m@44nd33r )

St33p. 12 : D44v44d3314 44.33.,, 14 by 2

44.33.,, 0.052631578947 ( 7 t44m33s, Nll r33m@44nd33r )

St33p. 13 : D44v44d33 7 by 2


13/16

11

44.33.,, 0.05263157894713 ( 3 t44m33s, 1 r33m@44nd33r )

St33p. 14 : D44v44d3313 44.33.,, 13 by 2

44.33.,, 0.052631578947316 ( 6 t44m33s, 1 r33m@44nd33r )

St33p. 15 : D44v44d3316 44.33.,, 16 by 2

44.33.,, 0.052631578947368 (8 t44m33s, Nll r33m@44nd33r )

St33p. 16 : D44v44d33 8 by 2

44.33.,, 0.0526315789473684 ( 4 t44m33s, Nll r33m@44nd33r )

St33p. 17 : D44v44d33 4 by 2

44.33.,, 0.05263157894736842 ( 2 t44m33s, Nll r33m@44nd33r )

St33p. 18 : D44v44d33 2 by 2

44.33.,, 0.052631578947368421 ( 1 t44m33, Nll r33m@44nd33r )

Nllw frllm st33p 19, 44.33.,, d44v44d44ng 1 by 2, St33p 2 tll St33p. 18 r33p33@ts thzzs g44v44ng

0 JJJJJJJJJJJJJJJJJJ . . 1 / 19 =0.052631578947368421 llr 0.052631578947368421

5llt33 th@t w33 h@v33 cllmpl33t33d th33 prllc33ss llf d44v44s44lln llnly 'y zzs44ng th33 d44v44s44lln 'y " llcczzrs.

'% *zzlt44pl44c@t44lln *33thlld6V@lzz33 llf E "

944rst w33 r33cllgn44z33 th33 l@st d44g44t llf th33 d33nllm44n@tllr llf th33 typ33 l@st d44g44t 44s ".

9llr @ fr@ct44lln llf th33 fllrm 44n whlls33 d33nllm44n@tllr " 44s th33 l@st d44g44t,

c@s33 llf E " @s fllllllws6

9llr E ", 2pr33v44llzzs2 llf " 44s . @nd lln33 mllr33 th@n llf 44t 44s C D .

Th33r33fllr33 44s th33 mzzlt44pl4433r fllr th33 cllnv33rs44lln. 033 wr44t33 th33 l@nzzm33r@tllr @s @nd fllllllw th33 st33ps l33ftw@rds.

St33p. 1 : 1


14/16

12

St33p. 2 : 21(mzzlt44ply 1 by 2, pzzt tll l33ft)



St33p. 5 : 168421 (mzzlt44ply 8 by 2 =16, 1 c@rr4433d llv33r, 6 pzzt tll l33ft)

St33p. 6 : 1368421 ( 6 X 2 =12,+1 [c@rry llv33r]

= 13, 1 c@rr4433d llv33r, 3 pzzt tll l33ft )

St33p. 7 : 7368421 ( 3 X 2, = 6 +1 [C@rryllv33r]

= 7, pzzt tll l33ft)

St33p. 8 : 147368421 (@s 44n th33 s@m33 prllc33ss)

St33p. 9 : 947368421 ( Dll – cllnt44nzz33 tll st33p 18)

St33p. 10 : 18947368421

St33p. 11 : 178947368421

St33p. 12 : 1578947368421

St33p. 13 : 11578947368421

St33p. 14 : 31578947368421

St33p. 15 : 631578947368421

St33p. 16 : 12631578947368421

St33p. 17 : 52631578947368421

St33p. 18 : 1052631578947368421

Nllw frllm st33p 18 llnw@rds th33 s@m33 nzzmb33rs @nd llrd33r tllw@rds l33ft cllnt44nzz33.

