I make myself feel better by telling me that posts like that are actually the work of trolls for the other side.
On the other hand, if your shopping trip consists of 3 gallons of milk, 2 loaves of bread, a dozen eggs, and five pounds of flour, then you will have to use order of operations, and the natural order of operations to use would put multiplication before addition.
Sadly, there are a lot of people who don’t understand the context of equations they write. That’s what math teachers are really trying to get through to students. If you don’t remember a formula, you can look it up. But a formula will be useless to you if you don’t have the skill of understanding context.
Knowing what things to group together is exactly what order of operations is.
That said, there are some cases where order of operations can be ambiguous. If I write “x/2\pi”, do I mean \frac{x}{(2\pi)}, or do I mean (\frac{x}{2})\pi? Almost certainly the former is meant, but the standard order of operations would interpret it as the latter. This is an extremely common calculator error, and in fact, different brands of calculator will interpret that input in different ways.
Oh, as to the asymmetric relationships one: One time, when I was in grad school, the weekly physics colloquium was about the gender gap in physics. The presenter stated, as one fact, that 90% of physicists are male, and then presented, as an independent fact, that female physicists were approximately ten times more likely than male physicists to be in a romantic relationship with another physicist. But of course, aside from same-sex relationships, this follows trivially from the first fact.
I’m wondering if perhaps there is a difference in what is meant by “use the order of operations” . If I want to total up my shopping list, I’m going to use something along the lines of (fake prices)
3 gallons of milk @$3.00/gal $9.00
2 loaves of bread at $2/loaf $4.00
1 dozen eggs @$2/dozen $2.00
1 5lb bag of flour @ $5/bag $5.00
Total $20.00
To me, that’s not using the order of operations , it’s setting things up so that I don’t need to worry about the order of operations. It doesn’t matter whether the order of operations calls for multiplication or addition to be done first, because only one operation is being performed in each line.
If was to do something more like (3 x 3) +(2 x 2) +(1 x 2) + (1 x 5) = X that would be sort of using the order of operations* - and if I were to write it in whatever way that could be expressed without using parentheses, that would definitely be using the order of operations. At this point in my life, I’m not certain how it could be expressed without parentheses, but I’m pretty sure it could be done without four sets of them.
* Only sort of, because all I have to remember is “parentheses first”
That’s similar to how I’d code a shopping App.
You want to multiply the number of items, 3 milk times the cost per item. Do that for each item and then sum.
You could use the memory on a calculator. Calculate the milk. Save to memory. Calculate the bread and add to memory. Calculate the eggs and add to memory. Etc.
I’d name that memory location in an App. CartTotal might be my variable. Its just a name for the memory button on a calculator.
That’s the purpose of the order of operations. It just organizes math.
Yes, of course you’re only doing one operation at a time. That means you’re doing some operations first, and then other operations. In what order are you doing those operations? You’re doing multiplications and then doing additions.
Yes, but my point is that when I’m doing it I’m not thinking “multiplication first then addition”. I’m thinking " How much is the milk, how much is the bread, how much are the eggs, how much is the flour? and then separately " How much is the total?"
I said
In other words, there isn’t any single line that says something like
3 x 3 + 2 x 2 with no context , where I could end up with either 13 or 33 depending on whether I remember the order of operations or not.
More than that, I am afraid:
\begin{align}
(3\times(3+(2\times2))) &= 21 \\
((3\times3)+(2\times2)) &= 13 \\
(((3\times3)+2)\times2) &= 22 \\
3\times((3+2)\times2) &= 30\\
\end{align}
The notation you want is Postfix notation, also known as Reverse Polish. It would look like this:
3, 3 * 2, 2 * + 1, 2 * + 1, 5 * +
or, equivalently,
3, 3 * 2, 2 * 1, 2 * 1, 5 * + + +
There are some calculators out there (notably, those made by Hewlett Packard), called RPN calculators, where you would press exactly those keys in that order. (The comma key may actually have a comma on it, but is more commonly marked ENTER.)
I was just posting elsewhere that an adding machine and a calculator are two different things. I think of them as different because the way you punch in numbers and functions is different. It was impossible to explain that difference to the innumerant artists I was tasked with training at my old art supply store. Dividing by a decimal less than one was an issue, too.
Anyway, before posting about inputs into an adding machine, I was reminded of my reluctant “training” as a Xennial child of how to use an HP calculator (my dad worked for Hewlett-Packard). The mental gymnastics needed to swap between those three probably helped my math skills. I only know “reverse Polish notation” because of Googling it recently.
In RPN, x/2\pi is unambiguously either x,2*\pi/=\frac{x}{2\pi} or x,2/\pi*=\frac x2 \pi. Note incidentally that when we do arithmetic with pencil and paper we have to give the arguments first and then the operations, so that RPN accords with the way we learned arithmetic. I can get confused with a “algebraic” calculator, but never with RPN.
I think you got confused with the RPN too. I think x, 2 * π /
is (2x)/π
ETA: Well, strictly speaking, (x2)/π
It has much to do with familiarity and practice.
A young woman who used to work for me was a long way from being even competent at maths. For example, when I asked her to calculate the average fuel consumption over a set period of a number of vehicles, she had no idea how to go about it.
On the other hand, she had spent two years working Saturdays on a market stall that sold cards (Christmas, Birthday, etc.). They had dozens of different prices and there was a discount for buying five or more. She had no difficulty at all totalling up a number of cards, applying the 10% discount and making change while exchanging banter with someone else.
Facility with algorithms is no indication a person understands what they are doing. I am not amazed by a person who can mentally add and subtract but does not understand “average fuel consumption”.
There is a story Feynman would tell about an arithmetic duel with an abacus guy. He had no chance, until it was his turn to pick a problem and he said something like, “cube root of 1729” or “cube root of 1729.003”
I just had an executive in Finance argue with me at great length that if 5% growth rate compounded over three years is 15.8% growth, not 15%, then how can -5% growth for three years result in cumulative -14.3% not -15% and certainly not -15.8%. So there must be something wrong with my formulas in Excel.
He also doesn’t seem to understand that -6% if lower than -5%, if you are using a formula to compare them. He is confused that when the formula says “IF x<y then B, else A”, it returns B when x = -6% and y=-5%.
This guy has a degree in Accounting, an MBA and about 15 years of experience in Finance. He manages a team of seven people, most of whom are as clueless as he is. He is approving capital expenditures in the tens of millions without being able to understand ANY of the analysis people are doing to evaluate the proposals. Every person he has hired in his tenure here is Asian, female and 25-35.
Are you hinting that female and/or young people tend to “know numbers” worse than average, or the other way around? (And I assume the story took place in Asia?) Age and a Y-chromosome did not help the manager.
No. This is in the US in an area where the hiring pool must be less than 2% Asian Female. For someone to hire four in a row cannot be coincidence.
The boss is a white male, who seems to have a fetish. He’s not hiring people based on academic and professional qualifications. One of the people he has hired had “Model/Actress/Influencer” as her last job and was working in Sales. If you read her LinkedIn profile you will not be able to make sense of her career progression. The hiring manager “passed” on her but was overridden by his boss, Mr Innumerate.
He himself is not even remotely competent. So he had no ability to know whether or not the people working for him are either. When I was on a panel interviewing candidates he kept asking prohibited questions.
He’s lost 57% of his team in eight months. He’s certainly helping our diversity statistics, but this is not how DEI is supposed to work!