Question Number 45014 by peter frank last updated on 07/Oct/18

$$\mathrm{6}\boldsymbol{\div}\mathrm{2}\left(\mathrm{1}+\mathrm{2}\right)?\:\mathrm{which}\:\boldsymbol{\mathrm{ans}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{correct}}\:\boldsymbol{\mathrm{between}}\:\mathrm{1}\boldsymbol{\mathrm{or}}\:\mathrm{9} \\ $$$$\boldsymbol{\mathrm{why}}\:\boldsymbol{\mathrm{do}}\:\boldsymbol{\mathrm{i}}\:\boldsymbol{\mathrm{look}}\:\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{mobile}}\:\boldsymbol{\mathrm{phone}}\:\boldsymbol{\mathrm{calculator}}\:\boldsymbol{\mathrm{is}}\:\mathrm{9}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{when}}\:\boldsymbol{\mathrm{i}}\:\boldsymbol{\mathrm{check}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{scientific}}\:\boldsymbol{\mathrm{calculator}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{ans}}\:\boldsymbol{\mathrm{is}}\:\mathrm{1}.\boldsymbol{\mathrm{i}}\:\boldsymbol{\mathrm{want}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{clarify}}\:\boldsymbol{\mathrm{correctly}}\:\boldsymbol{\mathrm{where}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{are}}. \\ $$
Commented by peter frank last updated on 07/Oct/18

$$\mathrm{ajfour},\mathrm{tanmay},\mathrm{MJS},\mathrm{Mr}_{\:} \underset{\mathrm{3}} {\mathrm{w}}. \\ $$
Commented by peter frank last updated on 07/Oct/18

$$\mathrm{why}\:\mathrm{are}\:\mathrm{the}\:\mathrm{answers}\:\mathrm{different}. \\ $$
Commented by MJS last updated on 07/Oct/18

$$\mathrm{there}'\mathrm{s}\:\mathrm{no}\:\mathrm{priority}\:\mathrm{for}\:\mathrm{implied}\:\mathrm{multiplication} \\ $$$$\mathrm{although}\:\mathrm{some}\:\mathrm{think}\:\mathrm{there}\:\mathrm{is} \\ $$$$\mathrm{6}\boldsymbol{\div}\mathrm{2}\left(\mathrm{1}+\mathrm{2}\right)=\mathrm{6}\boldsymbol{\div}\mathrm{2}×\left(\mathrm{1}+\mathrm{2}\right)=\mathrm{9} \\ $$$$\mathrm{to}\:\mathrm{get}\:\mathrm{1}\:\mathrm{you}\:\mathrm{must}\:\mathrm{write}\:\mathrm{6}\boldsymbol{\div}\left(\mathrm{2}\left(\mathrm{1}+\mathrm{2}\right)\right)\:\mathrm{or} \\ $$$$\mathrm{6}\boldsymbol{\div}\left(\mathrm{2}×\left(\mathrm{1}+\mathrm{2}\right)\right)\:\mathrm{or}\:\frac{\mathrm{6}}{\mathrm{2}×\left(\mathrm{1}+\mathrm{2}\right)} \\ $$
Commented by MJS last updated on 08/Oct/18

$$\mathrm{Sir}\:\mathrm{Peter}\:\mathrm{Frank},\:\mathrm{which}\:\mathrm{scientific}\:\mathrm{calculator} \\ $$$$\mathrm{do}\:\mathrm{you}\:\mathrm{use}? \\ $$
Commented by peter frank last updated on 08/Oct/18

$$\mathrm{casio}\:\mathrm{fx}\:\mathrm{Ms}\:\mathrm{991} \\ $$
Commented by MJS last updated on 08/Oct/18

