💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱 @ SmartmanApps @programming.dev

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“expand out” the multiplication first is a convention that the mathematics community agreed on, not a fact that can be proven

Order of operations proof - simple version

As a Maths teacher, almost daily 😂

8/2(1+3) even if they technically are meant to be evaluated the same

But 8/2(1+3) isn't a fraction. The / - the computing equivalent of ÷ (which can only be written using Unicode on a computer, so a bit of a pain to use compared to / )- is an operator, which means they're 2 separate terms. A fraction bar is a grouping symbol, which means it's 1 term. In this particular case it doesn't matter, but if it appeared in a bigger expression then it absolutely does matter. The way to write 8/2(1+3) as a fraction inline is to add extra brackets. i.e. (8/2(1+3)) - because brackets are also a grouping symbol.

And as for distributive law vs multiplication, maybe this is just taking for granted a thing that I learned a long time ago, but to me they’re just the same thing in practice

Bu they're not, for the same reason. Firstly, the Distributive Law isn't multiplication at all - which only applies literally to multiplication symbols - it applies to bracketed terms (i.e. is a single term which needs to be distributed) - and secondly it applies to a single term, whereas multiplication applies to 2 terms (one before and one after). Anyone who talks about 2(1+3) needing to be "multiplied" has already made the mistake that is going to lead to a wrong answer (unless they just happen to "multiply" before they divide, which is an accidental way to get the right answer).

if I was factoring something

Indeed, that is the precise reason the Distributive Law exists - they are the opposite operation to each other! Anyone who adds a multiplication symbol has broken up the factorised term, again leading to the wrong answer.

I’m just being a bit lose with the terminology

Yeah, and that's all I was pointing out in the first place - please don't use "implicit multiplication". The term itself - i.e. it includes "multiplication" - leads people to do it wrong (because they treat it as multiplication, not brackets, then argue about the precedence of "multiplication"!). It needs to die!

this can rapidly get unreadable once you nest more than a few parens,

Well that's why the rules of Terms ab=(axb) and The Distributive Law a(b+c)=(a*(b+c))=(ab+ac) exist to begin with - less brackets! :-) Imagine having to write a fraction as (1/(axb)) all the time!

(8)/(2(1+3)) is obviously different than (8/2)(1+3)

Correct, though a lot of people treat it as the latter (yet another way to do it wrong - doing division before brackets) because they figure the 8/2 is "outside the brackets", but in fact only the 2 is, because the slash separates them as being 2 terms.

We agree that the two situations are separate

Ok, that's a start.

but the first result is 4 ÷ a the second result is 4a

Exactly! So when a=2 then 4÷a=2, and 4a=8, which isn't the same thing. Welcome to why 2a and 2xa (and therefore also 2.a) aren't the same thing.

I use the dot as an expression of the same term rather than separate.

But that is incorrect. A dot is used for multiplication. i.e. it separates terms. If you use a . for 2.a, then you are writing the same thing as 2xa, not the same thing as 2a.

This is matter of my notational convention

Well, that's fine enough if you keep it to yourself, but don't use it in anything anyone else is going to read, or you're going to run into the issues I just pointed out

“Wrong answers” only according to our current order of operations

No, according to arithmetic.

math still works if you, for example, make additions come first

No, it doesn't - order of operations proof. The only way it could work with addition first is if we swapped the definitions of addition and multiplication around... but then we still have the same order of operations, all we've done is swapped around what we call addition and multiplication!

there is no ‘high truth’ behind it.

There is when it comes to order of operations.

That proof for the order of operations sure seems to rely a lot on our current order of operations

Doesn't use order of operations at all. It only uses the definitions of the operators. i.e. 3x4=3+3+3+3 by definition. i.e. nothing to do with order of operations.

