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Question Number 2917 by Rasheed Soomro last updated on 30/Nov/15
What is DMAS  rule? Where is it followed?  Do we follow this rule?
$$\mathcal{W}{hat}\:{is}\:{DMAS}\:\:{rule}?\:{Where}\:{is}\:{it}\:{followed}? \\ $$$${Do}\:{we}\:{follow}\:{this}\:{rule}?\: \\ $$
Answered by 123456 last updated on 30/Nov/15
if i understanded it about order of operation  like  1+2×3=1+6=7  1+2×3≠3×3=9  Division Multiplication Adition Subtration  48÷2 (9+3)= 288 or 2  however in general if you want avoid  confusion is beter to use patents, like  48 ÷[2(9+3)]=2  or  (48 ÷2) (9+3)=288
$$\mathrm{if}\:\mathrm{i}\:\mathrm{understanded}\:\mathrm{it}\:\mathrm{about}\:\mathrm{order}\:\mathrm{of}\:\mathrm{operation} \\ $$$$\mathrm{like} \\ $$$$\mathrm{1}+\mathrm{2}×\mathrm{3}=\mathrm{1}+\mathrm{6}=\mathrm{7} \\ $$$$\mathrm{1}+\mathrm{2}×\mathrm{3}\neq\mathrm{3}×\mathrm{3}=\mathrm{9} \\ $$$$\mathrm{Division}\:\mathrm{Multiplication}\:\mathrm{Adition}\:\mathrm{Subtration} \\ $$$$\mathrm{48}\boldsymbol{\div}\mathrm{2}\:\left(\mathrm{9}+\mathrm{3}\right)=\:\mathrm{288}\:\mathrm{or}\:\mathrm{2} \\ $$$$\mathrm{however}\:\mathrm{in}\:\mathrm{general}\:\mathrm{if}\:\mathrm{you}\:\mathrm{want}\:\mathrm{avoid} \\ $$$$\mathrm{confusion}\:\mathrm{is}\:\mathrm{beter}\:\mathrm{to}\:\mathrm{use}\:\mathrm{patents},\:\mathrm{like} \\ $$$$\mathrm{48}\:\boldsymbol{\div}\left[\mathrm{2}\left(\mathrm{9}+\mathrm{3}\right)\right]=\mathrm{2} \\ $$$$\mathrm{or} \\ $$$$\left(\mathrm{48}\:\boldsymbol{\div}\mathrm{2}\right)\:\left(\mathrm{9}+\mathrm{3}\right)=\mathrm{288} \\ $$
Commented by Rasheed Soomro last updated on 30/Nov/15
I claim that we don′t follow this rule.  This rule suggests  addition before subtraction,  but we don′t do so. For example  By dmas rule 12−2+3=12−(2+3)=12−5=7  Whereas  we conclude 13.      If we don′t follow this rule then where is it followed???
$$\mathcal{I}\:{claim}\:{that}\:{we}\:{don}'{t}\:{follow}\:{this}\:{rule}. \\ $$$$\mathcal{T}{his}\:{rule}\:{suggests}\:\:\boldsymbol{{addition}}\:\boldsymbol{{before}}\:\boldsymbol{{subtraction}}, \\ $$$${but}\:{we}\:{don}'{t}\:{do}\:{so}.\:{For}\:{example} \\ $$$${By}\:{dmas}\:{rule}\:\mathrm{12}−\mathrm{2}+\mathrm{3}=\mathrm{12}−\left(\mathrm{2}+\mathrm{3}\right)=\mathrm{12}−\mathrm{5}=\mathrm{7} \\ $$$${Whereas}\:\:{we}\:{conclude}\:\mathrm{13}. \\ $$$$\:\:\:\:\mathcal{I}{f}\:{we}\:{don}'{t}\:{follow}\:{this}\:{rule}\:{then}\:{where}\:{is}\:{it}\:{followed}??? \\ $$
Commented by Filup last updated on 30/Nov/15
When you have, addition or subtraction,  you can do it in any order if you make  all terms positive  −7=+(−7)  And you always work left to right
$$\mathrm{When}\:\mathrm{you}\:\mathrm{have},\:\mathrm{addition}\:\mathrm{or}\:\mathrm{subtraction}, \\ $$$$\mathrm{you}\:\mathrm{can}\:\mathrm{do}\:\mathrm{it}\:\mathrm{in}\:\mathrm{any}\:\mathrm{order}\:\mathrm{if}\:\mathrm{you}\:\mathrm{make} \\ $$$$\mathrm{all}\:\mathrm{terms}\:\mathrm{positive}\:\:−\mathrm{7}=+\left(−\mathrm{7}\right) \\ $$$$\mathrm{And}\:\mathrm{you}\:\mathrm{always}\:\mathrm{work}\:\mathrm{left}\:\mathrm{to}\:\mathrm{right} \\ $$$$ \\ $$
Commented by prakash jain last updated on 30/Nov/15
DM is Division or Multiplication from L to R  AS is Adition or Subtration from L to R  Rule does not give priority to addition  over subtraction. Also Same priority is  given to Multiplication and Division.  Some other acronymns  PEMDAS: Multiplication even though  it appears before division is given the same  priority. All expressions are always evaluated  from left to right.
