Arithmetic that makes you cringe

[cygnet]

Paired with brown rice, avocado and indecent amounts of lime, black beans work splendidly in a balanced, almost five-ingredient lunch.

So why not just call it four ingredients?

Isnt that a "Vege" burrito minus the tortilla?

[Cudzoziemiec]

Presentation of numbers: "Our revenue moved from two billion five hundred and eighteen to three billion and seventy-five"

would suggest to me 2.518 billion to 3.75 billion. The accompanying chart makes clear it's actually 3.075 billion. But the speaker is French (presenting in English) so il faut couper some slack, n'est ce pas-innit.

[Cudzoziemiec]

Paired with brown rice, avocado and indecent amounts of lime, black beans work splendidly in a balanced, almost five-ingredient lunch.

So why not just call it four ingredients?

Isnt that a "Vege" burrito minus the tortilla?

Burrito minus tortilla equals risotto!

[Pickled Onion]

Presentation of numbers: "Our revenue moved from two billion five hundred and eighteen to three billion and seventy-five"

would suggest to me 2.518 billion to 3.75 billion. The accompanying chart makes clear it's actually 3.075 billion. But the speaker is French (presenting in English) so il faut couper some slack, n'est ce pas-innit.

I initially read it as the second - 3.075 billion - assuming they'd missed out the word million, twice. But it's unclear.

Without the word "revenue", I'd read it as a scientist and assume the numbers were 2,000,000,518 and 3,000,000,075.

They've also made the cardinal sin of missing the units (unless that's your crop). Dollars or Euros neither here nor there nowadays, but if it's Yen...

[Cudzoziemiec]

The units were specified on the accompanying graph but not in the spoken presentation.

[Giraffe]

On a quiz show, something about 1st., 2nd. 3rd. are cardinal numbers; 1, 2, 3 are ordinal numbers.

[Cudzoziemiec]

In the first half of 2023, the average cost of an EV in China was US$33,000 (£26,040), more than half the US$70,700 (£55,800) people pay for EVs in Europe and the US$72,000 (£56,800) paid in the US.

It's more or less right.

[hellymedic]

The infant is the third to be found in Newham in four years, with a girl left in a park in February 2019, and a boy on the street in January 2020.

Methinks that’s five years…

https://www.bbc.co.uk/news/uk-england-london-68025769

[mrcharly-YHT]

6÷2(1+2) = 9

Some historical justifications tell that the answer is 1, but today we interpret that differently.

FFS.

This is from a website purporting to teach mathematics.

[Cudzoziemiec]

So which is correct, 9 or 1? I remember learning BEDMAS, but googled and found BODMAS, PEMDAS and GEMDAS; all pretty similar though. According to my vague memory it should be 9 but I can see how it could also be 1. I also found this:

I’m sorry to have to inform everybody, but there is not a universally recognized convention for evaluating this expression. The three come up with different results. There are three common conventions currently in practice:

