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Operations on Numbers

APPETIZER

Follow these directions.

Step 1. Write down the number of the month in which you were born.Step 2. Multiply the number in Step by !.Step 3. Add " to the result of Step !.Step 4. Multiply the result of Step # by "$.Step 5. Subtr%ct !"$ from the present ye%r.Step 6. Add the result from Step " to the result from Step &.Step 7. Subtr%ct the ye%r of your birth from the result from Step '.Step 8. (ircle the l%st two di)its of the result from Step *. The circled

number should )i+e your %)e on your birthd%y this ye%r,Step 9. The uncircled p%rt of the number from Step - should be the numberof your birth month,

/eTemple0 1 2on)0 33'0 p. &.4

Introduction

The our basic number operations are addition! subtraction! mu"tip"ication anddi#ision. There are #arious methods to compute these operations.

The four m%in methods th%t still rem%in tod%y %re menta"! abacus! $rittena"%orithms and ca"cu"ators methods. Which of these methods do you thin5 is themost popul%r %mon) people in this modern world6

2.1 &ddition! Subtraction! 'u"tip"ication and (i#ision on )ho"e Numbers

The four b%sic oper%tions %re +ery different in their me%nin)s. 2et7s in+esti)%te themnow,

&ddition and Subtraction

There %re different models w%ys of thin5in)4 to illustr%te the me%nin)s of %ddition %ndsubtr%ction. Addition c%n be thou)ht of %s %4 set model8 the combin%tion of discrete items9%nd b4 number:line model8 the combin%tion of continuous ;u%ntities. For the %dditionsentence " < ' = 9 " %nd ' %re the %ddends9 where%s is the sum.

" < ' =

&'

%ddend %ddend sum

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>n the other h%nd0 subtr%ction c%n be thou)ht of %s %4 t%5e:%w%y model9 b4 comp%risonmodel9 c4 missin):%ddend model9 %nd d4 number:line model. For the subtr%ction sentence3 ? # = '9 3 is the minuend9 # is the subtr%hend9 ' is the difference.

3 @ # = '

&ttachment 1Set 'ode" o &ddition

The word problem %nd di%)r%m below illustr%te the me%nin) of %ddition %s combinin)discrete obects of two sets.

Ahu h%s # red m%rbles %nd " blue m%rbles.Bow m%ny m%rbles does Ahu h%+e in %ll6

Number*+ine 'ode" o &ddition

The word problem %nd di%)r%m below illustr%te the me%nin) of %ddition %s combinin)two continuous ;u%ntities.

The temper%ture of w%ter in % be%5er is #-o (. After he%tin) for % while0 thew%ter temper%ture )oes up by &o (. Wh%t is the new w%ter temper%ture6

&*

minuend

dsubtr%hend difference

$ $ !$ #$ &$ "$ '$

#-(

&!(

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&ttachment 2Ta,e*&$a- 'ode" o Subtraction

The word problem %nd di%)r%m below illustr%te the me%nin) of subtr%ction %s t%5in)

%w%y one ;u%ntity from %nother.

F%iC h%s " chic5en. 2%st ni)ht0 ! of the chic5en were stolen.Bow m%ny chic5en %re left with F%iC6

omparison 'ode" o Subtraction

The word problem %nd di%)r%m below illustr%te the me%nin) of subtr%ction %s comp%rin)the difference between two ;u%ntities.

D%l5is h%s " %pples. (h%ndr% h%s ! %pples.Bow m%ny more %pples does D%l5is h%+e th%n (h%ndr%6

D%l5is

(h%ndr%

'issin%*&ddend 'ode" o Subtraction

The word problem %nd di%)r%m below illustr%te the me%nin) of subtr%ction %s findin) the%ddend needed to %dd up to % cert%in sum.

Dobie w%nts to 5eep " r%bbits. Bis uncle )%+e him ! r%bbits.Bow m%ny more r%bbits must Dobie buy6

Number*+ine 'ode" o Subtraction

The word problem %nd di%)r%m below illustr%te the me%nin) of subtr%ction %s findin) the

difference between two continuous ;u%ntities.

