9th Oct, 2020

Delft University of Technology

Question

Asked 12th Jun, 2014

Division, Multiplication, Addition and Subtraction (DMAS) is the elementary rule for the order of operation of the Binary operations. What is the scientific and technical reason behind this mathematical myth though Multiplication before Division also gives the same result mostly? DMAS, a nice tool but has less convincing/appealing to admit its order of operation.

DMAS and MDAS don't give same result in general.

As mentioned, both may give same result MOSTLY, not GENERALLY.

Hence DMAS matters.

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There isn't a "scientific" reason -- mathematics is not a science. In fact it isn't always presented as "DMAS", you can also see "MDAS" and, possibly "DMSA" and "MDSA" as well. The technical reasons are that given A, B and C

* division, A/B, is really A * multiplicative_inverse(B)

* multiplication (and hence division) distributes over addition, so there are "implicit parentheses in A * B + C, like so: (A * B) + C

* subtraction, A - B, is really A + additive_inverse(B)

In practice, someone who works with equations regularly will simplify parts of an equation as they go, possibly collecting *appropriate* addends first

f = 127 * 892 + 1 - 1

might be simplified to

f = 127 * 892

as a first step, since it is obvious that the one's cancel each other.

The point of the DMAS rule is to make the binding order of infix notation explicit for people who are learning to do more than just "sums" or "products".

Hope this helps.

How can we calculate 6÷3×2=?

Does it result into 4 or 1? And one more thing...which will be the right way to express this {6÷(3×2)} or {(6÷3)×2}...? DMAS still confusing...does'nt it?

1 Recommendation

These are just conventions that are agreed upon like why is the right side of a number line consist of the positive numbers.

In my calculator 6 / 3 x 2=4 so my calculator was programmed so that

6 / 3 x 2 =(6/3) x 2

Geoff's comment on conventions is spot on. In many countries the positive Z axis will come out of a page when the positive Y axis goes to the top of the page and the positive X axis goes to the right (such as Australia, UK, ...). In others, the Z axis goes *into* the page. Just conventions. Of course when you change conventions, you also have to change equations :-)

I'm quite sure that mathematicians would have no problem with the 'DMAS rule', if it was actually a rule, which it isn't, and any decent mathematician would know this! DMAS is just another made up memory aide. One of the many acronyms taught around the world to help students remember the agreed upon standard for solving mixed operator expressions, the Order of Operations. The Order of Operations is not the product of actual scientific reasoning; it is not a mathematical theory or law. It is a globally agreed upon convention comprised of a series of steps that serve as a hierarchy of mathematical operators and processes. The Order of Operations may not have been devised as the result of science but they definitely follow a good deal of mathematical logic.

The mathematics behind the Order of Operations makes a lot of sense and it is not difficult to understand if it is well taught. Unfortunately when it comes to mathematics, teachers are all too often looking for shortcuts. Teaching students a mnemonic or an acronym is seen as being easier than explaining the actual mathematical concepts. The problem with doing this and teaching students the Order of Operations with a mnemonic or acronym is the likelihood of it going sideways. Students taught the acronym DMAS have a tendency to move through their lives believing that division takes precedence over multiplcation and is uniformly calculated ahead of multiplcation. Likewise many students believe that addition must be calculated before subtraction. The lack of a letter in DMAS that corresponds to both of the grouping symbols and exponentiation steps is also a problem because it leaves many students confused about the correct order to calculate expressions containing these.

MDAS is no better. It sees many people go through life believing that multiplcation has precedence over division and is uniformly calculated ahead of it. MDAS also results in the same problem as DMAS with its lack of a letter to represent the step of grouping symbols and the step of exponentiation. The other commonly taught acronyms: BODMAS, BEDMAS, BIDMAS, PEMDAS, GEMDAS and GEMS also have the potential to mislead. One of the biggest problems with the six letter acronyms is the tendency for people to incorrectly end up believing that they represent six steps when in fact they only represent four steps. As with DMAS and MDAS above, the position of the D and the M in the acronym leads many people to believe the letter that comes first (DM or MD) represents the operation that must always be done first.

The acronyms with an E, an O or an I also have the potential to confuse. For starters, the O that stands for Orders, is often mistakenly thought to stand for Of, Open, Off or Others which often leads people to errorously equate this step with 'opening' or 'clearing' any brackets or parentheses. Whereas this step comes after that of calculating operations contained in grouping symbols and as such has nothing to do with them. Another problematic factor with the E, the O or the I is that just the one letter is used to represent a multitude of operations that are all done as part of this step. This E/O/I step includes exponentiation and its inverse operations of radicals/roots and logarithms as well as unary operations like percentages, ratios, fractions and factorials and also functions including the trigonometric functions; sin, cos, tan, arsin, arcos and cot. The fact that this step uses one letter for multiple operations means there is often a gap in people's knowledge as they are unaware that it includes more than just exponents.