PEMDAS/BODMAS:
This is a set of rules for order of expressions that is taught to a large number of students at the advance arithmetic and early algebra phases of their schooling. For the expression given in the question it directs:
6/2(1 + 2) = 6/2(3) —Do what is in the parentheses first.
= 6/2(3) — No change because next come handling exponents, of which there are none.
= 3(3) —Do multiplications and divisions in order from left to right.
= 9. —Repeating previous step since there is one division and one multiplication.
There are problems with this set of rules because they were designed for advanced arithmetic and early algebra. They do not handle more advanced expressions. For advanced arithmetic, typically all operations are expressed explicitly, so that 6/2(1 + 2) would not be given, but instead 6/2 × (1 + 2). We will see in the next convention how this distinction can be important, and standard PEMDAS/BODMAS do not distinguish the two expressions.
Traditional practice of professional mathematicians and physicists (which excludes pre-university mathematics and science teachers):
If we want to see how things are really done, let’s go to the professionals, rather than depending on overly simplified textbooks that cover only what you need to know now (which is what PEMDAS/BODMAS does). The basic rules are very similar to PEMDAS/BODMAS, with two exceptions: stacked exponentiation is regarded as to be done top-down (sometimes called right-associative), whereas PEMDAS/BODMAS usually does not specify a direction or, if they do, is left-to-right, which is backwards from professional practice; juxtaposed implicit multiplication has lower precedence than exponentiation (like PEMDAS/BODMAS) but higher precedence than all other multiplications and divisions (unlike PEMDAS/BODMAS). In other words, if you write 1/2a, the 2a is regarded as a tightly bound entity and to be treated as a single unit in the context of multiplications and divisions, so it means 1/(2a), not (1/2)a; PEMDAS/BODMAS would treat it as (1/2)a instead. It is necessary to distinguish juxtaposed multiplication from other expressions of multiplication to be able to handle properly formulas like:
sin 4u = 2 sin 2u cos 2u
as the first step in showing the expansion of sin 4u. Juxtaposed multiplications must be done before the trigonometric operators, which must be done before the non-juxtaposed multiplications. For the expression given in the question:
6/2(1 + 2) = 6/2(3) —Do what is in the parentheses first.
= 6/2(3) — No change because next come handling exponents, of which there are none.
= 6/6 —The juxtaposed multiplication 2(3) is to be done next.
= 1. —The only operation remaining.
Clarity reigns:
Because of the confusion that arises between methods 1 and 2, and confusion needs to be avoided, it has become in recent years standard practice among publishers of technical journals, the General Conference on Weights and Measures (responsible for defining the metric system), and several other standardization organizations to prohibit expressions that involve a division symbol followed on the right by a multiplication or another division within one term, unless explicit bracketing or use of vertical layout make it completely clear and explicit in which order the affected divisions and multiplications are to be done. For the expression given in the question, these rules regard 6/2(1 + 2) as undefined. We are not going to play some cutesy games like “I know PEMDAS/BODMAS and I’m going to see whether you do or I can trick you, so I am deliberately writing it hoping to confuse you.” For somebody who acts that way, you are being pretentious, acting like “I know more mathematics than you do and I am going to show you”, but in fact you are demonstrating that you know less mathematics, because you are showing that you do not realize that a key part of mathematics is to express your thoughts clearly and unambiguously without reliance on some convention not universally recognized. Therefore, the expression in the question cannot be validly evaluated unless it is rewritten in another form such as:
(6/2)(1 + 2) [9], 6/(2(1 + 2)) [1], 62(1+2)
 [9], 6(1+2)2
 [9], 62(1+2)
 [1]. [Brackets after each expression indicate the resulting value.]
“Rules of mathematics” are not some stagnant ideas thought up once, written down, and never changed again. As technology changes, the modes of conveying ideas change as well, and the different modes have different new abilities never available before in some cases and more restrictions in other cases. This sometimes requires that the rules change. The rules for how multiplications and divisions are handled have changed several times in the last 120 years. They are changing again. Convention 3 (to be explicit as to intent) is rapidly becoming the new dominant convention. In this regard, PEDMAS/BODMAS that is so treasured by some people will soon become outmoded—even dead—if it is not changed to accommodate this prohibition of multiplications and other divisions immediately following a division.

https://www.quora.com/What-is-the-answer-to-6-2-1+2

Which implies the answer is "be clearer in writing questions", but might equally be bullshit.

[mrcharly-YHT]

The juxtaposition rule applies. Otherwise you can't do algebra.


8 / 2(2+2) is the same as 8 / 2(x), where x=(2+2)
I'm pretty sure you would agree that 8 / 2(x) = 8/2x, NOT 8/2 * x
This is the juxtaposition and necessary to be able to do algebra.
writing 8 / 2(2+2)
is not the same as
8 / 2 * (2+2)

the problem is that the original equation isn't 'valid syntax' for arithmetic.

If you wrote
8 / 2(x), where x=(2+2)
then it is obvious that it resolves to

8 / (2*4)

[Cudzoziemiec]

Ah. I don't remember that from O level. But then that was in the days of handwriting. And I might just have forgotten it. But then they clearly aren't taking any notice of it in the example. So it should be either 6/2 * (1+2) = 9 or 6 / 2(1+2) = 1.

[mrcharly-YHT]

Ah. I don't remember that from O level. But then that was in the days of handwriting. And I might just have forgotten it. But then they clearly aren't taking any notice of it in the example. So it should be either 6/2 * (1+2) = 9 or 6 / 2(1+2) = 1.

6 / 2*(1+2) = 9
6 /( 2*(1+2)) = 1

When I did the equivalent of O levels, 8 / 2(2+2) would have been rejected. The use of a . to indicate multiplication was introduced with algebra, so

8 / 2.x where x = (2+2) would have been accepted.

