Why is the Order of Operations the way that it is?
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- čas přidán 29. 06. 2024
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Why do we have to evaluate operations in a particular order, instead of, for example, left to right? In this video, we outline some general reasons for this convention.
What is Dog × Tree? A dimensional analysis primer: • How To Multiply Dog × ...
00:00 Introduction
00:52 Convention, Not Truth
01:22 Parentheses
02:58 Argument 1: Precision
04:40 Division / Subtraction
05:10 Argument 2: Abstraction Levels
06:10 Argument 3: Dimensional Analysis
08:26 Argument 4: Algebraic Properties
10:58 Argument 5: Polynomials (Practicality)
11:58 Conclusion
My Patreon: / zhulimath
Intro riff taken from: Nikolai Kapustin - 8 Concert Etudes, Op. 40: III. Toccatina
Music Credit:
Forthcoming / Howard Harper-Barnes
Alexandria / Ebb & Flod
For Want of Means / Alfie-Jay Winters
Memory of Perception / Rand Aldo
Forever Be (Instrumental Version) / Victor Lundberg
courtesy of www.epidemicsound.com
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BRO YOU LISTEN TO KAPUSTIN???? BAAAASEDDDDDD ASFFFF
when i was young, i was taught a different order of operations:
in order from first operation to last
powers, multiplication, fractions, root, addition, subtraction...
i'm not quite sure... but i don't think that order of operations is really "wrong"...
it's probably that it's got brackets built into the way you write it down... like... fractions were written over top of one another and roots were with a tail over everything in the root...
at least... i think you don't need brackets if you write things like that (please do let me know if i'm mistaken)
Yep, the way you write fractions and roots implicitly contains parentheses. The only correction I would add here is that addition/subtraction should, in most all cases, be evaluated from left to right together. Everything else is common convention as I alluded to in the video!
I have discussed this with accountants who were active from the 80s and 90s and use desktop calculators. A few of them feel way more at home with reverse polish notation than standard PEDMAS. So there's also the whole issue of mechanical familiarity at play.
Indeed! People who are more used to our standard "infix" notation have more difficulty with the reverse Polish "postfix" notation.
In general, postfix notation has a lot of benefits over infix. Not only are parentheses and order of operations unnecessary, algebraic properties are still easy to work with, and it has been shown that people familiar with postfix notation make fewer computational mistakes!
@@zhulimath I think reverse Polish notation is really only good on a computer screen. When writing math on paper, all of the other features of standard mathematical notation are super helpful, like how division is typically done, and implicit multiplication is a godsend. Superscript exponentiation is also fantastic. Postfix notation sacrifices all of that, so it only really makes sense in text applications where implicit multiplication, superscript exponentiation, and top over bottom division are not possible.
It is really nice that postfix works well with polynomials. A string of one or more consecutive multiplication signs separates one term from another, and then a bunch of addition signs at the end. Easy.
Although I feel like prefix might be better in some situations. Mathematical operations are really just functions, and functions are prefix notated by default. However, the reason to write math in prefix is if you take particular interest in the functions you are using versus the operands you are providing. Since the standard mathematical operators are kind of boring compared to functions, it doesn't really make sense to write all of the operators up front the way we do it with functions.
I prefer unreversed polish notation because square roots and functions are still on the left
RPN is great, but NOBODY writes expressions on paper in RPN.
You still have to know what the expression means before you can enter it into your calculator in RPN order.
I prefer unreversed polish notation as well, as it keeps operations in line with other functions. i.e. f(x, y) lines up with + x y, as do f to x and y.
Here's how I think about it. Commutativity and associativity of addition are both properties that allow you to disregard the order in which the additions are done, and disregard the order of the things being added. The same is true for multiplication. These properties hold for addition and multiplication individually but not when they're mixed (e.g. the "mixed" version of associativity, (a + b) × c = a + (b × c), is false). That means that we need to fully evaluate one operation before moving on to the next one, in order to take advantage of these properties.
That explains why we do multiplication before addition instead of just doing everything left to right. But that doesn't explain why we do multiplication before addition instead of addition before multiplication. The reason for this is the distributive property. An addition-first expression can be rewritten as multiplication-first, because (a + b) × c = (a × c) + (b × c). This means that, if we do multiplication before addition as a rule, any expression involving multiplication and addition can be rewritten to use no parentheses. On the other hand, a multiplication-first expression, e.g. (a × b) + c, cannot be rewritten to be addition-first in general, so if doing addition before multiplication is the rule, not all parentheses can be eliminated. This is related to the point you made about polynomials.
To continue on the "addition is subtraction and multiplication is division" argument, radicals can also be rewritten as exponentiation. The nth root of a number is also the 1/nth power of that number.
to add another one the same is true for the reciprocal and a negative expoent; x^(-n) = 1/(x^n), all of which is very useful
Sometimes I use way too many parentheses to make sure I'm putting in my math right on a computer or calculator. I make every single operation explicitly contained within a set of them. I've become so used to doing it that I read it that way easier than with reduced parentheses in any other format.
I do the same thing with the calculator. For example, with the Texas Instruments calculator, in polynomials I do (3*((x)^15)) + (5*((x)^10)) (this is an example).
Are you a LISP programmer by any chance?
@@coborough c# and c++ are what I'm most familiar with. I've never heard of LISP, I'll look into it.
By "familiar with" I mean the biggest project I've made from nothing was a computer app of Conway's Game of Life, with a ton of features including saving and loading files and even changing the rules of the game itself by adjusting behaviors on each of the 8 different possible neighbor counts a cell could have, and I even played around with the rules to change the distance of the cells it would check for neighbor counts.
I learned how to use git, GitHub, learned a ton of design patterns, and my favorite was dynamic programming and algorithms. I love being able to take problems which normally would require an exponential time complexity solution and make it a linear time complexity one.
But I'm not certified on anything.
@@coboroughhahahaha
same
I don’t have dyslexia or anything (as far as I know) but it’s the only way I can keep track of what I’m calculating when there’s more than 3 numbers involved
This debate also exists in the world of programming languages; most languages use the convention established by C (which uses the PEMDAS order augmented with computer specific operations), but you also have languages on one extreme, like Forth, which uses strict left-to-right evaluation, and then on the other end of the spectrum you have Lisp dialects, in which *everything* is parenthesized, to the point where there are a zillion memes about it.
Even a language like C increases the number of prefix and infix expressions one has to consider: bitwise operators (& | ^ ~ >), boolean operators (! && ||), access ( . and -> ), prefix casting by type, function calls, and maybe a few others in newer standards. We deal with it because of the benefits of systems level languages, but there is a complexity cost in machines parsing source text for compilation too. I know Forth can get pretty low-level, but there is a trade off with the "stack dancing" that one needs to keep track of. As for Lisps, they haven't been really low level since Symbolics keeled over from lack of demand for Lisp Machines (competitors outfoxed them with workstation & unix compatible lisp interpreters) and some like Larry Wall deride the language family as looking "like oatmeal with toenail clippings" (the creator of *Perl* had the gall to criticize another language when his inspires "write once, run away")
Actually inherited from Fortran/Algol. APL is weird with right to left evaluation order.
@@georgerogers1166 it's generally accepted that the most prevalent style of language is descended from C, not Algol, etc, so that's why I said C. The APL style I like actually, makes everything consistent.
@@segganew Operator precedence wise, which is the expression language. The big difference is curlys vs begin end vs indentation. Fortran was the first programming language to have infix expressions.
I've written a parser for algebraic expressions... and I have some strong opinions about the terrible infix notation that everyone uses because of a historical accident centuries ago. I don't have time to write a long comment here right now, but some important things:
1. Parentheses are only necessary for infix notation. Prefix and postfix notation (reverse Polish) do not require parentheses at all.
2. The fundamental reason for all of this "order of operation" stuff is because *_binary operations compose with each other to form a binary tree_*
These notations are fundamentally just a serialization format for representing a binary tree. Usually I would draw some ASCII art trees for some simple algebraic expressions, but I don't have time to draw a picture right now.
The nested parentheses we use in the standard North American/European style infix notation tell you which leaves and nodes are at the bottom of the tree.
You evaluate the operations by starting at the bottom most level(s) of the tree, replacing each parent node with the output of the two child nodes. Then that value is one side of the parent's branch above it.... keep doing this, walking up the tree until you hit the root, and then you're done! Whatever the root node's value is, that's the final answer.
