I could adopt any arbitrary order of operations I want but it would be useless for communication if nobody else used it. What order of operations that is actually used by anybody would yield an answer of 1?
Left-to-right order.
…which is the mode used by Calculator in Windows 7 in “Standard Mode”.
KCalc (Linux/KDE calculator) gives the answer 7 even in “simple mode”.
Who uses that?
It was actually taught to me as “BODMAS” as an acronym to remember, which was useful until it just became second nature. This was in the late 80s in India, so it’s possible that it’s just another colonial artefact that isn’t used anywhere else, but I’m pretty sure I’ve seen the word in some contemporary maths textbooks here in the UK. I will investigate if possible.
On Google, “BODMAS” returns about 220,000 results, against 17,000 for “BOMDAS” and 6,000 for “BIDMAS” so I suspect “BODMAS” is fairly common. For example: Order of Operations - BODMAS
At least as far back as Windows 3.1, Windows calculator has had a “Standard Mode,” in which it behaves like a basic calculator and uses left-to-right order, and a “Scientific Mode,” in which it behaves like a scientific calculator and follows the rules for the algebraic order of operations.
Left-to-Right actually makes the actually process of doing math more complicated. With standard order of operations, something like 32+5=5+23, which isn’t true for left-to-right. And part of the power of standard order of operations is the concept of a term, which I’m not sure has an equivalent in left-to-right order. In many case, such as multiplying polynomials or integrating, you can treat terms separately, and you would lose that with left-to-right order of operations.
Some calculators and programming languages. There is a picture going around the tubes showing four calculators giving four different answers to the same problem. When in doubt, use parentheses.
This discussion reminds me of my first math class in college, which was the first class I ever had where we were allowed to use calculators (calculators were forbidden in math classes when I was a lad). On the first day of class the teacher was telling us how to choose a calculator. He said to punch in something like 1 + 2 * 3, and if the answer it gives is 9, don’t buy it.
On a side note, he said the reason we were allowed to use them in college was that by the time we graduated high school we should already know how to do basic math by “brute force”.
The OP is not, strictly speaking, a “math question”. There is nothing inherently, objectively, “right” about the widely adopted order of mathematical operations, it is a convention, nothing else. It’s not “math”, it can’t be proven. In any case, relying on this, both casually and in stuff like programming, is not a good idea. When I code, I always disambiguate with parentheses.
Yeah, for graphing or other calculators that let you punch the whole line in. If it’s a simple calculator that only does one calculation at a time, and you get 7, well I’m not even sure if that’s possible.
I thought it was re-instated…
Would you mind cross-posting your comment to the current thread on mnemonic sentences?
I have a 30 year old calculator that isn’t a graphing calculator. If I type in 1 + 2 + 3 =
The display will show as follows:
type 1, display shows [ 1]
type +, display shows [ 1 +]
type 2, display shows [ 2 +]
type +, display shows [ 3 +]
type 3, display shows [ 3 +]
type =, display shows [ 6]
1 + 2 X 3 = yields this:
1 [ 1]
- [ 1 +]
2 [ 2 +]
X [ 2 X]
3 [ 3 X]
= [ 7]
Of course it’s a convention. “Convention” means that everyone agrees to use it. The way we write numbers, and the symbols we use to represent mathematical operations are also conventions. It’s not like there are competing schools of thought on how to represent and parse mathematical expressions. Alternate notations didn’t come into play until electronic calculators where things like reverse Polish notation were introduced as a convention, but a convention only for entering expressions into a machine. I agree with the idea of using parentheses in code to make things clearer, but not because there is ambiguity–because people make errors.
No, this is the confusion of superficial notational choices with the underlying mathematics.
A term (in the sense of elementary algebra) is a product of variables and scalars; this exists as a concept independently of how you choose to notate it. Left-to-right vs. conventional order-of-operations has no bearing on this.
Similarly, in left to right order, the equation you intended would instead be written 3 * 2 + 5 = 5 + (2 * 3), expressing the same true fact. Nothing is lost; it’s just notated differently. Yes, parentheses would come up, just as they come up when you want to indicate certain things under the current convention. (You can eliminate parentheses altogether if you move away from infix notation [e.g., if we notated addition and multiplication the same way we denote general functions, we’d write +(*(3, 2), 5) = +(5, *(2, 3)), or, just as well, + * 3 2 5 = + 5 * 2 3, as the parentheses and commas would no longer be necessary. Perhaps ideally, we’d write things in their direct tree form, rather than linearizing them…], but if you cling to infix notation, you’re stuck with these conflicts of precedence, and whatever convention you pick for deciding them, you will sometimes want to move away from its default)
Yes, I was taught BODMAS in high school in the UK, although that was back in the 1960s. In British English, “brackets” usually means what Americans call parentheses. We would not have understood the word “parentheses”. (These things - - we call “square brackets”, on the rare occasions we need to refer to them.)
I was told the O of BODMAS stands for “of”, and was equivalent to multiplication, but I suppose it is really just there for pronouncability. (At that stage of things, I don’t think we had been introduced to the term “exponent” at all, though we probably knew about “powers”.)
48,000 Google hits for BEDMAS, which is what I was taught in New Zealand.
The only other order of operations convention that I know of is Polish notation, where the operators are placed to the left of the operands. So 1+2 in conventional notation is expressed as + 1 2. But the equaiton in the OP doesn’t make sense in Polish notation.
Oh and I think the equation in the OP expressed in Polish notation would be:
- 6 * 1 0 + / 2 2