SuCi h%s " liters of or%n)e uice. She )%+e ! liters of or%n)e uice to her sister.Bow m%ny liters of or%n)e uice %re left6

Dy now0 you should h%+e noticed th%t %ddition %nd subtr%ction %re +ery closely

rel%ted. In f%ct0 subtr%ction is the in+erse oper%tion of %ddition. This me%ns th%t if' < # = 30 then 3 ? # = ' or 3 ? ' = #.

&-

$ ! # & " '

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/asic 0acts or &ddition and Subtraction

& basic act or addition is the sum o an- t$o $ho"e numbers "ess than 1.

Since there %re $ such numbers0 therefore there %re % tot%l of $$ b%sic f%cts of%ddition.

Since subtr%ction is the in+erse oper%tion of %ddition0 therefore the in+erse of %ny%ddition b%sic f%ct is % b%sic f%ct for subtr%ction. As %n e%mple0 since 5 7 12 isa basic addition act! thereore 12 5 7 and 12 7 5 are t$o basic acts orsubtraction.

epresentin% &ddition and Subtraction on a Number +ine

umber line is one +ery simple %nd e%sy w%y to do %ddition %nd subtr%ction.

Fi)ure . Represent%tion of ' < ! on % number line.

Fi)ure !. Represent%tion of ' ? ! = & on % number line.

'u"tip"ication and (i#ision

Multiplic%tion c%n be thou)ht of %s8 %4 repe%ted %ddition0 %nd b4 rect%n)ul%r %rr%ys.For the multiplic%tion sentence " - = &$9 " %nd - %re the f%ctors9 where%s &$ is theproduct.

&3

" - = &$

f%ctor f%ctor product

0 1 2 3 4 5 6 7 8 9 10 11 12 13

6+ 2

0 1 2 3 4 5 6 7 8 9 10 11 12 13

'

@ !

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>n the other h%nd0 di+ision c%n be thou)ht of %s8 %4 repe%ted subtr%ction0 %nd b4p%rtiti+e di+ision where % cert%in number of obects %re distributed e;u%lly into cert%innumber of )roups %nd %s5in) how m%ny obects %re in e%ch )roup.

For the di+ision sentence0 !- G & = *9 !- is the di+idend9 & is the di+isor9 * is the;uotient.

!- G & = *

di+idend di+isor ;uotient

&ttachment 3'u"tip"ication as epeated &ddition

The word problem %nd di%)r%m below illustr%te the me%nin) of multiplic%tion %s repe%ted%ddition.

There %re # p%c5%)es of tom%toes. & tom%toes %re in % p%c5%)e.Bow m%ny tom%toes in %ll 6

3 4 means 4 4 4

'u"tip"ication as ectan%u"ar &rra-

The word problem %nd di%)r%m below illustr%te the me%nin) of multiplic%tion %srect%n)ul%r %rr%y.

There %re # rows of tom%toes. & tom%toes %re in e%ch row.Bow m%ny tom%toes in %ll6

3 4 means 3 ro$s 4 in each ro$

"$

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&ttachment 4(i#ision as epeated Subtraction

The word problem %nd di%)r%m below illustr%te the me%nin) of di+ision %s repe%ted

subtr%ction.

There %re ! tom%toes. >ne child e%ts # tom%toes.Bow m%ny children h%+e tom%toes to e%t6

12 3 means 3 can be subtracted rom 12 or 4 times 12 3 3 3 34

(i#ision as :;ua" (istribution

The word problem %nd di%)r%m below illustr%te the me%nin) of di+ision %s e;u%ldistribution.

There %re ! tom%toes. There %re # children.Bow m%ny tom%toes for e%ch child6

12 3 means 12 can be distributed e;ua""- to 3 %roups $ith 4 in each %roup.

B%+e you notice %ny simil%rity between the %ddition:subtr%ction rel%tionship %nd themultiplic%tion:di+ision rel%tionship6 >f course0 di#ision is the in#erse operation omu"tip"ication. This means that i 3 4 12 then 12 3 4 or 12 4 3.