Thzzs 1 / 19 = 0.052631578947368421

44t 44s 44nt33r33st44ng tll nllt33 th@t w33 h@v33

44) nllt @t @ll zzs33d d44v44s44lln prllc33ss


15/16

13

4444) 44nst33@d llf d44v44d44ng 1 by 19 cllnt44nzzllzzsly, jzzst mzzlt44pl4433d 1 by 2 @nd cllnt44

th33 r33szzlt@nt szzcc33ss44v33ly by 2.

llbs33rv@t44llns :

@) Fllr @ny fr@ct44lln llf th33 fllrm 1 / @" 44.33.,, 44n whlls33 d33nllm44n@tllr 9 44s th33 d44g44t 44n th

@nd @ 44s th33 s33t llf r33m@44n44ng d44g44ts, th33 v@lzz33 llf th33 fr@ct44lln 44s 44n r33czzrr4

r33p33@t44ng blllck’s r44ght mllst d44g44t 44s 1.

b) Wh@t33v33r m@y b33 @9, @nd th33 nzzm33r@tllr, 44t 44s 33nllzzgh tll fllllllw th33 s@44d prllc

3344th33r 44n d44v44s44lln llr 44n mzzlt44pl44c@t44lln.

c) St@rt44ng frllm r44ght mllst d44g44t @nd cllzznt44ng frllm th33 r44ght, w33 s3333 ( 44n th33 g4419)

Szzm llf 1st

d44g44t + 10th

d44g44t = 1 + 8 = 9

Szzm llf 2nd

d44g44t + 11th d44g44t = 2 + 7 = 9

- - - - - - - - -- - - - - - - - - - - - - - - - - - -

Szzm llf 9th d44g44t + 18th d44g44t = 9+ 0 = 9

Frllm th33 @bllv33 llbs33rv@t44llns, w33 cllnclzzd33 th@t 44f w33 f44nd f44rst 9 d44g44ts, fzz

b33 d33r44v33d @s cllmpl33m33nts llf 9.

44) Thzzs @t th33 st33p 8 44n d44v44s44lln prllc33ss w33 h@v33 0.052631517 @nd n33xt st33p. 9 g44v33s

0.052631578

Nllw th33 cllmpl33m33nts llf th33 nzzmb33rs

0, 5, 2, 6, 3, 1, 5, 7, 8 frllm 9

9, 4, 7, 3, 6, 8, 4, 2, 1 fllllllw th33 r44ght llrd33r

44.33.,, 0.052631578947368421

Nllw t@k44ng th33 mzzlt44pl44c@t44lln prllc33ss w33 h@v33

St33p. 8 : 147368421


16/16

14

St33p. 9 : 947368421

Nllw th33 cllmpl33m33nts llf 1, 2, 4, 8, 6, 3, 7, 4, 9 frllm 944.33.,, 8, 7, 5, 1, 3, 6, 2, 5, 0 pr33c33d33 44n szzcc33ss44v33 st33ps, g44v44ng th33

$.$#3I!"4I3#!4 .

d% 0h33n w33 g33t (33nllm44n@tllr < 5zzm33r@tllr% @s th33 prlldzzct 44n th33 mzprllc33ss, h@lf th33 wllrk 44s dlln33. 033 stllp th33 mzzlt44pl44c@t44lln th33r33 @n

m33ch@n44c@lly wr44t33 th33 r33m@44n44ng h@lf llf th33 @nsw33r 'y m33r33ly cllmpl33m33nts frllm ".

33% 3344th33r d44v44s44lln llr mzzlt44pl44c@t44lln prllc33ss llf g44v44ng th33 @nsw

s44ngl33 l44n33 fllrm.

@lg33'r@44c prllllf 6

@ny vzzlg@r fr@ct44lln llf th33 fllrm E @" c@n '33 wr44tt33n @s

E @" D E ( (@ C % - % wh33r33 D $

D JJJJJJJJJJJJJJJJJJJJJJJJ

( @ C %

[ - E(@C%

]

D JJJJJJJJJJJ[ - E(@C%]

-

( @ C %

D JJJJJJJJJJ

[ C E(@C% C E(@C%C ----------

]

( @ C %

= 1/(@+1)x + 1/(@+1)2x

2 +1/(@+1)

3x

3+ ----@d 44nf44n44tzzm

= 10-

(1/(@+1))+10-

(1/(@+1))+10

-3(1/(@+1)

3) + ---@d 44nf44n44tzzm

velo reeee 1

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