$$\mathrm{I}\:\mathrm{just}\:\mathrm{read}\:\mathrm{the}\:\mathrm{manual}\:\mathrm{online}\:\mathrm{and}\:\mathrm{obviously} \\ $$$$\mathrm{the}\:\mathrm{calculator}\:\mathrm{is}\:\mathrm{programed}\:\mathrm{to}\:\mathrm{first}\:\mathrm{calculate} \\ $$$$\mathrm{terms}\:\mathrm{like}\:\mathrm{2}\pi\:\mathrm{or}\:\mathrm{5}\sqrt{\mathrm{3}}\:\mathrm{which}\:{seems}\:\mathrm{to}\:\mathrm{be}\:\mathrm{a} \\ $$$$\mathrm{simplification}\:\mathrm{but}\:\mathrm{it}\:\mathrm{leads}\:\mathrm{to}\:\mathrm{mistakes}. \\ $$$$\mathrm{what}\:\mathrm{does}\:\mathrm{this}\:\mathrm{mean}: \\ $$$$\mathrm{1}/\mathrm{2}\pi+\mathrm{1}/\mathrm{3}\pi \\ $$$$\mathrm{standard}\:\mathrm{math}\:\mathrm{is}\:\mathrm{definite}\:\mathrm{and}\:\mathrm{precise} \\ $$$$\mathrm{1}/\mathrm{2}\pi+\mathrm{1}/\mathrm{3}\pi=\frac{\mathrm{1}}{\mathrm{2}}\pi+\frac{\mathrm{1}}{\mathrm{3}}\pi=\frac{\mathrm{5}}{\mathrm{6}}\pi \\ $$$$\mathrm{1}/\left(\mathrm{2}\pi\right)+\mathrm{1}/\left(\mathrm{3}\pi\right)=\frac{\mathrm{1}}{\mathrm{2}\pi}+\frac{\mathrm{1}}{\mathrm{3}\pi}=\frac{\mathrm{5}}{\mathrm{6}\pi} \\ $$$$\mathrm{1}/\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{5}=\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{3}}+\mathrm{5}\:\mathrm{not}\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}}+\mathrm{5}\:\mathrm{and}\:\mathrm{not}\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{5}} \\ $$$$\mathrm{1}/\mathrm{2}\left(\pi+{k}\right)=\frac{\mathrm{1}}{\mathrm{2}}\left(\pi+{k}\right) \\ $$$$\mathrm{sorry}\:\mathrm{to}\:\mathrm{say}\:\mathrm{that}\:\mathrm{casio}\:\mathrm{calculators}\:\mathrm{are}\:\mathrm{often} \\ $$$$\mathrm{wrong}.\:\mathrm{buy}\:\mathrm{a}\:\mathrm{texas}\:\mathrm{instruments}\:\mathrm{or}\:\mathrm{a}\:\mathrm{hp}\:\mathrm{unit} \\ $$$$\mathrm{or}\:\mathrm{use}\:\mathrm{a}\:\mathrm{good}\:\mathrm{app}\:\mathrm{like}\:\mathrm{HiPER}\:\mathrm{calc}\:\mathrm{pro} \\ $$
Answered by Joel578 last updated on 07/Oct/18

$$\mathrm{Bracket},\:\mathrm{then}\:\mathrm{division},\:\mathrm{then}\:\mathrm{multiplication} \\ $$$$\mathrm{6}\boldsymbol{\div}\mathrm{2}×\mathrm{3} \\ $$$$=\mathrm{3}×\mathrm{3} \\ $$$$=\mathrm{9} \\ $$
Commented by MJS last updated on 08/Oct/18

$$\mathrm{sorry}\:\mathrm{but}\:\mathrm{that}'\mathrm{s}\:\mathrm{not}\:\mathrm{standard}\:\mathrm{mathematic} \\ $$$$\mathrm{where}\:\mathrm{have}\:\mathrm{you}\:\mathrm{learned}\:\mathrm{this}? \\ $$
Commented by Rasheed.Sindhi last updated on 08/Oct/18

$$\mathrm{6}\boldsymbol{\div}\mathrm{2}\left(\mathrm{1}+\mathrm{2}\right)\:\mathrm{and}\:\mathrm{6}\boldsymbol{\div}\mathrm{2}×\left(\mathrm{1}+\mathrm{2}\right)\:\mathrm{are}\:\mathrm{different}. \\ $$$$\mathrm{In}\:\mathrm{6}\boldsymbol{\div}\mathrm{2}\left(\mathrm{1}+\mathrm{2}\right),\:\:\mathrm{2}\left(\mathrm{1}+\mathrm{2}\right)\:\mathrm{is}\:\mathrm{like}\:\mathrm{a}\:\mathrm{single}\: \\ $$$$\mathrm{term}\:\mathrm{which}\:\mathrm{should}\:\mathrm{be}\:\mathrm{calculated}\:\mathrm{first}. \\ $$$$\left(\mathrm{In}\:\mathrm{other}\:\mathrm{words}\:\:\mathrm{x}\boldsymbol{\div}\mathrm{yz}\neq\mathrm{x}\boldsymbol{\div}\mathrm{y}×\mathrm{z}\right. \\ $$$$\:\:\ast\:\mathrm{x}\boldsymbol{\div}\mathrm{yz}=\frac{\mathrm{x}}{\mathrm{yz}} \\ $$$$\left.\:\ast\:\:\mathrm{x}\boldsymbol{\div}\mathrm{y}×\mathrm{z}=\frac{\mathrm{x}}{\mathrm{y}}×\mathrm{z}=\frac{\mathrm{xz}}{\mathrm{y}}\:\right) \\ $$$$\mathrm{So}, \\ $$$$\mathrm{6}\boldsymbol{\div}\mathrm{2}\left(\mathrm{1}+\mathrm{2}\right)=\mathrm{6}\boldsymbol{\div}\mathrm{2}\left(\mathrm{3}\right)=\mathrm{6}\boldsymbol{\div}\mathrm{6}=\mathrm{1} \\ $$$$\left(\mathrm{9}\:\mathrm{is}\:\mathrm{not}\:\mathrm{correct}.\right) \\ $$
Commented by Rasheed.Sindhi last updated on 08/Oct/18

