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| caption1 = The Casio fx-9860 GII SD returns the correct answer to the equation as given, and . The next row provides the answer that a less sophisticated calculator would return.

'''6÷2(1+2)=?''' is the pre-algebra question that a majority of many Internet respondents seem to get wrong. If you are in the camp that knows the correct answer is 1, read no further. If you believe the answer is 9, you should probably review this document -- {{highlight|there is only one number that 6 can be divided by to result in 9, and that number is 2/3.}}

We substitute x for 2(1+2) from the given equationand backtest the equation with the result of 9:

there is a Common Denominator common divisor of 3 so we reduce the fraction...

Now we compare the value given in the equation to the value we find found (above) for x.

<p style==Processing rules== In the majority of incorrect proofs"text-align:center;"> {{highlight|Without fail, the respondent 2(1+2) will cite PEMDAS or BODMAS, however these solvers incorrectly resolve the [P]arentheses or [B]rackets of PEMDAS - BODMASalways equal (2*1 + 2*2), ignoring this is the '''Distributive Law. For some reason, they believe that they can solve within the parentheses, but ignore the adjacent and dependent coefficient. '''}}</p> == Proof of correct solution ==''Encyclopedia Britannica'':

Therefore, in 6÷2(1+2)=?, we start with the <span style="color:green;"> Distributive Law </span> in the <span style="color:green;">Parenthesis </span> step of the Order of Operations.

::'''6÷2(1+2)=?'''::'''6÷<span style="color:green;">(2*1 + 2*2)</span>=?'''

==Processing rules== In the majority of incorrect proofs the respondent will cite PEMDAS or BODMAS, we all agree that parentheses or brackets are processed first. However, some respondents incorrectly resolve the [P]arentheses or [B]rackets of PEMDAS - BODMAS, for some reason, these solvers believe that they can resolve within the parentheses, but neglect to multiply the adjacent and dependent coefficient. === PEMDAS ===<blockquote> ::P parenthesis::E exponentiation::M multiplication::D division::A addition::S subtraction ::&ndash;[https://mathworld.wolfram.com/PEMDAS.html Wolfram PEMDAS]</blockquote> === Parentheses ===''Wolfram Mathworld'' also confirms that this is the way we handle on parenthetical expressions:

::&ndash;[http://mathworld.wolfram.com/Parenthesis.html Wolfram Parenthesis]</blockquote> ===Distributive law===''Encyclopedia Britannica'' on parenthetical expressions: <blockquote>::'''''Distributive law''''', ''in mathematics, the law relating the operations of multiplication and addition, stated symbolically, a(b + c) = ab + ac; that is, the monomial factor a is distributed, or separately applied, to each term of the binomial factor b + c, resulting in the product ab + ac.''::&ndash;[https://www.britannica.com/science/distributive-law Encyclopedia Britticana]

::<s>'''<div style="color:red;">3*(3)</div>'''</s>-->== The Fundamentals of Algebra (1983) = Parenthetical expressions === 

''Parenthetical Expression. The parenthesis was described in Chapter 1 as a grouping symbol. When an algebraic expression is enclosed by a parenthesis it is known as a parenthetical expression. '''When a parenthetical expression is immediately preceded by coefficient, the parenthetical expression is a factor and <u>must be multiplied by the coefficient.</u>''' This is done in the following manner.''

Standing alone== Explicit vs Implicit (implied) multiplication ==<div style="text-align:center">{{Quote|text='''''Implied multiplication''' has a higher priority than explicit multiplication to allow users to enter expressions, 6 will always equal 6 and 2in the same manner as they would be written.''<br/> &ndash;[https://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110 Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators] }}</div> Explicit operations have explicit operators (1* ÷ +2- ) will also always equal sixwhich are delimiters that divide equations into separate terms. Implied multiplication is notation that informs us that the value of a variable, so dividing these two bracketed function or exponent are connected and not separate terms will . ''Implied multiplication is everywhere, 1x = x and x/1=x. Any number times 1 is that number and any number divided by 1 is that number.'' Implied multiplication is also implicit multiplication. {{Quote|text='''im·plic·it''' ''adjective'' :1. implied though not plainly expressed. :2. essentially or very closely connected with; always equal to be found in. '''im·plied''' ''adjective'' :1. We would suggested but not write 2directly expressed; implicit. '''ex·plic·it''' ''adjective'' :1. stated clearly and in detail, leaving no room for confusion or doubt. &ndash;Oxford}} 6÷2(a1+b2) as has two EXPLICIT operators, division and addition. However, IMPLIED multiplication tells us to IMPLICITLY multiply, [2(2(a+b)1)], in the P step of PEMDAS. In the division step, nor would we re-write must apply division to the given equation as 6÷2*(a+bentire)implicitly connected parenthetical expression. Granted, We cannot apply division to the [https://www.wyzant.com/resources/blogs/14831/the_obelus_for_division obelus coefficient (÷2)] is an archaic symbol for and then not apply division, but this is not a postgraduate level equation, this is middle-school level mathto the factor (2+1).