$$\mathrm{DM}\:\mathrm{is}\:\mathrm{Division}\:\mathrm{or}\:\mathrm{Multiplication}\:\mathrm{from}\:\mathrm{L}\:\mathrm{to}\:\mathrm{R} \\ $$$$\mathrm{AS}\:\mathrm{is}\:\mathrm{Adition}\:\mathrm{or}\:\mathrm{Subtration}\:\mathrm{from}\:\mathrm{L}\:\mathrm{to}\:\mathrm{R} \\ $$$$\mathrm{Rule}\:\mathrm{does}\:\mathrm{not}\:\mathrm{give}\:\mathrm{priority}\:\mathrm{to}\:\mathrm{addition} \\ $$$$\mathrm{over}\:\mathrm{subtraction}.\:\mathrm{Also}\:\mathrm{Same}\:\mathrm{priority}\:\mathrm{is} \\ $$$$\mathrm{given}\:\mathrm{to}\:\mathrm{Multiplication}\:\mathrm{and}\:\mathrm{Division}. \\ $$$$\mathrm{Some}\:\mathrm{other}\:\mathrm{acronymns} \\ $$$$\mathrm{PEMDAS}:\:\mathrm{Multiplication}\:\mathrm{even}\:\mathrm{though} \\ $$$$\mathrm{it}\:\mathrm{appears}\:\mathrm{before}\:\mathrm{division}\:\mathrm{is}\:\mathrm{given}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{priority}.\:\mathrm{All}\:\mathrm{expressions}\:\mathrm{are}\:\mathrm{always}\:\mathrm{evaluated} \\ $$$$\mathrm{from}\:\mathrm{left}\:\mathrm{to}\:\mathrm{right}. \\ $$
Commented by Filup last updated on 30/Nov/15
DMAS works using parentheses laws  That means, everything is made to work  with a addititve opperator ′+′.    e.g.  −2+5×−6+12−7  =+(−2)+(5×(−6))+12+(−7)  =5×(−6)+12+(−7)+(−2)  =−30+12−7−2  =−9
$$\mathrm{DMAS}\:\mathrm{works}\:\mathrm{using}\:\mathrm{parentheses}\:\mathrm{laws} \\ $$$$\mathrm{That}\:\mathrm{means},\:\mathrm{everything}\:\mathrm{is}\:\mathrm{made}\:\mathrm{to}\:\mathrm{work} \\ $$$$\mathrm{with}\:\mathrm{a}\:\mathrm{addititve}\:\mathrm{opperator}\:'+'. \\ $$$$ \\ $$$${e}.{g}. \\ $$$$−\mathrm{2}+\mathrm{5}×−\mathrm{6}+\mathrm{12}−\mathrm{7} \\ $$$$=+\left(−\mathrm{2}\right)+\left(\mathrm{5}×\left(−\mathrm{6}\right)\right)+\mathrm{12}+\left(−\mathrm{7}\right) \\ $$$$=\mathrm{5}×\left(−\mathrm{6}\right)+\mathrm{12}+\left(−\mathrm{7}\right)+\left(−\mathrm{2}\right) \\ $$$$=−\mathrm{30}+\mathrm{12}−\mathrm{7}−\mathrm{2} \\ $$$$=−\mathrm{9} \\ $$
Commented by Rasheed Soomro last updated on 30/Nov/15
On the comment of Sir prakash  I  can′t reject your interpretation of DMAS .  But some words about my interpretation:        Some time ago I had a chance to study a text book  of maths Count Down Series for class V(?) by  Oxford University Press. There I had read the   interpretation of DMAS which was that of mine.  I−E  addition before subtraction. The book  had given a clear example. Since that I began to  think that DMAS is a different  convention.