Oh yeah, I have another long rant about how mathematical notation is an ambiguous and inconsistent mess just like how sheet music notation and English orthography... all for exactly the same historical reasons. (But I don't have time for CZcams right now. )
@@juliavixen176 What's the name of your parser if it's accessible online, and where could someone who's interested in this topic read more?
I don't like the term "order" in order of operations because it makes people think they have to evaluate the expression in a specific order. As you said order of operations is a set of rules to know how to go from a linear expression to a tree expression. But then evaluation doesn't have to happen from the deepest leaves to the root or any specific order, or using any fixed algorithm for that matter.
Are there studies about ease of reading postfix or prefix notation (or other alternatives) though? The usual notation is at least quite readable without too much time to have accustomed to it. Also as it sometimes needs patentheses, it’s not a big leap of imagination to use parentheses as an aid for readability and ease of easily-testably-correct manipulations of complicated expressions. On the other hand, using economic notation with no parentheses might end up harder to perform with the same error rate. Using expression trees is quite inefficient, I’d say, not just that it uses more physical space (or physical paper) but maybe also for eyeing things out. Though factoring _some_ subexpressions in this way might very well be beneficial! But also we already have temporary variables and notations for that right now.
@@markopanev3317 I assume by "parser" the OP means "interpreter" - with parsing and evaluation and no intermediate representation(s) in between. You can find lots of examples online by searching for "expression parser", "expression interpreter", or "expression evaluator" along with the computer language of your choice. It's a common exercise in an "into to compiler theory" CS course.
I've written hand-coded parsers for many languages, and always use a Pratt parser at the expression level and plain recursive descent for the rest. Pratt parsers are nice, among other reasons, because you specify precedence and right/left associativity explicitly and can change them by just changing a constant. Java, for examples, has 15 levels of mathematical precedence and one more level for ".", "[", and "::". Coding all that with recursive descent would be cumbersome, error-prone, and annoying to modify. With a Pratt parser, if they add a new precedence level at some point, I can just bump the precedence of everything above it if it needs to fit between by changing precedence numbers in a table (this happened when they added the lambda operator at a new lowest precedence).
I'll bring up a classic:
1/2a
Most people would say this is obviously 1/(2a), but many would say this is not true when using an obelus and replacing a with an expression in parentheses like this:
1-:-2(1+1)
Implicit multiplication was shown off in the video without stating it can have a different priority than explicit multiplication.
This is a very important convention that is rarely tought, but incredibly useful when writing expressions inline instead of using fractions.
This is not even implicit multiplication, the 2 is normally considered a coefficient of a. So some people consider 1/2a to be 1/(2a), whilst considering 1/2(a+b) as (a+b)/2. Confusing, because coefficients and implicit multiplication is basically the same thing.
This really is "What is your highest level of math class that was not required by The State?"
1/2a its half of a since division 1/2 its before multiplication
@@gregstunts347 A coefficient s an implicit multiplication. Everyone treats implicit multiplication at higher priority naturally. Only if they start applying some rules they might not treat it so. The problem is that te rules are taught first and implicit multiplication is introduced years later. There is no reason to assume it should be treated like the explicit one.
Professional mathematicians and physicists give it higher priority. Most calculator manufacturers do also. It is essentially only US math teachers who do not and therefore TI calculators do also not. TI even said that it should have higher priority but the teachers say no.
Nobody would interpret Sin 2x as Sin(2)*x.
@@okaro6595 I know, and I perfectly agree with you. I was just pointing out at how others make things more confusing by differentiating implicit multiplication and use of coefficients, even though they are basically the same thing.
Back in highschool, my math teacher hated pemdas with a burning passion. He hammered GEMA into us, because it forces you to understand shared priority. It isn't as useful at this stage since most of what you're doing from here on is solving for variables, or taking derivatives, but it does still come up with stuff like logarithms and limits. I forget that teacher's name, but I'm glad he made me throw out pemdas since gema's implicit structure forces you to learn an intuition for math
GEMA?
= Groups, Exponents, Multiplication, Addition?
Same, his name was Mr. Sibert for me
In Germany we just do "dot before dash" Since it's multiplication and division (* and :) first, then addition and subtraction (+ and -). You don't need to mention parentheses since their whole point is grouping and you don't need to mention exponents since they're tacked on to a number so it intuitively makes sense that that is a single thing now.
@@mesplin3i believe it was groups, exponents, multiplicative, additive
im so glad there's a term for this way of thinking, and i agree 100% with your teacher. Understanding multiplication and division (also addition and subtraction) as fundamentally the same kind of operation was a massive development for my mathematical ability. Like you said it teaches an intuitive understanding of what these operations are doing and having that opens tons of doors for further study
as a kid, I never understood the order of operations, so when I wrote down any maths questions, I would surround everything with brackets to know what came first
I too used to first go and replace every subtraction with a plus negative. Ex. "3x^2 - 2x - 4" would become "3x^2 + (-2x) + (-4)". That helped me get past some confusion when distributing, or factoring, or subtracting a negative, etc. Eventually I got comfortable enough to drop that, but it was helpful for the first little while
@@bigpopakap i always hated when the math teachers marked people down for not solving the problems their way, when I could only do the things that made sense to me. Solving the equation is the important part, so why should working out matter, you should be allowed to do what helps you get the right asnwer
I've always done that as well. always got confused without them
@@mikubrotIt's a binary tree. Composing multiple binaries operations together makes a binary tree. I don't know why on Earth they don't teach this in general math classes. (They teach it in computer science when you get to parsers, it should really be taught in grade school!)
@@tiaanvanrensburg1032 can you give an example of this?
I think the most cursed thing that surprised me in school, is that summation notation with Sigma usually includes everything up until the first Add/Subtract.
So, Sigma 2x * y means Sigma (2x * y), but Sigma 2x + y means Sigma (2x) + y.
Yeah, that's pretty confusing. The sigma is like a weird function symbol, but it doesn't have any parentheses around its argument.
@@volblaYou are free to add parentheses to clarify the structure of your equation.
that’s more of a convenience thing tbh, you’d always use brackets around the argument if it included addition or subtraction. as with anything, if it’s not otherwise handled with brackets, addition or subtraction denotes a new term, so you’re just getting rid of a redundancy by not including the brackets around the argument for a single termed expression
@@user-gd9vc3wq2h Of course, but you also have to know that you _need_ to do that if you're typing a sum into a calculator.
Relative to computation and programming, sigma and pi notations for summing and multiplying over a sequence can be encoded as higher-order functions, where the expression being summed/multiplied is a function argument passed in along with the bounds of the sequence. At least, that's how functional programmers would view it; imperative programmers usually write a for loop and the expression becomes part of an assignment statement
I appreciate that you clarify *"in this context"* when talking about addition and subtraction being interchangeable by choosing a corresponding element, most inverse operations in general do not have this property, and in many cases where both addition and subtraction are defined they can lack this property too (e.g. the natural numbers) and of course even in the usual real numbers multiplying by 0 cannot be represented by division.
Multiplication by zero can for sure be represented using division
0/5 is equivalent to 0*.2 for example.
@@VivBrodock that is not what is meant by 'multiplying *by* 0'.
Partially bind the operator so it becomes unary (a function in R -> R, given by binding y to 0 in x*y but leaving x free in R). Since its codomain ({0}) is singleton but the domain (R) is not, it does not have an inverse.
Concretely '1/0' does not exist. But, we needed this because to replace 'x*0' with division we need to find a y such that for all x 'x*(1/y) = 0', which in R would imply '1/y = 0' which cannot be since in R, 1/y is unique non-zero or DNE.
I love how your video starts with the last second of Nikolai Kapustin's Concert Study "Toccatina" op. 40 No. 3, that only true Kapustin fans recognize
A long time ago I realized that you can summarize the motivation behind order of operations very concisely:
Order of operations is the way it is, to make polynomials correct.
if you write, say, 2x^3 + 3x^2 - 7x + 4, only the "correct" order of operations allows us to write this without parentheses at all
Except for postfix and prefix notation, which require no parentheses at all. Only reason infix is better on paper is that only infix notation permits implicit multiplication, superscript exponentiation, and top over bottom division, among other features of standard mathematical notation. Which is why many calculators use postfix notation, since they can't do some of those things anyways.
Although thinking about it, there's no reason why postfix or prefix notation _couldn't_ do implicit multiplication, superscript exponentiation, or top over bottom division. So maybe the only reason we still use infix is that it is entrenched.
@@J7Handle Look, Ma, the polynomial without parentheses: 1 × x + 2 × x + 3 × x + 4 × x + 5.