The use of B and P in their corresponding acronyms is potentially limiting too. Brackets and parentheses are just two out of a range of signs and symbols used to group operations. Other types of grouping symbols include braces, angle brackets, the modulus sign in absolute values, the vinculum as an extension of a radical to group any operations under it and the vinculum when used horizontally to separate any operations above it (the numerator) from operations below it (the denominator). Given the range of symbols used in this first step and their purpose of grouping the operations in an expression that need to be calculated as the first priority, I feel that the letter G, if an acronym is to be used, and the term 'grouping symbols' is a more appropriate representation of Step 1 in the Order of Operations.

As far as I'm concerned though, get rid of the acronyms and teach the Order of Operations in relation to the mathematical logic it follows.

Step 1 - Grouping Symbols

Calculate all operations grouped together and placed within an opening grouping symbol and a closing grouping symbol.

Step 2 - All Other Operations

Exponents and their inverse operations radicals roots and logarithms from left to right, factorials, percentages, fractions, ratios, functions. Multiple exponents uniquely go from right to left.

Step 3 - Multiplcation AND Division

Multiplcation and its equivalent inverse operation of division across the page in the order they appear from left to right.

Step 4 - Addition AND Subtraction

Addition and its equivalent inverse operation subtraction across the page in the order they appear from left to right.

Teaching the four steps above in this way should not be hard nor should it be hard for students to understand, at least not if they have been given a solid foundation in mathematics. A knowledge of inverse operations is clearly a prerequisite but that is easily taught and it should be taught regardless. Understanding that inverse operations are a version of each other, that they undo each other's calculations and are thus of the same precedence and therefore calculated with equal priority is essential to developing an actual understanding of the Order of Operations' structure. Another key concept to know is that multiplication is repeated addition. It is more powerful because it speeds up the addition process. The following addition: 2+2+2+2+2+2+2+2+2+2 uses 10 addends and 9 operators to get to the answer of 20. Multiplication is more powerful because of its repeated addition quality, it can reach the answer of 20 in the above using just 2 multiplicands and 1 operator like so: 10x2.

Exponentiation is to multiplication as multiplication is to addition. Exponentiation is repeated multiplication and as such it is more powerful again. The following multiplication: 2×2×2×2×2×2×2×2×2×2 uses 10 multiplicands and 9 operators to get to the answer of 1,024. Whereas the use of exponentiation is more powerful because it is repeated multiplication. It gets to the answer is 1,024 using just 1 number and 1 exponent like so: 2¹⁰.

The obvious logic is the the more powerful the operation, the higher it is in the Order of Operations hierarchy. And operations and their inverses logically go together too.

All you have to do is look around the web to see the mass confusion that results from incorrect knowledge of the Order of Operations and even more so the misuse of each of the acronyms discussed here.

6÷2(1+2)

-2²

4+4-4

3÷3+3-3+3÷3

3+3×0+3+3÷3

These questions are frequently asked, hotly debated and more often than not the majority of answers given are incorrect. If only the Order of Operations was explained to students by knowledgeable teachers at the start of their mathematical education!!!

2 Recommendations

In the case of the arithmetic operations (addition and subtraction, division and multiplication), the priorities of the arithmetic operations are divided according to the processes in the magnitude, and if the magnitude is devoid of arcs, roots, and exponents, the arrangement will be as follows: division and multiplication, division and multiplication are stronger than addition and subtraction, and in If they are in one of the ingredients, the priority is for them first and then the addition and subtraction operations, and the matter did not end here, so the division and multiplication are still in the same cuff and one of them must be determined ahead of the other process, and the arrangement and priority takes place according to their presence in the matter, if the amount is written in the language Show Yeh, priority from the right, but if the amount is written in English, priority from the left, that is the priority of the right process (multiplication, division) which is written first. Addition and subtraction, the addition and subtraction operations are considered in the second order after multiplication and division, and if the two operations exist together in the same matter, then the priority is according to their location in the amount, and if the magnitude is written in Arabic then the priority is from the right, but if the magnitude is written in English then the priority From the left side, meaning that the right of the process (addition, subtraction) is written first

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