/asic 0acts o 'u"tip"ication and (i#ision

& basic act o mu"tip"ication is the product o an- t$o $ho"e numbers "essthan 1. Hust li5e b%sic f%cts of %ddition0 there %re 1 basic acts omu"tip"ication too. Also0 %s di+ision is the in+erse oper%tion of multiplic%tion0 thein+erse of %ny multiplic%tion b%sic f%ct is % b%sic f%ct of di+ision. As %n e%mple0since6 7 42! thereore 42 6 7 and 42 7 6 are t$o basic acts o

di#ision.

"

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epresentin% 'u"tip"ication and (i#ision on a Number +ine

Hust li5e %ddition %nd subtr%ction0 multiplic%tion %nd di+ision c%n %lso be representedon % number line. Fi)ure # %nd Fi)ure & show one e%mple for multiplic%tion %ndone for di+ision.

Fi)ure #. Represent%tion of ' ! = ! on % number line.

Fi)ure &. Represent%tion of ' G ! = # on % number line.

2.2 +a$s o Number Operations

There %re three import%nt properties of number oper%tions which )i+e rise to threel%ws of number oper%tions8 %4 commut%ti+e0 b4 %ssoci%ti+e0 %nd c4 distributi+e.

The ommutati#e +a$ T

Accordin) to this l%w0 ch%n)in) the order of the numbers does not ch%n)e the resultof the oper%tion. Addition %nd multiplic%tion %dhere to this l%w. So0 for %ny numbersa%nd b0

a< b= b< a

a b= b a

>n the other h%nd0 subtr%ction %nd di+ision do not %dhere to this l%w. Therefore0 for%ny numbers a%nd b0

"!

0 1 2 3 4 5 6 7 8 9 10 11 12 13

< ! < ! < ! < ! < ! < !

0 1 2 3 4 5 6 7 8 9 10 11 12 13

@ !@ !@ !

'

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a? bJ b? a

a G b J bG a

:ercise 3

%4 /r%w % di%)r%m to show th%t " < * = * < ".

b4 /r%w % di%)r%m to show th%t & ! = ! &.

The &ssociati#e +a$ S:=

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The (istributi#e +a$ T&/

The third l%w of number oper%tions is %pplic%ble to %4 multiplic%tion o+er %ddition0%nd b4 multiplic%tion o+er subtr%ction. For %ny numbers a0 b%nd c0

a b< c4 = a b4 < a c4 %nd b< c4 a = b a4 < c a4

a b@ c4 = a b4 @ a c4 %nd b@ c4 a = b a4 @ c a4

2.3 )ritten! 'enta" and a"cu"ator omputation

D%sic%lly you c%n do % comput%tion usin) %ny of the three modes of c%lcul%tions8%4 written comput%tion0 b4 ment%l comput%tion0 %nd c4 c%lcul%tor comput%tion.

)ritten omputation? &"%orithms or &ddition and Subtraction

Written comput%tion is norm%lly c%rried out by followin) % st%nd%rd %l)orithm. An%l)orithm is % system%tic0 step:by:step procedure used to find %n %nswer. Thecommon %l)orithm for %ddition in+ol+es two m%in procedures8 %4 %ddin) sin)le di)its0%nd b4 re)roupin). Fi)ure " shows the st%nd%rd %ddition %l)orithms for ##&* < !"$%nd "'- < #3&.

1 1

# # & *< ! " $

" ' -

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# " *

< ! * &

! $

< " $ $

' #

Fi)ure '. An %ltern%ti+e %ddition %l)orithm.

The common written %l)orithm for subtr%ction in+ol+es two m%in procedures8%4 subtr%ction of numbers th%t in+ol+es b%sic f%cts of %ddition0 %nd b4 re)roupin).Fi)ure * shows the st%nd%rd subtr%ction %l)orithms for #"* ? !# %nd #!# ? '&.

2 11 13

# " * @ ! #

# ! #@ ' &

! # & ! " 3

Fi)ure *. St%nd%rd subtr%ction %l)orithms.