If I have 1 2l bottle of milk, and 4 3l bottles of milk, how many litres of milk do I have? It can be solved by simply adding them up - again, nothing to do with order of operations here, just simple addition. Now, write it out as a mathematical expression which uses multiplication, and tell me which order of operations gets you the right answer. Voila! Welcome to how we worked out what the order of operations rules had to be.

That’s an after the fact justification

You got some sources with dates in them to show it was "after", and not, you know, before?

That would be 8/(2x(2+2)) if we were keeping it all in the denominator

(2x(2+2)) is the same thing as 2(2+2)

I have to respectfully disagree with your analysis

Which means you disagree with how Maths textbooks teach how to do this (see previous link).

Please see this section of Wikipedia on the order of operations

That section is about multiplication, and there isn't any multiplication in this expression.

The “math” itself might not be ambiguous, but how we write it down absolutely can be

Not in this case it isn't. It has been written in a way which obeys all the rules of Maths.

This is why you don’t see actual mathematicians arguing over which one of these calculators is correct

But I do! I see University lecturers - who have forgotten their high school Maths rules (which is where this topic is taught) - arguing about it.

it is not either calculator being wrong

Yes, it is. The app written by the programmer is ignoring The Distributive Law (most likely because the programmer has forgotten it and not bothered to check his Maths is correct first).

US - PEMDAS vs UK - BODMAS

Those aren't the rules. They are mnemonics to help you remember the rules

notice division and multiplication swapped places

Yes, that's right, because they have equal precedence and it literally doesn't matter which way around you do them.

you can’t actually do all of the multiplication and division at one time

Yes, you can!

Some are taught to simply work left to right

Yes, that's because that's the easy way to obey the actual rule of Left associativity.

we are all taught to use parentheses correctly to eliminate ambiguity

Correct! So 2(2+2) unambiguously has to be done before the division.

Something about the way this thread was written was kind of confusing,

Ok, sorry about that. I'm more than happy to update it if you want to give me some constructive feedback on what was confusing about it. Note though that this was the 3rd part in the series, and maybe you didn't go back and read the previous 2 parts? They start here

Is it just that the terminology is wrong? Or am I missing something?

Yes and yes. :-) The 2 actual rules of Maths that apply here are Terms and The Distributive Law. These are 2 different rules of Maths - neither of which is "multiplication" - and so when lumping them together as "implicit multiplication" you end up with unpredictable, and usually wrong, answers. The only way to always get the right answer is to follow the actual rules of Maths.

a x b, a*b, ab, and a(b) are all acceptable notations to describe the operation “multiply a and b.”

No, they're not. The first two are multiplications, the second two are Terms. Note that a Term is a product, the result of a multiplication. In the mnemonics, "Multiplication"" refers literally to multiplication signs, and nothing else.

Calculate 8 ÷ 2a where a = 4. Then,

Calculate 8 ÷ 2 × a where a = 4.

See how in the first form a is implied to be part of the fraction where in the second it isn’t?

It's not implied, it's explicitly because of the definition of Terms.

P.S. now substitute a=2 and you'll see why it matters.

A dot • could be between 2 and a and it would still follow the first example

No, it wouldn't. Inserting a dot (multiplication) makes it the same as your second example. i.e. 3 Terms, not 2 Terms.

In vector multiplication, dot and cross products produce different results.

This isn't vector multiplication. This is Year 7 algebra.

Implied multiplication coming before explicit multiplication/division is what’s recent.

1. "implicit multiplication" is not taught, because there's no such thing as implicit multiplication
2. the "controversy" over it isn't recent either - we can see Lennes complaining about it more than 100 years ago! The more things change the more they stay the same (sigh).

you’re in for a surprise

I'm not actually. A lot of people don't want to confront evidence that they're wrong.

She didn’t reference any math textbooks because she made the video for commoners, aka not math majors.

Did you notice she's a Physics major? In other words, she doesn't have any Maths textbooks to reference.

Her explanations make sense

So, even when she couldn't explain why one calculator "sometimes obeys juxtaposition, sometimes doesn't", that still made sense to you?

technically wrong

Bingo!