2'''Incorrect:''' (1+2) ''because division is understood to be a function, placing this function anywhere in a larger equation must always resolve not EXPLICITLY applied to the same value. entire parenthetical expression'')<div style="color:red;"><div style="width:60px; text-align:center; float:left;">6 ---- The equation, 6÷2(1+2) </div><div style=? is illustrative to why we cannot substitute explicit multiplication for implied multiplication, in other words, 6÷2(1+2) is not equal to 6÷2"width:60px; text-align:center; float:left;"><br/><nowiki>*(1+2).</nowiki><br/></div>

{{Quote|<div style="width:60px; text=''Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written.''-align:center; float:left;"><br/>(2+1)<br/></div>

&ndash<div style="width:60px;[httpstext-align:center; float:left;"><br/>=<br/></epsstore.ti.comdiv> <div style="width:60px; text-align:center; float:left;"><br/>3 * (1+2) <br/> </div></OA_HTMLdiv></csksxvmdiv><div style="clear:both;"></div> '''Correct:''' (''when EXPLICIT division is applied, it is applied to the entire term.jsp?nSetId'')<div style=103110 Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators]"color:green;"><div style="width:60px; text-align:center; float:left;">6 ---- }}1 </div><div style="width:60px; text-align:center; float:left;"><br/>÷<br/></div>

<div style==Correctly solving "width:60px; text-align:center; float:left to right==;">2(1+2) ---- 1 </div>

If we insist on a <div style="width:60px; text-align:center; float:left to right solution, ignoring both PEMDAS and the Distributive Law, we can do this with the Least Common Denominator rule;"><br/>=<br/></div>

<div style="width:60px; text-align:''6÷2center; float:left;">2 * (1+2)=?'' ---- ::''3÷(6 * 1+2)=?'' </div></div></div>::''3÷(3)<div style=?'':"clear:''3÷3=1''both;"></div>

<div style==Solution written as a fraction =="color:green;"><div style="width:100px60px; text-align:center; float:left;">

<div style="width:100px60px; text-align:center; float:left;">

<div style="width:100px60px; text-align:center; float:left;">

1*(1+2)

<div style="width:100px60px; text-align:center; float:left;">

<div style="width:100px60px; text-align:center; float:left;">

<div style="width:100px60px; text-align:center; float:left;">

= 1

<!--As we can see, we cannot add an EXPLICIT operator to 6÷2(1+2) without maintaining the IMPLICIT relationship that was stated with IMPLIED multiplication. 6÷2(1+2) == The calculator problem ==6÷(2*(1+2)).-->

The [https://www.desmos.com/scientific Desmos Scientific Calculator] handles our given equation correctly. The error checking appears ==Correctly solving left to be built into the division key.right==

I realized that shorthand parenthetical notation could become a problem when I was taking a ::''6÷2(1+2)=?''::''3÷1(1+2)=?'C' programming course in college. Unfortunately, we are still using code that was written for computers that had severe memory limitations. ::''3÷(1+2)=?''::''3÷(3)=?''::''3÷3=1''

Unlike humansWhen we reduce by the common divisor, mathematics modules do this does not look ahead to identify a parenthetical expression embedded within an equation. In complete the 'C' languagedivision operation, it simply reduces the coefficient to process our given equation, we need explicit operators adjacent one. Division must be applied to both the first parenthesis; additionally, we must maintain factor and the grouping coefficient of our function(s). ''Still to this day, many programming languages / spreadsheets cannot process a the parenthetical expression without an explicit operator.''

Entering the ==The equation into a programming languageis not ambiguous =====6÷2(1+2) is not equal to 6÷2*(1+2)===Standing alone, or low quality calculator require the explicit multiplication symbol 6 will always equal 6 and outer parenthesis2(1+2) will also always equal 6, so dividing these two terms will always equal 1.