The dot was quickly abandoned, as it really doesn't work well with people who have crappy handwriting.

[T42]

the problem is that the original equation isn't 'valid syntax' for arithmetic.

I'd go with that.  Normally (traditionally or whatever) a division would be written as

dividend
--------
devisor

meaning that anything below the line is effectively in brackets (ditto above the line).  Write everything in the same line and you have to insert the brackets, otherwise you've got arithmetic à la Facebook where the loudest Karen wins.

WRT BODMAS and such pontes asinorum, they're all inadequate because multiplication and division have equal priority, as do addition and subtraction. A better notation might be BE[M,D][A,S] where comma-separated items in brackets have equivalent priority, but it wouldn't trip as lightly off the tongues of infants.

[mrcharly-YHT]

Agreed.

Do away with the dividing line and you have to use parentheses for clarity.

[Kim]

Yup.

I'd write that as either:

  6
------
2(1+2)

or:

6
- × (1+2)
2

Or 6/(2(1+2)) or (6/2)*(1+2) inna computer.

'÷' is a primary school thing for carefully curated simple arithmetic questions (and less carefully curated quiz questions on newspaper websites and the like), and tends not to appear in real world mathematics.

In this context it's being used as a trap for teaching BODMAS, which is fair enough (it's what I was taught as a snotty 11 year old), but I also expect things to be right-associative, which I don't remember being explicitly taught - either because I've recycled the neurons (likely) or because I did enough SCIENCE to learn it by osmosis.

[Kim]

The dot was quickly abandoned, as it really doesn't work well with people who have crappy handwriting.

I didn't come across the dot until scalar products.  Which might have been the first year of university, now I think about it.  Crappy handwriting was pretty much compulsory at that point.

[T42]

As I recall, when converting to reverse Polish you have to parse expressions from right to left.  Long time since I mucked about with compilers' guts, though.

[MattH]

I wonder what percentage of people here instinctively came up with 1 or 9?
I thought 1, due to the lack of a * before the parenthesis. If the * was there, I'd have said 9.

When writing code, I tend to be very explicit and put in more () than are strictly needed in formulae, to make it clear what is going on and less likely to make an error later when I or someone else changes the code.

The original way the formula was presented is simply sloppy and seemingly designed specifically to trick people by being ambiguous.

[rafletcher]

As I recall, when converting to reverse Polish you have to parse expressions from right to left.  Long time since I mucked about with compilers' guts, though.

Indeed. The 2 results might be considered to be the the difference between HP calculators (RP notation, gives 1) and IBM Commodore calculators (9) that did the functions as you entered them. Sort of.

FWIW I instinctively got 1 as the answer. Parenthesis first, then multiply before divide.

[mrcharly-YHT]

I wonder what percentage of people here instinctively came up with 1 or 9?
I thought 1, due to the lack of a * before the parenthesis. If the * was there, I'd have said 9.

When writing code, I tend to be very explicit and put in more () than are strictly needed in formulae, to make it clear what is going on and less likely to make an error later when I or someone else changes the code.

The original way the formula was presented is simply sloppy and seemingly designed specifically to trick people by being ambiguous.

I thought '1', then saw the alternative calculation promoted by Neil Does Maths and the Art of Mathematics, so did some research.

Cross with both of them, because by saying '9', they will confuse the heck out of students when those students are introduced to algebra.

Really, they should have rejected the syntax and explained why.

[HTFB]

To throw in another example of how the choice of notation affects the conventional order. In arithmetic "and" and "plus" are the same operation, just pronounced differently. So one hundred plus five means the same as one hundred and five: but three times one hundred plus five is not the same as three times one hundred and five.

[Kim]

Except when 'and' is a bitwise boolean operation.  100 and 5 is of course 4.

[tom_e]

I'm gonna nominate this quite interesting article about Geothermal power in Kenya

for what appears to be blind copying from an obviously-wrong bit of source material:

Geothermal energy had its start in the small settlement of Larderello, Italy in 1904. The small plant provided a mere 10kW of energy, just enough to power five lightbulbs.

The mind boggles slightly.

[Giraffe]

Leaving aside the omission of 'h', Kenya must have very viscous darkness to need 2kW lamps.

BTW, "32ºF (1.6ºC)" - <-- there's a free minus sign for you.