@@J7Handle I mistakenly tagged you. My comment was an answer to the big comment.
@@Bolpat My apologies, then.
You can actually write polynomials in a really nice way if you're strictly going from left to right:
ax^3+bx^2+cx+d = ((ax+b)x + c)x + d =~ a * x + b * x + c * x + d
(where =~ means that the thing to the right of the tilde is going by strict LTR evaluation order)
This doesn't work if you have more than one variable in the polynomial, like x^2 + 2xy + y^2, since there are no factors of y in x^2. But it is not bad for one variable polynomials.
yes, very sloppy of this video to ignore this obvious reframing, and even worse that the author saw your comment and chose to like the defensive reply to it without acknowledging the oversight in any way
I apologize for giving that kind of impression. There have been many comments about this way of expressing polynomials, and I didn't feel the need to reply to every single one of them.
I am very happy that everyone is thinking about the different ways we can play with our operations. In my video, I mention that our choice of the order of operations is simply convention. I want to encourage people to understand the underlying reasons for our conventions, rather than accepting and memorizing it, because it allows us to not only understand these ideas in a deeper way, but also teaches us where to creatively break the rules when it provides a benefit.
This particular way of representing polynomials is admittedly a way that I have overlooked, but it is partially because I don't personally see or use it often, so if there are good uses of this, I am not aware of them. If there aren't many good uses for it, then it doesn't really motivate changing the order of operations to facilitate its expression. If there are good uses, then great, but it wouldn't be my place to acknowledge it since I don't know enough. My goal is simply to get people thinking, and I'm happy that people are.
For what it's worth, I try to reserve my heart reactions for things that go a little bit beyond things I simply agree with, which I give a thumbs up. I have given a thumbs up to every comment mentioning this alternate way of expressing polynomials.
@@SeanTBarrett i didn’t find it sloppy nor the reply defensive. that caveat is important and all-in-all the current precedence rules are better. i just intended to share the relevant fun fact, not to contradict the video thesis
@@zhulimath as to if there are good uses for it, the context i’ve encountered it is as a digital representation for polynomials in some libraries for hybrids between symbolic and numerical computer algebra. while both representations allow polynomials to be represented by an array of numbers, the one i mentioned have some specific properties that makes it more useful for certain applications
for example, you only need to multiply by x n times in total, where n is the degree of the polynomial
i don’t remember what exactly, but i believe that the context was either automatic differentiation, where the library can calculate the derivatives of any functions you’ve written in normal code, or something related to Taylor series
I took a class on Gödel's theorems and while during the course we stuck to standard notations we showed that we can write logical formulae and sentences in polish notation and then restricted it even more, later similar stuff was done to arithmetic
While it made stuff nigh unreadable (hence why we stuck to the standard notations outside of proofs where we were messing with the formulation of formula) it did make a lot of proofs much simpler, especially the Gödel Numbering could be reduced to only needing 3 primes instead of the full prime factorization theorem
Yep that’s very good stuff. Even better would be inductive types (a la “typed trees”) but that needs its own course to explain from the ground unfortunately.
Standard of expression would be judged by criteria of use:
1. Minimize extra markup
2. Maximize readability, often vocalized form
But this evaluation depends of environment and application. Pencil and paper can do things simple digital text can't and for example computer programming doesn't understand several forms of the same operation differing in appearance. We will gravitate to different natural standards of notation based on our tools and task.
Just like bird speciation on the Galapagos the conventions become incompatible by isolation. Then we get to argue on facebook.
Correction: Our computers absolutely understand one operation having different appearance. That's easy. What's hard is the same appearance having different meanings depending on context.
computers have no agency. They can be programmed however we like, to perceive to be either capable of "understanding" or "not understanding" any specific mathematical concept or equivalence@@dojelnotmyrealname4018
About dimensional analysis, you _can_ add apples and oranges, but you end up with just that. In mathematics, the complex numbers are an example for that: You _can_ add 1 and i, and the result is 1+i. It just doesn’t simplify.
Technically true!
My previous video on dimensional analysis covers this in the abstract briefly in the conclusion, but I didn't dive super deep into this idea because I think it is not at the heart of the lesson underlying dimensional analysis, I intentionally omitted it.
@@zhulimath After watching the video to the end, I realized that as well. In physics, mixed units in summation usually makes no sense.
3 apples + 2 oranges = 5 fruits.
Awesome video. I think it's important that you mentioned that the order of operations is not a mathematical truth, just a standard. I see a lot of debate online about things such as 8/2(2+2), and a lot of people incorrectly justify their arguments by saying that mathematics cannot be wrong, or that there simply cannot be multiple answers because mathematical truth is absolute.
I think it is a bit more nuanced, mathematical truth is absolute given you take your axioms of the mathematical system to be absolutely true. In other words math is just a set of rules we set and say are true based on what rules seem to be most practical in a given application.
Yep, that kind of arguing always weirded me out, like WHAT WHY FOR WHOSE SAKE all this mayhem, why so heated, that’s so dumb. If somebody actually thinks that’s about mathematical truth, I at least start to understand the phenomenon.
Mathematics cannot be wrong, but it can definitely be used incorrectly. That's why it's so easy to tell false naratives using statistics.
The Problem with 8/2(2+2) is that there is no agreed on convention in regards to implied multiplication ( multiplication denoted by juxtaposition).
Sometimes it is thought to have a slightly higher priority than "regular" multiplication/Division
The same way we see X×Y as one unit of XY or 2×x^2 as one unit of 2x^2 , "2(2+2)" can be seen as a unit of 2x (with X=(2+2) and the question becomes 8/2x with simplifies to 4/X and thus becomes 4/(2+2) => 4/4
On the other hand, if we don't regard implied multiplication, then 8/2(2+2) becomes "8 over 2" times (2+2) or 4/1 × 4 => 4×4
@@GrandProtectorDark thus that's where confusion begins. Trust me, I've done the "viral" problems on this site, and showed my work. It's rough, but I do understand it clearer now.
a situation i've seen a sorta left to right (or right to left) use in practice is in computing pade approximants, rather than doing each power in each series, chaining (constant + x * (constant + x)). it's the same as doing constant + x * constant + x * x, but it uses one fewer multiplication, making it faster, and usually the top and bottom could go up to fifth, sixth, seventh power depending on the approximation quality needed. the compiler could do it for you but if you are writing software that needs to run something in real time on a variety of computers with varying performance, it's useful to optimize whenever you can. and yes, it's using parentheses and still using the pemdas subset of whatever OoO the compiler has, but it's a situation where x + c * x + c ... (or (x+c)*x+c to put it in pemdas) would be easier.
dude glad I found this channel! This makes a lot more sense when dealing with higher math like calculus. Thank you you've referenced 11:04 ! It explained when dealing with limits, the taylor series way of solving pops up!
In Europe, multiplication and division are usually represented by ⋅ and : and the equivalent of PEMDAS is _Punkt vor Strich_ in German and _punto prima del trattino_ in Italian; it translates to “periods before dashes” because it’s the only non-trivial case. Mentioning parentheses and exponentiation is silly. “Periods before dashes” works so nicely because the symbols actually are just periods and dashes.
In Finland multiplication was done before division until the mid 80s. It then was changed to match what calculators do.
This is so alien to me, i do not even recall anyone referring to a ":" as a "dash", i thought "dash" was a "-".
@@elio7610 The colon (:) isn't considered a dash. I really don't understand where you read that in my comment. It's the division symbol and clearly a dot symbol. It is dots (⋅ and :) before dashes (+ and -).
That is true, and you're missing the point a little. Colons(:) and periods(.) are point-based symbols, while plusses(+) and minus(-) are line-basedsymbols. So "Points before lines" means do things written with points before things written with lines. @@elio7610
Your channel got my subscription the second you used the ending of Kapustin’s Toccatina Etude as intro music.
