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# Order of Operations

This post is meant to poll your opinion.

I have received clarifications about how an equation should be interpreted, which indicates that the student doesn't understand how Order of Operations work.

Example 1: In the problem regarding $$a + b \times c = 100$$, some students are confused about whether it means $$a + ( b \times c) = 100$$ or $$(a+ b) \times c = 100$$.

Example 2: In the problem regarding $$6 \div 2 \times 3$$, some students are confused about whether it means $$( 6 \div 2) \times 3$$ or $$6 \div ( 2 \times 3)$$.

Are there countries which use a different system of order of operations, in which you would choose the second expression in either of the above examples?

Note by Calvin Lin
3 years, 6 months ago

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I choose the former for both · 3 years, 6 months ago

Operations in parentheses take precedence over all else. Unary operators take precedence over binary operators. Exponentiation takes precedence over multiplication, which takes precedence over addition. Operations of equal precedence are executed from left to right. This is generally the way computer algebra systems and programming languages work, but I would go one step further and say that an expression such as $$6 \div 2 \times 3$$ is ill-formed. A mathematician would always write this in an unambiguous fashion, e.g., $$\frac{6}{2 \times 3}$$ or $$\frac{6}{2} \cdot 3$$.

Note that multiplication and division should be treated with equal precedence--they are both multiplicative operations on the field of complex numbers. That is to say, $$a \div b$$ for $$b \ne 0$$ is defined as $$a \cdot b^{-1}$$, where $$b^{-1}$$ is the (unique) multiplicative inverse of $$b$$ in the group structure. Similarly, addition and subtraction are additive operations and also have equal precedence. · 3 years, 6 months ago

I would agree with you, with the exception of the latter half of the first paragraph. It seems contradictory to say that $$6 \div 2 \times 3$$ is ill formed for the following reasons
I. I believe that $$6 - 2 + 3$$ is not considered ill formed.
II. As you mentioned, it is equivalent to $$6 \times \frac {1}{2} \times 3$$, which is not considered ill formed.

The underlying question that I was getting at, was if division and multiplication (or equivalently subtraction and addition) were treated with equal precedence amongst most high school kids, who were taught about order of operations in the past 10 years.

[I'm avoiding the older generation who learn the previous system of "Do brackets first, then everything from left to right", which result in Facebook troll posts asking for the value of $$1 + 1 + 1 \times 0$$.] Staff · 3 years, 6 months ago

sir mainly we use bodmas in india b=brakett o=of like 23(31) d=division m=multiply a=add s=substract

we use this list from up to down · 3 years, 6 months ago

In our country, we use the mmenomics PEMDAS, which means: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. We are provided that Parentheses we be in the order of parentheses ( ) , square brackets [], then braces {} (Sometimes, instead of using square brackets, and braces, we use parentheses several times to enclose the operation.). Multiplication and Division are of equal precedence, therefore a division can be used first than the multiplication. Also, addition and subtraction are of equal precedence.

For example $$1 + 2 \times 3 \div 4$$ = $$1+ ((2 \times 3) \div 4)$$ and $$1 + 2 \div 3 \times 4$$ = $$1+ ((2 \div 3) \times 4)$$

Also: $$1 + 2 - 3$$ = $$(1+2)-3$$ and $$1-2+3$$ = $$(1-2)+3$$

Answering your question: I choose the former of the both: $$a + ( b \times c) = 100$$ and $$(6\div 2) \times 3$$ · 3 years, 6 months ago

In my teaching, and learning, I have found the easiest, and safest way to explain is via the 'left-to-right rule' or the (AS) then the (MD) operations respectively. In example 1, I think a+b x c must always be interpreted as the latter choice, multiplication must take place first. Example 2, perhaps the less clear one, should be interperated as the first choice, the first operation that comes is calculated first, 'no looking ahead' sometimes I will say to the younger students I have.

On a more conceptual level, I think understand the relationship of inverse operations helps to further clarify the rules of order of operations. Within all pairs of inverse (i.e mult/div, add/sub, log/exp) hold equal order of precedent in order of operations, because they are in essence the same operation, just "backwards".