$$\mathrm{In}\:\mathrm{other}\:\mathrm{words}\:\:\mathrm{implicit}\:\mathrm{multiplication} \\ $$$$\mathrm{has}\:\mathrm{priority}\:\mathrm{over}\:\mathrm{division}\:\mathrm{and}\:\mathrm{division} \\ $$$$\mathrm{have}\:\mathrm{priority}\:\mathrm{over}\:\mathrm{explicit}\:\mathrm{multiplication}…. \\ $$$$\mathrm{Although}\:\mathrm{at}\:\mathrm{the}\:\mathrm{moment}\:\mathrm{I}\:\mathrm{haven}'\mathrm{t}\:\mathrm{a}\:\mathrm{reference}. \\ $$
Commented by MJS last updated on 08/Oct/18

$$\mathrm{multiplication}\:\mathrm{and}\:\mathrm{division}\:\mathrm{have}\:\mathrm{equal} \\ $$$$\mathrm{priority},\:\mathrm{similar}\:\mathrm{to}\:\mathrm{addition}\:\mathrm{and}\:\mathrm{subtraction} \\ $$$$\mathrm{5}−\mathrm{3}+\mathrm{7}−\mathrm{8}=\left(\left(\mathrm{5}−\mathrm{3}\right)+\mathrm{7}\right)−\mathrm{8}=\mathrm{1} \\ $$$$\mathrm{5}/\mathrm{3}×\mathrm{7}/\mathrm{8}=\left(\left(\mathrm{5}/\mathrm{3}\right)×\mathrm{7}\right)/\mathrm{8}=\frac{\mathrm{35}}{\mathrm{24}} \\ $$$$ \\ $$$$\mathrm{implicit}\:\mathrm{multiplication}\:\mathrm{is}\:\mathrm{just}\:\mathrm{multiplication} \\ $$$$\mathrm{where}\:\mathrm{we}\:\mathrm{left}\:\mathrm{the}\:“×''\:\mathrm{sign}\:\mathrm{because}\:\mathrm{it}'\mathrm{s}\:\mathrm{less} \\ $$$$\mathrm{writing}\:\mathrm{work}.\:\mathrm{there}'\mathrm{s}\:\mathrm{no}\:\mathrm{extra}\:\mathrm{rule}\:\mathrm{for}\:\mathrm{it}. \\ $$$${ab}/{cd}={a}×{b}/{c}×{d}\:\mathrm{must}\:\mathrm{be}\:\mathrm{calculated}\:\mathrm{from} \\ $$$$\mathrm{left}\:\mathrm{to}\:\mathrm{right}.\:\mathrm{everything}\:\mathrm{else}\:\mathrm{is}\:\mathrm{sloppy}\:\mathrm{habit} \\ $$$$\mathrm{people}\:\mathrm{write}\:{ab}/{cd}\:\mathrm{instead}\:\mathrm{of}\:\frac{{ab}}{{cd}},\:\mathrm{that}'\mathrm{s}\:\mathrm{the} \\ $$$$\mathrm{initial}\:\mathrm{problem}.\:\mathrm{but}\:\mathrm{we}\:\mathrm{have}\:\mathrm{to}\:\mathrm{have}\:{one}\:\mathrm{rule} \\ $$$$\mathrm{if}\:\frac{{termA}}{{termB}}={termA}/{termB}\:\Leftrightarrow\:\frac{\mathrm{4}{x}+\mathrm{3}}{\mathrm{3}{x}−\mathrm{4}}=\mathrm{4}{x}+\mathrm{3}/\mathrm{3}{x}−\mathrm{4} \\ $$$$\mathrm{and}\:\mathrm{confusion}\:\mathrm{is}\:\mathrm{complete} \\ $$$$\mathrm{instead}\:\mathrm{it}'\mathrm{s}\:\frac{{termA}}{{termB}}=\left({termA}\right)/\left({termB}\right) \\ $$$$\Rightarrow\:\mathrm{6}\boldsymbol{\div}\mathrm{2}\left(\mathrm{1}+\mathrm{2}\right)\neq\frac{\mathrm{6}}{\mathrm{2}\left(\mathrm{1}+\mathrm{2}\right)} \\ $$
Commented by MJS last updated on 08/Oct/18