$$\mathcal{O}\boldsymbol{{n}}\:\boldsymbol{{the}}\:\boldsymbol{{comment}}\:\boldsymbol{{of}}\:\boldsymbol{\mathcal{S}{ir}}\:\boldsymbol{{prakash}} \\ $$$$\mathcal{I}\:\:{can}'{t}\:{reject}\:{your}\:{interpretation}\:{of}\:{DMAS}\:. \\ $$$$\boldsymbol{{B}}{ut}\:{some}\:{words}\:{about}\:{my}\:{interpretation}: \\ $$$$\:\:\:\:\:\:{Some}\:{time}\:{ago}\:{I}\:{had}\:{a}\:{chance}\:{to}\:{study}\:{a}\:{text}\:{book} \\ $$$${of}\:{maths}\:\boldsymbol{{Count}}\:\boldsymbol{{Down}}\:\boldsymbol{{Series}}\:\boldsymbol{{for}}\:\boldsymbol{{class}}\:\boldsymbol{{V}}\left(?\right)\:\boldsymbol{{by}} \\ $$$$\boldsymbol{{Oxford}}\:\boldsymbol{{University}}\:\boldsymbol{{P}\mathrm{ress}}.\:{There}\:{I}\:{had}\:{read}\:{the}\: \\ $$$${interpretation}\:{of}\:{DMAS}\:{which}\:{was}\:{that}\:{of}\:{mine}. \\ $$$${I}−{E}\:\:\boldsymbol{{addition}}\:\boldsymbol{{before}}\:\boldsymbol{{subtraction}}.\:{The}\:{book} \\ $$$${had}\:{given}\:{a}\:\boldsymbol{{clear}}\:{example}.\:{Since}\:{that}\:{I}\:{began}\:{to} \\ $$$${think}\:{that}\:{DMAS}\:{is}\:{a}\:\boldsymbol{{different}}\:\:{convention}. \\ $$$$ \\ $$
Commented by Filup last updated on 01/Dec/15
Also, Rasheed, in your earlier comment  you mentioned how by if you follow  dmas you do addition first making:  12−2+3=12−(2+3).  That is incorret. Following dmas  you obtain:  12−2+3=12+(−2+3)
$${Also},\:{Rasheed},\:\mathrm{in}\:\mathrm{your}\:\mathrm{earlier}\:\mathrm{comment} \\ $$$$\mathrm{you}\:\mathrm{mentioned}\:\mathrm{how}\:\mathrm{by}\:\mathrm{if}\:\mathrm{you}\:\mathrm{follow} \\ $$$$\mathrm{dmas}\:\mathrm{you}\:\mathrm{do}\:\mathrm{addition}\:\mathrm{first}\:\mathrm{making}: \\ $$$$\mathrm{12}−\mathrm{2}+\mathrm{3}=\mathrm{12}−\left(\mathrm{2}+\mathrm{3}\right). \\ $$$$\mathrm{That}\:\mathrm{is}\:\mathrm{incorret}.\:\mathrm{Following}\:\mathrm{dmas} \\ $$$$\mathrm{you}\:\mathrm{obtain}: \\ $$$$\mathrm{12}−\mathrm{2}+\mathrm{3}=\mathrm{12}+\left(−\mathrm{2}+\mathrm{3}\right) \\ $$
Commented by Filup last updated on 01/Dec/15
I was always taught that dmas is:  ′division and multiplication′ THEN  ′subtraction and addition′.  You can do × and ÷ in any order similar  to + and − in any order.    2×3÷4=(((2×3))/4)=2×((3/4))    2−3+4=(2−3)+4=2+(−3+4)