5:10 I'd like to note that stepping from succession to addition is not exactly the same as the other steps. In the S->A step, we start with an argument and apply the unary operation to it the other argument of times. In the other steps, we start with the identity element and use the second argument of the new operation as the second argument of the old operation. So:
A(n, k) = R[S](n, k)
M(n, k) = R[A(n, _)](0, k)
P(n, k) = R[M(n, _)](1, k)
T(n, k) = R[P(n, _)](1, k)
Note: 0 is used for A->M because it's the identity element for addition. If a binary operation doesn't have an identity element, you can't think of the next operation as applying it k times. You can only think of it as applying it k-1 times
Forgive my lack of precision, I'm up quite late - You're absolutely right that Successor's behavior seems odd, because it is actually a hidden binary operator, and numbers are functions. "1 apple" applies the function "have one of X" to the object "apple". That means "1" is a function. 0 is the function "no X", and Successor is "Y then another X". So (writing Successor as "S"), 1 is "S(0)" meaning "no X then another X", 2 is "S(S(0))" or "no X then another X then another X", etc. But, I just said Successor is a binary operator! Yes it is, and we "curry" (Mr. Haskell Curry) the last variable X as part of the output function - again, numbers are functions ("have one of X"), so if Successor outputs numbers, Successor must output functions in terms of X.
To tie better with your note, the idea of applying succession as a unary operation to a number rather than a binary operation with an identity element seems odd because you're straddling the conceptual boundary between "numbers are corpuscles of quantity that I can manipulate" (1, 2, 3) and "numbers are functions that I can compose" [S(0), S(S(0)), S(S(S(0)))].
In case I'm sleep-deprived and none of this makes sense, read about Church Numerals for numbers-are-functions, and maybe Lambda Calculus for functions-of-functions-with-currying.
Cheers!
The biggest reason I can think of for using the standard order of operations is that its... standard.
Nobody will be confused. Everyone assumes the standard, so we're not constantly having to re-state which order we're using.
The problem is that there is no standard... a lot of people still use "implicit multiplication" or "Multiplication by juxtaposition" but there is also a lot of people who don't use it. If there was a standard there wouldn't be a problem.
Yeah, that's why I speak in English, too.
This kind of thinking is a bit narrow minded. In different cultures they write right to left, top to bottom. In the three most notable asian cultures they use entirely different forms of script (fun fact, Korean uses an alphabet!). So the problem is that standards are often not so standard, especially if you cross borders. I think the best solution is to just make your standard explicit in your work. Just write down what the rules are!
Great video and great explanation!
I think the part that so many people get wrong, is about _implicit multiplication by parentheses._ The statements "6(3)", "6*(3)", and "6*3" evaluate to the same thing, but they are technically _not_ functionally identical! Implicit multiplication by parentheses is _supposed_ to be elevated to Parentheses tier, _not_ delayed to Multiplication tier. I'm struggling to remember the exact equation used, but there was a "you probably get this wrong" post on -Twitter- -X- _the Birds-Aren't-Real App_ a few months ago... and most of the replies were actually _wrong,_ since it had an implicit multiplication by parentheses that changed the entire result if not evaluated in the Parentheses stage. The entire point of PEMDAS, etc is that at the end of a "phase", there should be _no items left_ of that step, and to eliminate all parentheses, you _must_ do the implied multiplication. Only when there's an actual [insert multiplication symbol here] does it actually count as capital-m Multiplication and have to wait for its own step. Only in the case of "6*(3)" would the multiplication be delayed to the M-stage, the parens would just evaluate and drop, leaving "6*3" at the end of the P-stage.
I've usually seen this described as juxtaposition, since typically this convention uses implicit parentheses around any group of juxtaposed (i.e., implicitly multiplied by virtue of being adjacent) terms, even if none of those terms have parentheses. For example, this convention interprets AB/CD as (A*B)/(C*D).
@@xxgn That's actually a good idea and sounds familiar, yeah. It _enforces_ "multiplication by parentheses takes place before regular multiplication", by turning it into a parenthetical statement itself 👍
in standard PEMDAS/BEDMAS/GEMA convention, implicit multiplication is multiplication, and should be in the multiplicative phase. 6(3) = 6*(3) = 6*3. the parentheses tier is confusing in PEMDAS (& thus renamed in GEMA, which is literally the exact same order just renamed) because it specifically points to parentheses, and not to their /function/, which is to group operations. under GEMA, you evaluate Groups first -- 6(3) evaluates the group (3) first, to 3. then in the Multiplicative phase, we evaluate 6*3.
juxtaposition convention, which is what you're using, implicitly parenthesizes these implicitly multiplied terms (say that 5 times fast lmfao), so 6(3) is interpreted as (6(3)) - that does not mean that 6(3) in standard convention requires you to evaluate the 6(3) first.
the reason the parentheses in 6(3) disappear at the end of the parentheses phase is because 6(3) is correctly evaluated to 6*3, after which the multiplication is evaluated in the multiplicative phase.
standard PEMDAS/GEMA convention evaluates a/b(c+d) as a/b*(c+d).
juxtaposition convention evaluates this phrase as a/(b(c+d)). however, this convention is not commonly used -- generally, explicit parentheses are required. try typing, for example, a/b(c+d) into an online calculator such as wolfram alpha, and you'll see the standard order of operations applied unambiguously, evaluating it to a/b*(c+d) aka (a/b)*(c+d).
EDIT: also, evaluating implicit multiplication during the parentheses stage would imply that 6(3)^3 = 18^3, since you evaluate the implicit multiplication BEFORE the exponentiation -- which isn't true even in juxtaposition convention. juxtaposition convention just evaluates implicit multiplicative operations before explicit ones -- so a/b(c) = a/(bc) but a/b(c)^2 = a/(b(c)^2) = a/(b*c^2)
This absolutely depends on, at the very least, which country you were taught mathematics in. In the United States, high-school Algebra 1 teachers will mark you incorrect if you do what's being described in this comment, even though most mathematicians and scientists might agree with it. I think in most of Europe, on the other hand, it's considered correct.
The way that i thout (before this video) was that multiplication is a "shortcut" to adition.
For exemple, 2x3 = 2+2+2 (3 times) so we say 3 times 2 or 2 times 3.
Aplying that, take a look:
2x3+1 =? 2x3+1
2x3+1 =? 2x4
Lawful evil and chaotic evil are definitely the wrong way round in the thumbnail
Also, 11:52 this can be written as x + 2x + 3x + 4x + 5 (although you should probably expand the multiplications for readability)
Also, the primary argument given here about reducing the number of brackets necessary is a great argument in favour of 6÷2(1+2) being 1, not 9 - 9 can be expressed with an additional × whereas 1 would (otherwise) only be expressible either with extra brackets or with commutation
Why do you say lawful evil and chaotic evil are the wrong way around?
You yourself give an example of why left-to-right evaluation order could be considered _good._
@@Tzizenorec brackets everywhere leaves no room for ambiguity, and is therefore not chaotic. Left to right evaluation, especially with multiplication by juxtaposition, is chaotic as it's very easy to misinterpret due to it not being even remotely common in actual usage, and being unintuitive due to juxtaposition and exponentiation 'feeling' like they should happen first
Note that in the video he notes that he had to switch exponentiation to up-arrow notation to prevent it from becoming too confusing, and note also that tetration syntax (³4 = 4 tetrated by 3) now completely breaks the flow of reading the 'left to right' expression
@@Starwort All of your arguments for left-to-right being chaotic also work for PEMDAS being chaotic. So this argument can only inevitably lead to "his attempt to classify the various methods onto the alignment chart is totally bogus" (and everything probably actually goes into Lawful Neutral because that's what mathematics itself is).
Which would be awful disappointing... so maybe we should just leave his joke alone, and not expect any technical accuracy from that particular bit.
@@Tzizenorec Sure, LtR being chaotic and BODMAS (:P) being chaotic are both reasonable stances - but specifically *bracketing everything* is the most lawful approach, as it *cannot* be misinterpreted, unlike any method without brackets
Very nice video. Was wondering about it in my third grade and then forgot about it.
10:59 I would argue that the polynomial structure we use is a product of the order of operations we use, not the other way around. A polynomial in a left-to-right order of operations could look like this:
5*x+4*x+3*x+2*x+1
(Equivalent to 5x⁴ +4x³+3x²+2x+1 in PEMDAS)
Notice how no exponentiation is needed in that structure? Arguably, that's an advantage. (Or you could argue that the advantage of having index numbers right next to the terms helps humans to understand it at the cost of making a calculator's job harder.)
@Tzizenorec Either you made a typo, or to use this structure one must make an assumption. In the latter case the assumption is that if the exponent is "+1", then it MUST NOT be stated. For ONLY THEN can the "+1" be seen as the addition of "1" to the value of the rest.
This raises another issue, namely if the "2*x+1" is instead "2*x-1". Then one can validly ask whether this is the addition of a negative quantity (-1), or raising the "x" to the power "-1". Also, how is one to express the power of "0"? Another also: how is one to express absolute values?
The point is the usual point: that of allowing the possibility of assumptions.