Hust li5e %ddition0 subtr%ction c%n %lso be c%rried out with %ltern%ti+e %l)orithms.Fi)ure - shows one e%mple of such %l)orithms for #'! ? -*.

10

2 5 10

# ' !

@ - *

* "

Fi)ure -. An %ltern%ti+e %l)orithm for subtr%ction.

""

* ? #

" ? !

# ?

# ? &

? '

! ? $

$ ? * < !

$ ? - < "

! ?

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It7s time to put on your thin5in) c%p,

)ritten omputation? &"%orithms or 'u"tip"ication and (i#ision

>ne con+ention%l w%y is to compute multiplic%tion in the +ertic%l form.

Simil%r to %ddition %nd subtr%ction0 there %re some other multiplic%tion %l)orithms.Fi)ure 3 shows one e%mple of such %l)orithm0 while Fi)ure $ shows %nother.

! *

L # '

& ! ' *

! $ ' !$! $ #$ *

' $ $ #$ !$

3 * !

Fi)ure 3. An %ltern%ti+e multiplic%tion %l)orithm.

!$ * !* = !$ < *

#$ '$$ !$ #$ !$ = '$$9 #$ * = !$

' !$ &! ' !$ = !$9 ' * = &!

#' = #$ < '

Fi)ure $. Another %ltern%ti+e multiplic%tion %l)orithm.

>ne common %l)orithm for di+ision is done throu)h % process c%lled "on% di#ision.

>ne %ltern%ti+e di+ision %l)orithm is shown in Fi)ure .

"'

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&

# $ #&

$ $

* 3 &

* $ $ Subtr%ct $$ *

! &

! $ Subtr%ct #$ *

#

! - Subtr%ct & *

# rem%inder

Fi)ure . An %ltern%ti+e di+ision %l)orithm.

@ou must bear in mind that these a"%orithms are a"" e;ua""- app"icab"e andmathematica""- correctA

See how fleible m%them%tics is,

TI/DIT

The word %l)orithm deri+es from %l:Nhow%riCmi0 the n%me of the ei)hth centuryAr%b scribe who wrote two boo5s on %rithmetic %nd %l)ebr% )i+in) % +ery c%reful%ccount of the Bindu system of numer%tion %nd methods of c%lcul%tion. Thou)h %l:Nhow%riCmi m%de no cl%im to in+entin) the system0 c%reless re%ders of 2%tintr%nsl%tions of his boo5 be)%n to %ttribute the system to him %nd c%ll the newmethods of c%lcul%tion %l:Nhow%riCmi or0 c%relessly %l)orismi. >+er time the wordbec%me %l)orithm0 %nd c%me to me%n %ny orderly0 repetiti+e scheme.

/eTemple 1 2on)0 33'0 p. -'.4

'enta" omputation

&"thou%h a"%orithms ma- pro#ide -ou $ith a $a- to compute in an order"-

manner! menta" arithmetic can be #er- useu" $hen -ou need to speed up -ourcomputation.

There %re tric5s th%t you c%n %lw%ys le%rn to simplify these t%s5s.

:ercise 6

Perform these comput%tions by ment%l c%lcul%tion.

%4 ! < & < * < - < '

b4 $ < !$ < "$ < '$ < &$c4 " < " < #"0 < !" < "

"*

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d4 ! * ? # !e4 & # !"f4 !*# ? #3)4 '- G #

&ttachment 5Tric,s or 'enta" omputations

Trick 18 (omp%tible umbers %nd Properties of umber >per%tions

ompatib"e numbers%re numbers whose sums0 differences0 products0 or ;uotients %ree%sy to c%lcul%te ment%lly. P%irs of %ddends with sums in the multiples of $ %re )oode%mples of comp%tible numbers for %ddition. Some e%mples include # < * = $9 %nd-' < & = $$. E%mples of comp%tible numbers for other oper%tions %re !$$ ? $ =3$9 !" & = $$9 %nd '$$ G #$ = !$.

