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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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