@@user-gx1rk8yw6l Eh? There are no exponents in my left-to-right version of the polynomial. Here, have it in "parenthesis everywhere" format:
((((((((5*x)+4)*x)+3)*x)+2)*x)+1)
I realize I might have confused you a bit by using the numbers 5, 4, 3, 2 and 1 for the terms of the polynomial, but none of those are meant to be exponents. They're all numbers that get added or multiplied.
Horner's method to evaluate a polynomial works pretty well with "left to right" order:
Usual notation: x^4 + 2x^3 + 3x^2 + 4x + 5
Usual notation, Horner's method: (((1x+2)x+3)x+4)x+5
"Left to right": 1*x+2*x+3*x+4*x+5
Maybe it's not easy to manipulate polynomials such as addition or subtraction?
One rule that was not well explained to me, or to some students I have tutored, the line in a rational expression acts as a bracket.
One that I cannot reproduce here due to typesetting limitations, you can have fractions over fractions. Evaluate the one with the smaller line first.
One that is slightly controversial, multiplication by concatenation has priority over multiplication or division with a division operator. e.g. 3x/yz is (3x)/(yz) and not 3(x/y)z. Wolfram-alpha might get this wrong, but if you see this in most math/science publications that is how you should read it.
Great video! I struggle learning without understanding aka memorization so this helps a lot.
Great stuff. We always never get theory with all this test driven teaching
The one thing this video omits - which always causes angry debate on the internet - is "implied multiplication" or "multiplication by juxtaposition" which _some_ place before signed multiplication and division - ie 3÷2x would be 3÷(2x) and not 3÷2*x evaluated left to right.
that's because there's no such thing as multiplication by juxtaposition. the "implied multiplication" before a parenthesis is just omision for simplification. the same way you don't write the 2 when writing a square root √ compared to any other root. for example ∛ or ∜.
@@DanielRossellSolanesHow can you say that there is no such syntax?
@@drdca8263 that's easy. exactly the same way you can write that there is. you write one character after another. or speak one syllable after another.
@@drdca8263 Very simple: just because a layperson insists a form of syntax exists does not mean they should be taken seriously. Laypersons have a penchant for explicitly contradicting what they are taught solely due to a whim that they have, and this is true not just when discussing mathematics, but in all other topics too. Anti-education and anti-intellectualism are serious problems we are facing, especially in the U.S.A.
@@DanielRossellSolanes "No such thing" as implied multiplication?
Are you nuts?
It's whenever a multiplication is IMPLIED without the use of an explicit multiplication operator symbol.
I really liked the plane, car, walk and the increasing opperations examples for the order of opperations, and the video was just really interesting as a whole. I am going to start tutoring math soon and I'm definitely keeping this video in my back pocket
@smartmanapps5588 Well, yes, but this is a good visual for that exact topic. Especially for people who struggle more in math.
@smartmanapps5588 Last warning, please refrain from being argumentative with everyone in my comment section. We have gone over this many times in many comment threads already.
3:37 in the ideal world, if the distance is long enough, you can take a high speed rail. once you are in a smaller range, you can take the subway. finally, you can ride a bike or walk in a mixed-use landscape
Obviously the best order is left to right with reverse polish notation.
I agree, but good luck getting anyone to use it when writing things down on paper.
We can still do 2*apples + 3*oranges, it just equals 2*apples + 3*oranges, like how 2x + 3y doesn't simplify any further, but that doesn't make the expression invalid.
I remember a guy I knew insisted that PEMDAS was wrong. Instead he would use… PEDMAS. He insisted that you couldn’t do multiplication and division in the same step (same with addition and subtraction). No idea what his problem was, but I’m glad I don’t see him regularly anymore.
it's stupid to just do it left to right. it defeats the purpose of an order of operations!
I have a feeling that it has something to do with implied multiplication
Excellent video that forced me to think about something I otherwise wouldn't have really questioned.
Useful properties aside, PEMDAS is part of natural language! If you order fries, a coke, and a quarter pounder with cheese, the cheese is only for the quarter pounder. It's not a separate item, nor is it put on the fries and coke. Cheesing up a burger isn't quite the same thing as multiplication, but our languages are full of modifiers that apply only to the thing they're next to.
For a more direct example, if a shopping list has "🍎×3 + 🍌×2", nobody in their right mind would take this to mean 6 apples.
And for the left-to-right crew that really wants cheese on everything, fret not! You can ask for fries, a coke, and a quarter pounder, *all* with cheese. The "all" is like the natural-language version of parentheses. (But whether the place will actually comply and put cheese in your soda is another question entirely.)
Just a small philosophical grumbling from a non-mathematician:
You can definitely compare apples and oranges. Apples have smooth skins, oranges have rough skins. There, I made a comparison between apples and oranges. Furthermore, if I add 3 apples and 2 oranges together, even though they are two different units, they give me an answer that is in a third type of unit: I have five fruits.
A question arises: can mathematics represent the addition of two different units that produce a third unit? On the surface it would seem that letters could be such designators, maybe one could write a+b=c but I have no idea if that works. Set theory could help, the addition of set a to set b produces the superset of ab.
Anyway, I'm tired and can't be bothered to think further. I would be glad to hear perspectives on this.
Your comments are thoughtful and critical!
In math, when we say we are comparing two things, there is usually a technical definition for "comparison". For instance, in this case, we know that 3>2, so 3 apples is considered more than 2 apples. Apples and oranges are not comparable because we don't know which is "more", an apple or an orange (at least, you can't know without more context). You can totally just treat them as a more general "fruit" unit in this example, but this is not doable or at least not useful in general. What is 1 second + 1 meter? For a more detailed explanation on how unit arithmetic works, I recommend my previous video introducing dimensional analysis: czcams.com/video/IvQ5ag2cEx8/video.html
As for sets, "addition" isn't a well-defined operation without more context. You can, for example, do a union b, or ab as a Cartesian product, but a+b doesn't have a standard definition.
*Furthermore, if I add 3 apples and 2 oranges together, even though they are two different units, they give me an answer that is in a third type of unit: I have five fruits.*
What you did here is not addition. It is set union. Those are very different operations.
*A question arises: can mathematics represent the addition of two different units that produce a third unit?*
The answer is no, and I am currently typing a paper for my thesis, where I develop classical physics using a measure-theoretic approach, where I explain why this is impossible.
*On the surface, it would seem that letters could be such designators, maybe one could write a + b = c.*
This does not work at all. All you have done is say "some quantity, added to some other quantity, is equal to yet some other quantity." Nothing about this enables anything with regards to measurement units. At the very very very minimum, you need measure theory to even begin to try to develop a notion of "measurement units."
Reducing parentheses is partly why there are proponents for the order of operations to put multiplication by juxtaposition ab=a*b above the normal left to right precedence for division and multiplication. 1/2x being treated as 1/(2x) means less parentheses while also distinguishing between (1/2)x as x/2
Computer science has somewhat changed the common needs and use cases with regard to notation and operation order. For example fractions and exponents are harder to communicate when typing than they would be in writing. So there's an argument to bring back juxtaposition as it's own layer of precedence for "implicit" multiplication (which would come before explicit). For example, should 1/xy be treated as 1/x * y or should it be 1/(xy)? There's definitely an argument that the latter is often more convenient. In fact, historically they actually did use this convention
historically each paper used it's own convention. even the same mathematician has used the letter pi to mean the ratio between the perimeter and the diameter or the ratio between the perimeter and the radius of the circunference.
that's why they began with "let be π the ratio between the..." and why they made a step by step on operations. to avoid confusion on what they mean on that paper.
Every textbook or paper I've ever seen means 1/(xy) whenever they write 1/xy.
It's rather unfortunate that most programming languages defaulted to infix notation when the literal purpose of expressing arithmetic in most code is to express the order of the operations we want to happen, rather than expressing algebraic equations. Either Polish Notation is vastly superior for the purposes of expressing the actual sequence of events in the registers that we want to happen as programmers.
We load a value into A, load a value into B, then add. Thats RPN, and thats what assembly generally looks like for a reason.
This is not an accurate representation of what is happening. There has never been such a thing as "implicit multiplication with juxtaposition." The real issue is, as you have mentioned, with communicating fractions and exponents, with the reason being the following: when handwriting expressions using these concepts, we do *not* use infix notation. The notation for denoting fractions is vertical, and the notation for exponents is superscript, also a form of vertical notation. When typing on CZcams, infix notation is used, because vertical notation is impossible. Thus, the traditional order of operations applies. When typing on LaTeX, you use vertical notation: for fractions, you employ the vinculum bar, which separates numerator from denominator, and it works less like an operation symbol and more like a punctuation symbol, akin to a comma or a semicolon. In all other contexts, the order of operation applies if the notation is horizontal and infix.