$${I}\:\mathrm{was}\:\mathrm{always}\:\mathrm{taught}\:\mathrm{that}\:\mathrm{dmas}\:\mathrm{is}: \\ $$$$'{division}\:{and}\:{multiplication}'\:{THEN} \\ $$$$'{subtraction}\:{and}\:{addition}'. \\ $$$$\mathrm{You}\:\mathrm{can}\:\mathrm{do}\:×\:\mathrm{and}\:\boldsymbol{\div}\:\mathrm{in}\:\mathrm{any}\:\mathrm{order}\:\mathrm{similar} \\ $$$$\mathrm{to}\:+\:\mathrm{and}\:−\:\mathrm{in}\:\mathrm{any}\:\mathrm{order}. \\ $$$$ \\ $$$$\mathrm{2}×\mathrm{3}\boldsymbol{\div}\mathrm{4}=\frac{\left(\mathrm{2}×\mathrm{3}\right)}{\mathrm{4}}=\mathrm{2}×\left(\frac{\mathrm{3}}{\mathrm{4}}\right) \\ $$$$ \\ $$$$\mathrm{2}−\mathrm{3}+\mathrm{4}=\left(\mathrm{2}−\mathrm{3}\right)+\mathrm{4}=\mathrm{2}+\left(−\mathrm{3}+\mathrm{4}\right) \\ $$
Commented by Rasheed Soomro last updated on 01/Dec/15
Mr Filup       When you write 12−2+3 as 12+(−2)+3  ′−′ remains no longer binary operation.In the  latter form ′−′ is unary operation and DMAS  rule is  about order of binary operations. You   suggest an alternate whereas I think what  the rule require in actual expression.       Actually matter isn′t it how could we calculate  an expression.Matter is what is an interpretation  of DMAS.         One interpretation is that Mr prakash has  mentioned. Other interpretation is what I had  read some time ago.         The interpretation mentioned by Mr prakash  is consistent with our way of calculation whereas  interpretation mentioned by me is not compatible.           Anyway DMAS is matter of convention and  my problem its actual interpretation.That′s all.
$${Mr}\:{Filup} \\ $$$$\:\:\:\:\:{When}\:{you}\:{write}\:\mathrm{12}−\mathrm{2}+\mathrm{3}\:{as}\:\mathrm{12}+\left(−\mathrm{2}\right)+\mathrm{3} \\ $$$$'−'\:{remains}\:{no}\:{longer}\:{binary}\:{operation}.{In}\:{the} \\ $$$${latter}\:{form}\:'−'\:{is}\:{unary}\:{operation}\:{and}\:{DMAS} \\ $$$${rule}\:{is}\:\:{about}\:{order}\:{of}\:{binary}\:{operations}.\:{You}\: \\ $$$${suggest}\:\boldsymbol{{an}}\:\boldsymbol{{alternate}}\:{whereas}\:{I}\:{think}\:{what} \\ $$$${the}\:{rule}\:{require}\:{in}\:{actual}\:{expression}. \\ $$$$\:\:\:\:\:{Actually}\:{matter}\:{isn}'{t}\:{it}\:{how}\:{could}\:{we}\:{calculate} \\ $$$${an}\:{expression}.{Matter}\:{is}\:{what}\:{is}\:{an}\:{interpretation} \\ $$$${of}\:{DMAS}.\: \\ $$$$\:\:\:\:\:\:{One}\:{interpretation}\:{is}\:{that}\:{Mr}\:{prakash}\:{has} \\ $$$${mentioned}.\:{Other}\:{interpretation}\:{is}\:{what}\:{I}\:{had} \\ $$$${read}\:{some}\:{time}\:{ago}. \\ $$$$\:\:\:\:\:\:\:{The}\:{interpretation}\:{mentioned}\:{by}\:{Mr}\:{prakash} \\ $$$${is}\:{consistent}\:{with}\:{our}\:{way}\:{of}\:{calculation}\:{whereas} \\ $$$${interpretation}\:{mentioned}\:{by}\:{me}\:{is}\:{not}\:{compatible}. \\ $$$$\:\:\:\:\:\:\:\:\:{Anyway}\:{DMAS}\:{is}\:{matter}\:{of}\:{convention}\:{and} \\ $$$${my}\:{problem}\:{its}\:{actual}\:{interpretation}.{That}'{s}\:{all}. \\ $$