The properties of number oper%tions %nd comp%tible numbers c%n be used to simplifycomput%tion. When comput%tions %re simplified0 you c%n do them ment%lly,

E%mple 8 3 8 7> = # < * < -4 commut%ti+e= # < *4 < - %ssoci%te comp%tible numbers= $ < -= -

E%mple !8 86 B 15 = -' $ < "4= -' $4 < -' "4 distributi+e= -'$ < $ -'" = O -' $4 = $

= !3$

Trick 28 (ompens%tion

When you don7t see %ny p%ir of comp%tible numbers0 you c%n still speed up yourcomput%tion by reormu"atin%% sum0 difference0 product or ;uotient to one th%t ismore re%dily obt%ined ment%lly. Two e%mples %re illustr%ted below8

E%mple 8 43 36 19 = < #' 14 < 3 * 14 reformul%te #' %nd 3= < #*4 < - comp%tible numbers= -$ < -= 3-

E%mple !8 48 B 5 = &- 24 " B 24 h%lf %nd double= !& $= !&$

Trick 38 2eft:To:Ri)ht Methods

Kou m%y be +ery used to perform oper%tion from ri)ht to left i.e. from sm%ller tol%r)er pl%ce +%lues40 but sometime )oin) the opposite w%y c%n help you to performf%st ment%l comput%tion. Rese%rch h%s found th%t people who %re ecellent inperformin) ment%l c%lcul%tions use this left:to:ri)ht method to reduce memory lo%d.Two e%mples %re described below8

"-

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E%mple 8 To compute #&! < #'0 you c%n first compute #$$ < $$0 then &$ < #$0%nd then ! < ' to obt%in &*-.

E%mple !8 To compute "- < !*3. you c%n thin5 %s follow8 $$ < !$$ = #$$9#$$ < "$ < *$ = &!$9 &!$ < - < 3 = *Q

E%mple #8 To compute & !"#0 you c%n thin5 %s follow8 & !"$ = $$$0 then$$$ < & #4 = $!.Q

"3

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a"cu"ator omputation

a"cu"ators can be in e"ectronic or non*e"ectronic orms. The most commonnon*e"ectronic orm o ca"cu"ator is the abacus

2i5ewise0 electronic c%lcul%tors0 bein) the most popul%r comput%tion%l tool of thepresent d%ys .

:ercise 7

. Perform these c%lcul%tions by %b%cus0 %nd then chec5 your %nswersby %n electronic c%lcul%tor.

%4 !" < & < "- < '' < &- < "3b4 -- < & < "& < "'&- < ---"c4 "&! ? &'"-d4 '$!& ? &*#3

!. se %n electronic c%lcul%tor to compute these c%lcul%tions.%4 !# &" &!b4 !3 *'c4 !!# "'-d4 -&- G 'e4 "'&G!f4 $ G #")4 #" G $

Summar-

. Addition c%n be illustr%ted %s %4 combinin) two sets of discrete items9 %ndcombinin) two continuous ;u%ntities on % number line.

!. Subtr%ction c%n be illustr%ted %s %4 t%5e %w%y % ;u%ntity from %nother9 b4comp%rin) the difference between two ;u%ntities9 c4 findin) the %ddend to%dd up to % cert%in sum9 %nd d4 findin) the difference between twocontinuous ;u%ntities.

#. Multiplic%tion c%n be illustr%ted %s %4 repe%ted %ddition9 %nd b4 rect%n)ul%r%rr%y.

&. /i+ision c%n be illustr%ted %s %4 repe%ted subtr%ction9 %nd b4 p%rtiti+e

di+ision.

". The rel%tionships of the b%sic oper%tions c%n be summ%riCed in the followin)di%)r%m.

&ddition is the in+erse of Subtractionis repe%ted is repe%ted

'u"tip"ication is the in+erse of (i#ision'. Addition %nd multiplic%tion %re %ssoci%ti+e %nd communic%ti+e0 but

subtr%ction %nd di+ision %re neither.

'$

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*. Multiplic%tion %nd di+ision %re distributi+e o+er %ddition %nd subtr%ction.

/ESSERT

The word add comes from the 2%tin word adhere0 which me%ns to put to.QWidm%n first used Q