$$\mathrm{another}\:\mathrm{argument}: \\ $$$$\mathrm{implied}\:\mathrm{multiplication}\:\mathrm{is}\:\mathrm{not}\:\mathrm{defined}\:\mathrm{for} \\ $$$${a},\:{b}\:\in\mathbb{R}.\:\mathrm{or}\:\mathrm{how}\:\mathrm{can}\:\mathrm{we}\:\mathrm{write}\:{ab}\:\mathrm{for}\:{a}=\mathrm{7}.\mathrm{43} \\ $$$$\mathrm{and}\:{b}=−\mathrm{2}.\mathrm{56}? \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{just}\:\mathrm{writing}\:\mathrm{shortcut},\:\mathrm{not}\:\mathrm{an}\:\mathrm{operation} \\ $$
Commented by ajfour last updated on 08/Oct/18

$${where}\:{have}\:{you}\:{been},{all}\:{this}\:{while},\: \\ $$$${Rasheed}\:{Sir}? \\ $$
Commented by MJS last updated on 08/Oct/18

$$\mathrm{there}\:\mathrm{are}\:\mathrm{some}\:“\mathrm{bad}\:\mathrm{spellings}'',\:\mathrm{most}\:\mathrm{of}\:\mathrm{them} \\ $$$$\mathrm{sloppiness}.\:\mathrm{but}\:\mathrm{we}\:\mathrm{learned}\:\mathrm{in}\:\mathrm{school}\:\mathrm{things} \\ $$$$\mathrm{like} \\ $$$$\frac{\mathrm{35}}{\mathrm{4}}=\mathrm{8}\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{just}\:\mathrm{for}\:\mathrm{children}\:\mathrm{to}\:\mathrm{make}\:\mathrm{it}\:\mathrm{visible}\:\mathrm{for} \\ $$$$\mathrm{them} \\ $$$$\left.\mathrm{standard}\:\mathrm{math}\:\mathrm{for}\:\mathrm{adults}\:;−\right) \\ $$$$\mathrm{8}\frac{\mathrm{3}}{\mathrm{4}}=\mathrm{8}×\frac{\mathrm{3}}{\mathrm{4}}=\mathrm{6} \\ $$$$\mathrm{because}\:\mathrm{otherwise}:\:\mathrm{confusion}.\:\mathrm{solve} \\ $$$${x}\frac{\mathrm{3}}{\mathrm{4}}=\mathrm{6} \\ $$$$\mathrm{hopefully}\:\mathrm{it}'\mathrm{s}\:{x}=\mathrm{8}\:\mathrm{not}\:{x}=\frac{\mathrm{21}}{\mathrm{4}} \\ $$$$\mathrm{also}\:\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{calculate}\:\mathrm{with}\:\mathrm{these} \\ $$$$\mathrm{mixed}\:\mathrm{fractions} \\ $$$$\mathrm{8}\frac{\mathrm{3}}{\mathrm{4}}\pm\mathrm{3}\frac{\mathrm{5}}{\mathrm{6}}= \\ $$$$\mathrm{8}\frac{\mathrm{3}}{\mathrm{4}}×\mathrm{3}\frac{\mathrm{5}}{\mathrm{6}}= \\ $$$$\mathrm{8}\frac{\mathrm{3}}{\mathrm{4}}\boldsymbol{\div}\mathrm{3}\frac{\mathrm{5}}{\mathrm{6}}= \\ $$$$\mathrm{we}\:\mathrm{have}\:\mathrm{to}\:\mathrm{use}\:\mathrm{standard}\:\mathrm{fractions}\:\mathrm{and}\:\mathrm{standard} \\ $$$$\mathrm{maths} \\ $$
Commented by Rasheed.Sindhi last updated on 08/Oct/18

$$\mathcal{T}{hanks}\:{Sir}\:{ajfour}\:{for}\:{missing}\:{me}!!! \\ $$$${Actually}\:{unfortunately}\:{I}\:{lost}\:{my}\:{interest} \\ $$$${for}\:{mathematics}!\:{I}'{ll}\:{try}\:{to}\:{participate} \\ $$$${in}\:{the}\:{forum}\:{again}.\: \\ $$$$\mathcal{T}{hanks}\:{again}! \\ $$
Commented by MJS last updated on 08/Oct/18