Commented by 123456 last updated on 01/Dec/15
sum is associative and comutative  so you can take it in any order  (x+y)+z=x+(y+z)=x+(z+y)  however subtraction dont is comutative in geeral  x−y≠y−x  and dont is associative in general  x−(y−z)=x−y+z  (x−y)−z=x−y−z  x−(y−z)≠(x−y)−z
$$\mathrm{sum}\:\mathrm{is}\:\mathrm{associative}\:\mathrm{and}\:\mathrm{comutative} \\ $$$$\mathrm{so}\:\mathrm{you}\:\mathrm{can}\:\mathrm{take}\:\mathrm{it}\:\mathrm{in}\:\mathrm{any}\:\mathrm{order} \\ $$$$\left({x}+{y}\right)+{z}={x}+\left({y}+{z}\right)={x}+\left({z}+{y}\right) \\ $$$$\mathrm{however}\:\mathrm{subtraction}\:\mathrm{dont}\:\mathrm{is}\:\mathrm{comutative}\:\mathrm{in}\:\mathrm{geeral} \\ $$$${x}−{y}\neq{y}−{x} \\ $$$$\mathrm{and}\:\mathrm{dont}\:\mathrm{is}\:\mathrm{associative}\:\mathrm{in}\:\mathrm{general} \\ $$$${x}−\left({y}−{z}\right)={x}−{y}+{z} \\ $$$$\left({x}−{y}\right)−{z}={x}−{y}−{z} \\ $$$${x}−\left({y}−{z}\right)\neq\left({x}−{y}\right)−{z} \\ $$
Commented by 123456 last updated on 01/Dec/15
2×3÷4=2×(3/4)=((2×3)/4)=(3/2)  however if you interchange × with ÷  the order will be important  2÷3×4  2÷(3×4)=(2/(3×4))=(2/(12))=(1/6)  (2÷3)×4=(2/3)×4=(8/3)
$$\mathrm{2}×\mathrm{3}\boldsymbol{\div}\mathrm{4}=\mathrm{2}×\frac{\mathrm{3}}{\mathrm{4}}=\frac{\mathrm{2}×\mathrm{3}}{\mathrm{4}}=\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\mathrm{however}\:\mathrm{if}\:\mathrm{you}\:\mathrm{interchange}\:×\:\mathrm{with}\:\boldsymbol{\div} \\ $$$$\mathrm{the}\:\mathrm{order}\:\mathrm{will}\:\mathrm{be}\:\mathrm{important} \\ $$$$\mathrm{2}\boldsymbol{\div}\mathrm{3}×\mathrm{4} \\ $$$$\mathrm{2}\boldsymbol{\div}\left(\mathrm{3}×\mathrm{4}\right)=\frac{\mathrm{2}}{\mathrm{3}×\mathrm{4}}=\frac{\mathrm{2}}{\mathrm{12}}=\frac{\mathrm{1}}{\mathrm{6}} \\ $$$$\left(\mathrm{2}\boldsymbol{\div}\mathrm{3}\right)×\mathrm{4}=\frac{\mathrm{2}}{\mathrm{3}}×\mathrm{4}=\frac{\mathrm{8}}{\mathrm{3}} \\ $$
Commented by Rasheed Soomro last updated on 01/Dec/15
Matter is interpretating DMAS rule.