An alternate strategy would be to go through a problem filled with addition and subtraction: for example 3 - 2 + 6 - 8 and rewrite each subtraction as an addition of the negative: 3 + (-2) + 6 + (-8) ( equals -1 I hope =). From here, it seems more clear that addition would just be done in order, as we read it, left to right. You could do something similar with multiplication and division, 5 * 6 / 2 is just 5 * 6 * 1/2 (=15). · 3 years, 6 months ago

I would agree that division goes before multiplication, and multiplication before addition. Most of the time I include parentheses anyways just for extra clarity. · 3 years, 6 months ago

In the first case, I'd always use the former. The second case is, like others also have said, expressed in a bad way. Mathematics is all about rigorousness and unambiguity (and i know there are cases when this is not entirely true, but in this case it is), and therefore, a mathematician would never write division using $$\div$$ without appropriate brackets. · 3 years, 6 months ago

There is an easy way to remember the order of operations and that is BODMAS ( Brackets of Division, Multiplication, Addition, Subtraction).......... It is very clear that division comes first, multiplication 2nd, addition 3rd and subtraction 4th. So in both the cases the former one is right and don't worry, this is an internationally used system. · 3 years, 6 months ago

In example 1 the former is internationally (almost) accepted to be correct. However, there is a lot of confusion regarding the second one, which I believe to be the former. Some interpret the division symbol as a fraction bar. This is exactly why mathematicians omit multiplication symbols and use fraction bars instead of division symbols for clarification. The second one could be $$\frac{6}{2}\times 3$$ or $$\frac{6}{2 \times 3}$$ · 3 years, 6 months ago

I choose the latter for both · 3 years, 6 months ago

I would think that it is rare for someone to choose the latter for both. Can you explain your thought process? Staff · 3 years, 6 months ago

Probably left-to-right, interpreting the division symbol as a fraction bar. · 3 years, 6 months ago

We use "BODMAS": B-Bracket, O-Of, D-Divide, M-Multiplication, A-Addition, S-Subtraction.

Hence by this, both former answers are correct. :) · 3 years, 6 months ago

for Example 1, I choose a + (b x c) = 100 instead (a + b) x c = 100 for a + b x c = 100, because multiplication and division must be done first before addition and substraction. · 3 years, 6 months ago

The problem is that once students in India are thought BODMAS they presume Division always comes before multiplications and the same for addition.But as you suggested there should have been a bracket around D and M and also A and S. · 3 years, 6 months ago

NO, BODMAS rule is followed. BODMAS- bracket, off, division, multiplication, addition, subtraction · 3 years, 6 months ago

Although I would choose the first way for examples 1 and 2; I know many students in the United Kingdom are taught BIDMAS, the fact that the D is before the M leads them to think that the division should occur before the multiplication. Presumably students who had been taught BIMDAS would think they have to do multiplication first, which would make them choose option 2 for the second example. · 3 years, 6 months ago

Thanks. The problem with such mnemonics, is that it doesn't accurately reflect the procedure. There should be a parenthesis around MD, and around AS. Staff · 3 years, 6 months ago

BODMAS RULE SO a+b \times c multyply first · 3 years, 6 months ago

Another one would be $$12 \div 2(2+1)$$. By BEDMAS, some students argue on whether to compute it as $$12 \div 2 \times (2+1)$$ or compute $$2(2+1)$$ first before division(since brackets first in BEDMAS. Anyway I think 'B' mean inside the brackets, not this. But some teachers also compute $$2(2+1)$$ first). Different calculators, including scientific ones give different answers. So I would also like to know is it 18 or 2. · 3 years, 6 months ago

I think it is important to stress that writing 2(2+1) really is a shorthand way of writing 2(2+1). In your case, you should reduce the problem, step by step as follows: 12÷2(2+1), 12÷23, 6*3, 18. · 3 years, 6 months ago

I agree with that. But some people would say it becomes $$12 \div 2(3)$$ then $$12 \div 6=2.$$ · 3 years, 6 months ago