$$\mathrm{there}'\mathrm{s}\:\mathrm{an}\:\mathrm{international}\:\mathrm{standard},\:\mathrm{is}\:\mathrm{all}\:\mathrm{I}'\mathrm{m} \\ $$$$\mathrm{saying}. \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{is}\:\mathrm{an}\:\mathrm{entity},\:\mathrm{I}\:\mathrm{can}\:\mathrm{write}\:\mathrm{1}/\mathrm{2} \\ $$$$\mathrm{2}\pi\:\mathrm{is}\:\mathrm{an}\:\mathrm{entity},\:\mathrm{I}\:\mathrm{can}\:\mathrm{write}\:\frac{\mathrm{1}}{\mathrm{2}\pi}\:\mathrm{or}\:\mathrm{1}/\mathrm{2}\pi \\ $$$$\mathrm{in}\:\mathrm{1}/\mathrm{2}\pi,\:\mathrm{which}\:\mathrm{entity}\:\mathrm{has}\:\mathrm{priority}? \\ $$$$\mathrm{solve}\:\mathrm{this}: \\ $$$${x}^{\mathrm{2}} +\mathrm{1}/\mathrm{2}{x}+\mathrm{1}/\mathrm{4}=\mathrm{0} \\ $$$$\mathrm{is}\:\mathrm{it}\:\mathrm{obviously}\:\mathrm{a}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{degree}\:\mathrm{polynome}?\:\mathrm{or}\:\mathrm{is} \\ $$$$\mathrm{1}/\mathrm{2}{x}\:\mathrm{a}\:\mathrm{division}\:\mathrm{by}\:\:\mathrm{2}{x}? \\ $$$$\mathrm{and}\:\mathrm{this}:\:\int{dx}/{x}^{\mathrm{2}} +\mathrm{1}/\mathrm{2}{x}+\mathrm{1}/\mathrm{4} \\ $$$$\mathrm{or}\:\mathrm{sin}\:{x}+\mathrm{1}/\mathrm{2} \\ $$$$\mathrm{or}\:\sqrt{}\mathrm{2}{x}−\mathrm{3} \\ $$$$\mathrm{I}'\mathrm{ve}\:\mathrm{seen}\:\mathrm{similar}\:\mathrm{expressions}\:\mathrm{in}\:\mathrm{this}\:\mathrm{forum} \\ $$$$\mathrm{and}\:\mathrm{I}\:\mathrm{refused}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{them}\:\mathrm{sometimes} \\ $$$$\mathrm{because}\:\mathrm{it}'\mathrm{s}\:\mathrm{never}\:\mathrm{obvious}\:\mathrm{when}\:\mathrm{it}\:\mathrm{comes} \\ $$$$\mathrm{to}\:\mathrm{mathematics}.\:\mathrm{that}'\mathrm{s}\:\mathrm{the}\:\mathrm{reason}\:\mathrm{I}\:\mathrm{plead} \\ $$$$\mathrm{in}\:\mathrm{favour}\:\mathrm{of}\:\mathrm{the}\:\mathrm{standard}\:\mathrm{spelling}\:\mathrm{of}\:\mathrm{maths} \\ $$
Commented by MrW3 last updated on 08/Oct/18

$${Sir}\:{Rasheed}: \\ $$$${You}\:{are}\:{missed}\:{by}\:{me}\:{too}.\:{Welcome}\: \\ $$$${back}! \\ $$
Commented by Rasheed.Sindhi last updated on 09/Oct/18

$$\mathcal{T}{hanks}\:\boldsymbol{{sir}}\:\boldsymbol{{MrW}}\mathrm{3}. \\ $$$$\boldsymbol{{Sir}}\:\boldsymbol{{MJS}} \\ $$$${Same}\:{discussion}\:{is}\:{present}\:{on}\:{the} \\ $$$${following}\:{link}\left({see}\:{comments}\:{below}\right) \\ $$$${which}\:{suggest}\:{that}\:{there}'{s}\:{no}\:{agreed} \\ $$$${international}\:{standard}\:{in}\:{this}\:{connection}. \\ $$$${Both}\:{conventions}\:{are}\:{continue}\:{simultaneously}!! \\ $$
Commented by Rasheed.Sindhi last updated on 09/Oct/18

Commented by Rasheed.Sindhi last updated on 09/Oct/18