$${Matter}\:{is}\:{interpretating}\:{DMAS}\:{rule}. \\ $$$$ \\ $$
Commented by Filup last updated on 01/Dec/15
It is more of a matter of interperating  the parenthese laws. The dmas law  is heavily weighed onto the ability  to correctly apply parenthesise.
$$\mathrm{It}\:\mathrm{is}\:\mathrm{more}\:\mathrm{of}\:\mathrm{a}\:\mathrm{matter}\:\mathrm{of}\:\mathrm{interperating} \\ $$$$\mathrm{the}\:\mathrm{parenthese}\:\mathrm{laws}.\:\mathrm{The}\:\mathrm{dmas}\:\mathrm{law} \\ $$$$\mathrm{is}\:\mathrm{heavily}\:\mathrm{weighed}\:\mathrm{onto}\:\mathrm{the}\:\mathrm{ability} \\ $$$$\mathrm{to}\:\mathrm{correctly}\:\mathrm{apply}\:\mathrm{parenthesise}. \\ $$
Commented by Rasheed Soomro last updated on 01/Dec/15
′Multiplication has priority over addition′. No doubt it    is. But this is only by convention. It is only this  convention,for which we write 5+6×8 instead of   5+(6×8). This priority is man−made. If there were  a convention in which addition had priority over  multiplication then we had written 5+6×8 instead of (5+6)×8  and 5+6×8 were calculated as under                5+6×8=11×8=88   So priority  is purely man−made.  Dicision to parenthecise one operation and leave other  is purely man−made.  One example of priority to be man−made is as under:  −3^2  is considerd as −(3^2 ) in our convention,but in some  computer programs it is treated as (−3)^2 .
$$'{Multiplication}\:{has}\:{priority}\:{over}\:{addition}'.\:{No}\:{doubt}\:{it}\:\: \\ $$$${is}.\:{But}\:{this}\:{is}\:{only}\:{by}\:\boldsymbol{{convention}}.\:{It}\:{is}\:{only}\:{this} \\ $$$${convention},{for}\:{which}\:{we}\:{write}\:\mathrm{5}+\mathrm{6}×\mathrm{8}\:{instead}\:{of}\: \\ $$$$\mathrm{5}+\left(\mathrm{6}×\mathrm{8}\right).\:{This}\:{priority}\:{is}\:\boldsymbol{{man}}−\boldsymbol{{made}}.\:{If}\:{there}\:{were} \\ $$$${a}\:{convention}\:{in}\:{which}\:{addition}\:{had}\:{priority}\:{over} \\ $$$${multiplication}\:{then}\:{we}\:{had}\:{written}\:\mathrm{5}+\mathrm{6}×\mathrm{8}\:{instead}\:{of}\:\left(\mathrm{5}+\mathrm{6}\right)×\mathrm{8} \\ $$$${and}\:\mathrm{5}+\mathrm{6}×\mathrm{8}\:{were}\:{calculated}\:{as}\:{under} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}+\mathrm{6}×\mathrm{8}=\mathrm{11}×\mathrm{8}=\mathrm{88}\: \\ $$$${So}\:\boldsymbol{{priority}}\:\:\boldsymbol{{is}}\:\boldsymbol{{purely}}\:\boldsymbol{{man}}−\boldsymbol{{made}}. \\ $$$${Dicision}\:{to}\:{parenthecise}\:{one}\:{operation}\:{and}\:{leave}\:{other} \\ $$$${is}\:{purely}\:\boldsymbol{{man}}−\boldsymbol{{made}}. \\ $$$$\mathrm{O}{ne}\:{example}\:{of}\:{priority}\:{to}\:{be}\:{man}−{made}\:{is}\:{as}\:{under}: \\ $$$$−\mathrm{3}^{\mathrm{2}} \:{is}\:{considerd}\:{as}\:−\left(\mathrm{3}^{\mathrm{2}} \right)\:{in}\:{our}\:{convention},{but}\:{in}\:{some} \\ $$$${computer}\:{programs}\:{it}\:{is}\:{treated}\:{as}\:\left(−\mathrm{3}\right)^{\mathrm{2}} . \\ $$

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