I don’t think many people are going to see your reply

These comments are going to show up in search results for the rest of eternity, so I'm quite happy to debunk the disinformation in it.

you seem to have copy/pasted the reply on several comments

3 different people referred to the same video, so yeah I did something I don't normally do and copy/pasted for those 3 people. Read my other replies and you'll find they're all specific to the person I'm replying to.

It’s like you searched for the YouTube link

No, I've had multiple people tell me about it previously, as "proof" that Maths is ambiguous, hence why I wrote a thread debunking the claims she (and others) made.

It just seems suspect

It's all legit, so feel free to go back and read what I've written given that context.

because brackets are leftmost you do them first

No, not because leftmost (did I say leftmost? No, I did not), because brackets. Brackets are always first in order of operations.

2(4)^{2, wow we’re at a 2x}2

No, we're at x^{2, because 2(4) is a bracketed term, and order of operations rules is brackets before exponents, and to solve the brackets we have to **distribute the 2**, so 2(4)\}2=(2x4)2=8^2=64.

all sorts of properties. But they are not rules

Depends. The Distributive Property is a property, but The Distributive Law is a rule. Properties explain how/why things work, but rules have to be obeyed if you want to get the right answer. Terms is a rule, based on properties (similarly, The Distributive Law is a rule, which makes use of the Distributive Property).

they only apply when we have unknows

Are you referring to pronumerals? Textbooks are quite explicit that the same rules apply to pronumerals as to numerals (since pronumerals literally stand-in for numerals).

terms get prio because they are terms!?

Not priority, they are already fully solved because they are terms. If we have 2a, then there's literally nothing to be done (except substitute a value for a if you've been told what it is). 2xa on the other hand needs to be multiplied (2 terms separated by a multiplication).

Noted that you ignored where I pointed out why it makes a difference

There are no mention of term prio in the book.

Which book? I don't know what you're talking about now.

we have a simplified expression

AKA Terms. And Terms are not expressions. Expressions are defined as being made up of Terms and operators. See previous textbook screenshot. 2a is a Term, 2xa is an Expression. And yes, you are right that a Term is a simplified expression, and being simplified, there is no further simplification to be done.

2x^{2+3x+5 we call 2x}2 and 3x and 5 terms. And yes they get priority, not because we named them those, but because they are multiplications

No, they are Terms. There is no multiplication. Multiplication refers literally to multiplication symbols. A Term is a product. i.e. the result of a multiplication. That's why they don't have multiplication symbols in them - it has already been done.

using terms, as we just get a single number

EXACTLY!! When a=2 and b=3, ab=6, a single number. AKA a Term.

I totally understand why someone would use this, it’s easier

We use it because that's how Maths works, and is a rule taught in all the textbooks, and has been for more than a century.

I forgot the name/keywords but if you read a calculator’s manual there must be a chapter or something regarding this exact issue.

The name is Term. You can read about this exact issue in Maths textbooks.

Especially if you teach physics

I teach Maths, on which much of Physics is built.

As for your sources, you still linked a blog post

In other words, you didn't even read it. The sources are in it - there are Maths textbooks in it.

the whole issue can be avoided

...by following all the order of operations rules

You know sometimes both are correct

Nope. That's what the order of operations rules take care of.

the one on the right is correct

No, it isn't.

8/2×(2+2)

...isn't the same thing as 8/2(2+2). You separated the term in the denominator, leading the (2+2) to get flipped into the numerator, hence wrong answer.

Schools only started teaching that rule relatively recently

Recently? The order of operations rules have been taught for more than a century (we can see them in Lennes' letter).

Makes my programmer brain hurt when there’s no consistency and a lot of implicit rules.

All the order of operations rules of Maths are explicit

I’m with the right answer here

Apparently not.

if you wanted to treat 2(2+2) as a single unit

Yes, it is a Term subject to The Distributive Law, written just the way it is.