'''We would not write 2(a+b) as (2(a+b)), nor can we re-write the given equation as 6÷2*(1a+2b) == and maintain the same value. We can however, rewrite the full equation to include an explicit multiplier if we take care to maintain the parenthetical expression with additional bracketing 6÷(2*(1a+2b))'''.

When our given equation is correctly conditioned to read=== Obelus ===Granted, 6÷(2*the [https://web.archive.org/web/20210417145938/https://www.wyzant.com/resources/blogs/14831/the_obelus_for_division obelus (1+2÷)) the mathematics routine reserves additional memory to process ] is an archaic symbol for division, it visually represents a fraction with one dot being the parenthetical expression(s) numerator and then reassembles the equation according to the rules of other being the Order of Operations (PEMDAS)denominator. Conversely, in our given equation, 6÷2(1+2) Computers use the calculator processes 6÷2 and then discovers forward slash for division because the parenthetical expression too late. Instead of branching to fix the mistake, or declaring standard keyboard does not have an error condition, mathematics modules simply assume that the data-entry person is competent and adds explicit multiplication where implicit multiplication was givenobelus key.

=== The calculator solution =problem ==

Our The [https://www.desmos.com/scientific Desmos Scientific Calculator] handles our given equation is two terms, 6 is always equal to 6 and 2(1+2) is also always equal to 6, dividing these two numbers must always equal 1correctly. The equation is not ambiguous, it challenges error checking appears to be built into the theory that multiplication and division are equal in terms of the Order of Precedence. First, let's review fundamental fact that x÷1 = x and that 1*x = x. When we enclose a term within parentheses, implicit multiplication assumed, if the multiplier is not given, the multiplier is the number 1 because 0 times any number is 0key.

<div style="background-color:#E8E8E8; padding-left:30px;">Example that ignores Entering the Distributive Law: '''6÷2(1+2)=?''' ::becomes..equation into a programming language, or low quality calculator, requires the explicit multiplication symbol and outer parenthesis to maintain value of the parenthetical expression. '''3÷1(1+2)=?'''

As we see aboveOther calculators, including Google and Wolfram will simply strip the explicit division sustainsparenthesis and solve a different equation, as does 6÷2*3. This is because within the implicit multiplicationprogramming, we have simply reduced both sides of the equation by open parenthesis (bracketing) triggers a different function within the common denominator of 2programming, leaving us with ''3÷1(1+2)=?''an open parenthesis tells the compiler to find the innermost parenthesized term and work outwards. When we try Thus, to divide three by one, get the one is also a common denominatorcorrect answer from inferior calculators, finding a common denominator does not dismiss division, but if we dismiss the assumed one, we input must also dismiss the implicit multiplication. Continuing from above, we are left be formalized with the following equationcorrect bracketing.I.e.6÷(2*(1+2))

'''3÷==Spreadsheets==Entering the equation as =6/2(1+2)into a cell in a LibreOffice Calc spreadsheet will result in Err:509 (Missing operator), Google Sheets also returns an error (Formula parse error). This forces the user to format the equation using explicit multiplication. To avoid the left to right problem, the compiler must be instructed in advance to prepare for the function with an extra pair of parenthesis =?'''6/(2*(1+2)).

'''3÷(3)==See also==* [https://www.nytimes.com/2019/08/05/science/math-equation-pemdas-bodmas.html "That Vexing Math Equation?'''Here’s an Addition"] '''3÷3* [https://www.youtube.com/watch?v=1'''hsZCtgFcL40 "How to Solve 8÷2(2+2) Using Implied Multiplication"]<* [https://slate.com/technology/div>'''Therefore,''' if the calculator module has already processed the division 6÷2 by the time it reaches the parenthesis, the coefficient and the accompanying implicit multiplication has been eliminated2013/03/facebook-math-problem-why-pemdas-doesnt-always-give-a-clear-answer. The division must be sustained to apply to the remainder of html "What Is the function, in left Answer to right processing, it cannot be presumed that an implicitThat Stupid Math Problem on Facebook?"]* [https://implied operator always infers multiplication in computer mathematics moduleswww.inc. In com/dave-kerpen/this case, -basic-math-problem-is-breaking-internet.html This Basic Math Problem Is Breaking the processing must branch.Internet: How do you solve this simple arithmetic problem?]