*@[**09:27**]:* To be fair though, the 'adjacent variable' notation is most responsible for confusion here.
I learnt arithmetic including PEMDAS back before we had spreadsheets or computer programming. Before then it was just a convention to make sure everyone was on the same page, as it were. But now it's an essential part of knowledge if ANYTHING you do involves computers. Including modern screen-based calculators.
I'm glad you mentioned tetration in passing. I'm yet to be introduced to practical applications and how expressions involving tetration operations might ever arise in reality, but could there be a (probably inherently pathological) example where it makes a difference whether tetration is performed before exponentiaion or after it? If so, should theoretically higher-order hyperoperations be even higher priority still? (Or, rather, given that the video makes clear that order is a matter of convention, but the convention adopted in practice has useful properties, are there similar useful properties to be gained or lost by incorporating hyperoperations as having higher priority still?)
Programming languages tend to have even more complicated precedence rules. Especially functional languages.
One of the most interesting operators in haskell are the function application operators. Using a space as function application has tighter precedence than any other operator and goes from left to right. Using a $ for function application has looser precedence than any other operator and goes from right to left. So, "f y x + 2" would be the same as "f(y, x) + 2", "f $ y $ x + 2" would be the same as "f(y(x+2))",
I've always considered the parenthesis-only order of operations kind of great because it requires minimal time to teach and it requires no memorization. So it's easier to read for someone who isn't versed with our language, for example an alien.
Meaning that transcribing math equations into a language they understand would be easier for them, meaning it's closer to some "ultimate language" that's commonly accessible to all conscious beings no matter where they are in the universe.
Compared to something that's full of conventions or assumptions and requiring memorized data for reading(which may or may not be present in the manuscript being deciphered, and that crucial data about the assumptions being rarely available because we believe "it's common knowledge so there's no need to write it down over and over again" would make transcription more difficult for the aliens and delayed by decades even), and I think that's beautiful.
Also, you can get away with only one parenthesis in places where there need to be many if those parenthesis lie at the very start or end of the function
I mean the only disadvantage (and it's a big one) is that it takes a LOT more space. A lot of notation in math is introduced to shorted equations, at the cost of making them more obscure to someone unfamiliar. But when equations take multiple lines WITH all kinds of notation simplifications, learning the order of operations is a no brainer
Also lots of parentheses are confusing
@@HunsterMonter true it takes a lot of space. but i don't understand how lots of parenthesis are confusing. if you're analyzing an equation then sure, but if you're solving an equation you just have to look at the section where there is no parenthesis, solve it, remove one layer of brackets, then move on.
@@hrishikeshaggrawal Having lots of parentheses can make things confusing because having tons of opening or closing parentheses bunched together means you can easily miscount the number and change the expression accidentally, this is why we use {}, [] and () in my classes, even for expressions with only two or three sets of brackets. Now imagine an expression with 25 pairs of brackets (or more), either you invent a lot more delimiters or you get an unreadable expression
@@HunsterMonter nu uh. you can actually just make (((((( for example into just (. there is no need to have multiple in use sequentially(so there can be no miscounting, because there is no counting at all)(edit: wait no, you do need to be able to count, but what i said after this till works perfectly without requiring counting, but you have to use un-compressed brackets still).
you don't need to look at any other brackets other than the ones enclosing the very inner terms, solve those and then remove those two brackets with each step of solving.
how do you search for the inner most brackets you ask? just search for the first ) bracket and anything behind it till the last ( bracket is your current step. it's that easy. in fact it's so easy it would make for the most optimized equation solving algorithm for computing.
the only case where this is detrimental is if you especially need to be able to use the distributive property to progress(so you need to count) or when variables are present inside the brackets making those brackets nearly permanent throughout the course of the solution. but then again in most of those cases the brackets will probably pass on and into the very solution itself.
Converting multiple open parentheses in a row into a single one unfortunately produces ambiguous results.
Consider the expression:
(3+2(x+1)-2)^2)
This is ambiguous because you don't know if the expression should be:
((3+2(x+1)-2)^2)
or
(3+2((x+1)-2)^2)
And these are totally different expressions. Unfortunately, if you do use parentheses for everything, you will in fact need to count parentheses throughout the entire expression. You can always write shorthand for multiple parentheses in a row, but if they are spread out throughout the expression, then there is no way to eyeball which parentheses pair without counting.
Tolles Video 👍
Whats really funny is as a math major i dont use PEMDAS. Like idk maybe orders of opperation have more applicability for real world math like accounting, but i just use parentheses and fractions. Its way more clear in my brain what does what when the order is just laid out on the paper.
Great video
This is a great leader to the idea that implied multiplication can be a real thing. It’s slight higher precedence can be useful
I still do not understand why that is a thing.
because with math you have power and also, it is useful for writing some expressions without having to write many parenthesis (your way)@@elio7610
@@elio7610 Operator precedence reduces the number of symbols (especially parentheses) needed to represent an equation. Implicit multiplication reduces this even further. I'll note that this isn't purely a matter of convenience/laziness, though I'm sure that's a factor. However, those who master such rules are able to use less cognitive load to interpret such equations. An expert will understand 2XYZ as 2*X*Y*Z, but will only need to store it in their short-term memory using 4 tokens, rather than 7. While this sounds minor, it means that experts can fit "more" math in their head at once (or alternatively, can evaluate such equations with less cognitive load). However, mentally modeling 2XYZ without storing the multiplication tokens means you can no longer mentally disconnect the tokens, so you're forced to give implied multiplication higher precedence.
Implicit operators are also convenient for spoken mathematics. "5XY+3Z+2" *could* be spoken as "Five times X times Y plus 3 times Z plus 2," but that causes all the operators to run together and makes it harder to mentally track the equation. Whereas saying "Five X Y plus Z plus 2" has only two spoken operators (both plus), making it easier to track.
polynomials only look better with standard order of operations than left to right because it wasn't properly simplified first. (x^4+2x^3+3x^2+4x+5) in standard form is equivalent to (x+2*x+3*x+4*x+5) in left to right form. Left to right has its place. You just have to know how to use it.
YES!
4:44 there is no such thing as division or subtraction, they're only handy shortcuts for variations on multiplication and addition, respectively.
*only in this context* there are places where subtraction is an entirely distinct (non-total) operation such as in the natural numbers
If it can be defined, it's a thing.
@@MagicGonads If it's nontotal, then it isn't an operation to begin with.
Well, this isn't true. In a quasigroup, there is no associativity and there is no such a thing as inverse elements or even identity elements, yet quasigroup subtraction/division is still well-defined.
It exist but is EQUIVALENT to adding the negative, seriously why don’t people understand
I think it’s pretty intuitive…
When it comes to Arithmetic operations, there are only 2 clear options:
Least to greatest, or greatest to least (Pemdas).
Imagine addition is the base operation (+)^(1)
Multiplication is the repeated operation (+)^(2)
Exponentiation is repeated again (+)^(3)
Tetration … and so on: (+)^(N)
Pemdas is formatted greatest to least. The ‘P’ for parenthesis: encapsulates expressions, allowing us to modify the order of the operations freely.
Pasmde would’ve also been totally fine, probably:
Pemdas: ( 7 + 5 * 3 ) = (7 + 35) = 42
Pasmde: ( 7 + 5 * 3 ) = (12 * 3) = 36
*if you add exponents, these numbers get a lot bigger, so I didn’t use them here.
In general, these both work as a linguistic convention, for how we interpret Arithmetic expressions.
Another important thing to recognize, is exponentiation/ tetration/ etc… are no longer commutative.
It might even be better to use Pasmde because of commutativity:
( 11 + 7 * 5 ^ 3 ^^ 2 ) is read from left to right - the non communication operations are ordered from left to right - the operations are applied from left to right.
Just saying… but that can also be more confusing for kids to learn the math conceptually, if they assume it’s always being parsed from left to right… idk. I’m not a psychologist.
You are so underrated!
What are your thoughts on less common, but still often used, conventions like Reverse Polish Notation that the HP Voyager calculators used?
I think it's genius! The nice thing about this notation is that no expression requires the use of parentheses, so not only are parentheses extraneous, but so is the order of operations! Of course, you can still add parentheses anyways for clarity and readability, but it's not needed.
I really like the term "postfix" notation, because it implies the existence of the "prefix" notation as well! In retrospect I do wish I added a segment to the video briefly explaining this.
So just lisp?
@@bigzigtv706Right, LISP removes all parentheses by using prefix notation.
...
Oh, wait...
@@Hauketal LISP = "Lotsa Insane Silly Parentheses"
@@RealMesaMike scheme is worse
can you do a video on why we use infix notation vs postfix or prefix? This one choice to use infix affects how we think about order-of-operations.
I may do a video in the future on what the different notations are and the benefits of each, but the why in this case is mostly historical convention, which is generally outside of the scope of what I cover on this channel. I recommend reading about the history of mathematical notation if you want to learn more.
Function syntax with parenthesis only:
3x^2+1=+(*(3,^(x,2)),1)
Now solve your favourite cubic equation in this notation.
I prefer to think in Reverse Polish notation some times, because its stack based and in certain context makes sense. Especially when I have to write large parallelisable bits of code. in general I think stack based mathematics is interesting.
slight note: you don’t actually need parentheses to write polynomials with left to right order of operations
example
c3*x^3+c2*x^2+c1*x+c0
would be written as
c3*x+c2*x+c1*x+c0
which you’ll notice is actually even shorter than pemdas
1:58 no, this looks completely fine to me
The chaotic order of operations: PPPS.
Parentheses, Powers, Products, and Sums. Rolls right off the tongue doesn't it. :P
‘you cant compare apples to oranges’ caught me off guard and i just started laughing
Very well done. This is like the Lies of P version of a math video.
That thumbnail is perfect.
Math symbols are functional, which means we need to decide which functions have which arguments before we start evaluating the expression
Reverse polish notation is the parentheses only without the parentheses.
To all advocates of prefix or postfix notation: Do you use this for the binary comparison oparation usually denoted with = as well? This operation sends a pair expressions to one of the boolean constants TRUE or FALSE. A famous theorem of geometry would then read a 2 ^ b 2 ^ + c 2 ^ =
It's certainly funny, but it won't make things easier.
You could do pedmsa as far as i understand, literally doing d before m and s before a
6:55 I often do this: 3 apples + 2 oranges = 5 fruits
However, you shouldn't do that if they convert differently. For example, you can't do 1 second + 1 meter because 1 second + 16 desbimeters or 20 ticks + 1 meter would be inconsistent
See, now you're getting into category theory.
Object oriented math
*AHEMM*
people always forget that there's a rule that states that multiplication by juxtaposition goes before multiplication and division in the order of operations. for example, 21 / 3(5 + 2) = 1, not 49.
I have never heard of that rule. The way you wrote it you can rewrite it as 21*(1/3)*(5+2) because dividing by 3 and multiplying by 1/3 is the same operation. Which is in fact 49.
I read some more comments and it seems that this rule is pretty widespread. I'm honestly baffled but I guess if everyone would agree on that rule, then the correct answer would be 1, I learned something new today ^^'
Amusingly, this is a case where the common usage by most people diverges from professionals. Most professionals stick to the order of operations as I described in the video, and this "multiplication by juxtaposition" tends to be unheard of, since it complicates matters and is never taught in school. The general public adopted this rule, despite complicating matters, it was invented specifically to justify the multitude of people who find this alternate set of rules more intuitive. This divergence demonstrates how this is convention and why challenging even our most basic notions of mathematical concepts can potentially be meaningful.
This is also why professionals will almost never write division inline without being crystal clear using parentheses, and will use fractions when possible.
@@zhulimathIt is taught outside the USA. Almost all Japanese calculators, for example, explicity give it precedence and say so in the manual. And I firmly believe giving it precedence SIMPLIFIES things and makes equations look clearer. Unfortuantely, America decided to go it alone on this one and stick with the simplified PEMDAS religiously.
well, at least with the math people i know, they all follow this rule.
Since a hyperoperations can be said to be repeated operations of the lower operations,
Can’t you in theory, break all the operations down to repeated addition?
Basically replace any hyperoperation higher than the 0th with the equivalent 0th if that makes sense?
afaik hyperoperator exponents are only defined for integers
You can, but the entire point of defining higher hyperoperations is to avoid the comically large expressions that would result if you did such a thing.
Take 9↑↑4 for instance, which is applying the 4th hyperoperator, tetration. If you tried to write out the full base-10 digits of this number, there is no possible way you have enough space in the entire observable universe. Good luck expressing this in terms of repeated addition!
honestly in math i was so scared of getting my operations wrong that i super overused parenthesis
00:42 questions:
1. how can radicals (assuming radicals are square root, cube root) have order? their syntax is different from other operators
2. what about functions: sin x * y + q
3. what about factorial: q + y * x!
Radicals are essentially just functions, and functions generally apply order precedence through parentheses. For example, f(g(x)). There are some other notations for functions, but generally they either follow the same convention or their definitions are laid out. Factorials generally have very high priority, for many of the reasons listed in the video.
1. Radicals are just another notation for x^(1/2) etc., so *if* they have a precedence, it should be the same as exponents. You're right that in usual notation, they have a bar (called a viniculum) over the radicand that effectively puts parentheses around it. In fact, historically, the viniculum was used to indicate grouping instead of parentheses. The radical is just a special case where the viniculum notation stuck around. But in youtube comments (for example), I can't type a viniculum easily, so I have to type √(16+9) = 5, which is different from √16 + 9 = 13.
You could make similar arguments about division not needing a place in the precedence. The usual way to write division is to draw a horizontal fraction bar with the numerator and denominator unambiguous. But it's needed for the solidus (/) notation and for the uncommon-outside-of-elementary-school obelus (÷) notation.
2. Functions normally use parentheses for their arguments, so it doesn't matter. Trig functions and logarithms sometimes use more ambiguous notation for historical reasons. In the unparenthesized notation of these functions, they do not have an entirely consistent place in the order. "(sin(x))^2" is commonly written as "sin² x". But "sin x²" could mean either (sin(x))^2 or sin(x^2). "sin 2x" almost certainly means sin(2*x), but "sin x cos y" almost certainly means sin(x) * cos(y) rather than sin(x*cos(y)). I think almost everyone would agree it comes before addition, though. In your example, I would interpret it as ((sin(x)) * y) + q (even though I'd normally write that as "y sin x + q"), but if you had instead written "sin xy + q", I would interpret it as (sin(x*y)) + q.
In short, please just put parentheses around the arguments to trig and log functions, like every other function.
3. Equations involving factorials are consistently written to have factorials apply before multiplication. e.g., "nCr = n!/(r!(n-r)!)" the denominator is (r!) * ((n-r)!), not (r! * (n-r))!. It's a little less obvious whether they come before or after exponentiation, because the usual superscript notation usually makes that unambiguous: "3⁴!" is obviously (3^4)!, not 3^(4!); if we wanted that, the "!" would be superscripted also. But "3^4!" is probably 3^(4!). I have three reasons to think this: (1) it is natural to take unary operators as normally having higher precedence than binary operators; in programming languages, where explicit operator precedence is more important, this is usually the case, for example. The biggest exception is negation (unary minus), and that's mostly because the symbol is the same as the one for binary subtraction. (2) Using the speed/vehicle analogy in this video, for large n, n! grows faster than a^n for fixed a, and MUCH faster than n^a for fixed a. However, it does grow more slowly than n^n. (3) Wolfram MathWorld says so: mathworld.wolfram.com/Precedence.html
You say "the space of polynomials basically encodes the essence of addition and multiplication" at 11:20 ish. What does that mean?
For the sake of time, I was very loose in my language here, I apologize.
What this means is the following:
Start with a variable x or any real number. Add or multiply it to any other real value or x. Do this addition / multiplication as many times as you like, using any expressions that could be obtained in this way. No matter how you add or multiply your expressions, no matter how many times you do this, there will always be a unique polynomial that expresses all of the steps you have performed, altogether.
I hope that clarifies that!
@@zhulimath thanks, that makes sense. Inspiring video too!
Actually polynomials are even easier to write using left-to-right order, for example the polynomial shown in the video could be written as x+2⋅x+3⋅x+4⋅x+5.
Excellent observation! Being able to express a concept in multiple perspectives is a key skill for mathematicians and problem solvers.
To be technical, the reason I did rewrite the polynomial this way when explaining is because while you can think of there being three major "forms" of a polynomial (fully expanded standard form, factored form, and this one you've mentioned here), standard form and factored form are, at least for most math students, the most common and most useful forms, which helps explain why the order of operations was motivated this way, despite every polynomial trivially able to be expressed in this left to right order.
Ok, this might sound kinda weird but I personally don't think the "successor" operation makes sense as the 0th hyperoperation. Like its not a binary operation, and just doesn't really fit it well with the rest of the hyperoperations.
Like this sounds kinda stupid but I think it shoud be like 3+2 = ((2 ? 2) ? 2) where "?" Is the 0th hyperoperation. Also its kinda nice because with "?" The pattern of 4= 2?2 = 2+2 = 2*2 = 2^2 = ... Still holds.
The one major downside of having a binary 0th hyperoperation is that 3?2 is more or less undefined. Like 2?2 = 2+2 = 4 and 4?2 = (2?2)?2 = 3+2 = 5 but there isn't any good way to convert 3?2 into a bunch if the same number so you cant use additon to figure out what it is.
@zhulimath As a programmer, discrete math sounds so much simpler and clearer. Why can't all math be structured in this way? Not only the readability is superior in every way, but you can solve it naturally line by line, just like a programming language. You can even use variable names to make things even easier to read. It removes all possible confusion.
I understand how you feel, but unfortunately formal clarity is not necessarily the same as clarity in interpretation/communication and is not the same as pedagogical clarity or practical convenience, and throughout all of it you are fighting against historical convention. Be the change you want to see in the world!
Is there always a left-to-right version of a computation? For example (a+b)*(c+d) does not imo
I think you would have to use parenthesis, like a + b * (a + b), witch would become a lot more convoluted in bigger expressions =/
If you use RPN it becomes:
a,b,+,c,d,+,*
Good explanation for why the precedence of operations is what it is.
But...I wish teachers would quit calling it the "order of operations."
It's operator precedence!
As you noted, you can use the various properties to rearrange expressions so that you can evaluate them in an order you find more convenient.
Yes, you must respect operator precedence, but you DO NOT have to stick to some strict "order of operations." i.e., you don't necessarily have to "do what's in parentheses first" if there are other parts of the expression you can work on that don't involve the parentheses.
Yadda yadda yadda....
I know, I know... I'm tilting at windmills here.
I understand how you feel, and in retrospect I wonder if I should have added a comment talking about this distinction, or if that might be too pedantic for the purpose of this video...
@@zhulimath This is something that is actually driving me mad right now, as I have to teach kids this daily. I don't feel like the traditional order of operations we teach allows for the latitude of knowing when and how to do things out of order to make your life easier. For example something like (1/7) * 3 * 7 you would really want to do the 1/7 times 7 first, but it's really difficult to teach that you can do the multiplication and addition in any order you wish with associative and commutative properties rather than sticking to PEMDAS. I'd love to hear your thoughts on alternative takes to the traditional order of operations.
@@r4masami I encounter similar situations with my students as well!
My approach is usually to first explain the concept of inverses, whether negatives or reciprocals, for example that 5 + (-5) = 0, or that 3 * (1/3) = 1. Then I try to explain how problems that are exceptionally difficult to solve using traditional order of operations, such as 8!/7!, can be trivialized by applying these commutative/associative/distributive properties.
Once students see how much simpler these problems are by applying these properties, I get them to drill on all kinds of expressions where you must rely on these properties to make the problem practical to compute.
Some examples:
37*47 + 37*53
2^6 * 5^6
768*9999
This essentially forces the students to find these "shortcuts", and then they will develop an appreciation for how to think in this way.
@@zhulimath What an absolutely amazing response. Thanks for this, I'll try to keep this in mind.
I would like to think parenthesis only (the one my programming-ridden brain prefers since theres so many operators) is at least chaotic neutral :3
If you use Horners method for polynomials, polish notation is easier.
Please include implicit multiplication such as 2b.
Problems occur with a ÷ 2b
By PEDMAS, it should be (a ÷ 2) * b
But I'm convinced that it means a ÷ (2 * b)
@@smartmanapps5588 I disagree. I feel that you have missed the point of what you have read.
Web search "implicit multiplication", you'll find authoritative information.
2:29 the up arrows made it more confusing
Moving to a 'left to right' order of operations doesn't automatically mean that multiplication by proximity is broken. That would really be a separate thing that would have to be established in any such system. For example you could still treat 2b as a single quantity due to the implied multiplication by proximity, so that the expression 4+2b can be interpreted correctly using a 'left to right' order of operations, or it could be as shown in the video, where it is treated as (4+2)*b
This is true, but isn't this just the order of operations in disguise?
Haven't watched the video yet. Need to think about whether I want to. It's a topic I've always had strong views on (that the acronystic mnemonics are stupid because people should learn the deeper patterns instead), and if I did watch it, it would not be to learn things, but in order to (hopefully) nod along and say "I told you so", and then elaborate in the comments. And I probably won't have time for that.
If you want to perhaps learn some arguments in favor of your views to better convince others, perhaps skim through the video to see if there is anything of value to you!
I’d definitely swap chaotic evil and lawful evil in the thumbnail
Thumbnail was a bit rushed but in retrospect I think I agree. Might fix it sometime!
Patentheses only is literally the opposite of chaotic!
Before vid starts and my guess is that anything past addition and subtraction is just a lot of addition and subtraction with a fancy symbol.
6:49: Actually, not any unit holds this. If you assume x as 10kg U-235, then the right side collapses with a nuclear explosion.
What would be alignment for postfix notation?
Probably lawful good.
where does your intro sound come from
Nikolai Kapustin - 8 Concert Etudes, Op. 40: III. Toccatina
True neutral incremental function composition
i heard "parenthesis" and "eyesore" in the same sentence, immediately thought of lisp / racket
Why is succession the first hyperopator?
I would go the other way and use precession, as for natural numbers you can create addition out of repeated subtraction (e.g.: 3 - (0 - 1) = 3 + 1), but not the other way around
Succession is the zeroth hyperoperator, addition is the first, multiplication is the second, exponentiation is the third, and tetration is the fourth (hence the root "tetra-" in the name).
@@angelmendez-rivera351 that doesn't answer my question.
You can create multiplication out of repeated addition, so multiplication is a level higher. That makes sense.
You can create addition out of repeated subtraction, but you can't create subtraction out of addition.
So subtraction should be a level below addition in my opinion.
It's similar to Boolean Logic where all the other operations can be built from NOR. You can construct positives out of negatives, but you can't construct negatives out positives.
@@Domo3000 *You can create addition out of repeated subtraction, but you can't create subtraction out of addition.*
You can create multiplication out of division, but you cannot create division out of multiplication. See how your "logic" is completely illogical?
*So, subtraction should be a level below addition in any opinion.*
Yeah, that is not remotely how hyperoperator "levels" work at all.
*It's similar to Boolean logic where all the other operators can be built from NOR.*
Natural number arithmetic is not analogous to Boolean logic at all. Their axioms and signatures are completely different. This is like comparing a car to an space rocket.
@@Domo3000 The successor is the 0th hyperoperator, because it is the operator required in the Peano axioms to define the structure of the natural numbers axiomatically.
@@angelmendez-rivera351 I still don't see how it's illogical just because you call it illogical.
Subtraction can be used to create addition, but not the other way around.
Division is repeated subtraction. Division can be used to create multiplication, but not the other way around.
Roots are repeated division. With subtraction, division and roots you can create exponentation.
So these seem like more basic building blocks than their addition/multiplication/exponentation counterparts.
And that's why you can make an excellent case (especially in typing where you can't draw elaborate division lines easily) for "PEJMDAS" where implied multiplication takes precedence, so like "xy / 2x" is intuitively interpreted by "x*y" divided by "2*x" rather than x mujltiplied by y then divided by two then multiplied by x again. Basically to remove the brackets from the "correct" PEDMAS expression where it would be (xy) / (2x). In this convention, implied multiplication by juxtaposition also implies brackets around the expression.
It's generally much harder than you'd think to come up with accurate rules for basically all cases. For instance, if I write "log xy!", what are you evaluating first? Most people would read that as log(x(y!)), but what's the rule?
In general, we tend to evaluate postfix operators first (factorials, exponents, etc.), then implicit infix operators (normally multiplication, although the implied operation can be something else in other contexts), then prefix operators (minus signs, functions, etc.), then explicit infix operators (+, ×, and so on). It's funny how much of this ends up relying on intuition in more complex cases.
@@smartmanapps5588 Factorials, like exponents, aren't grouping symbols. There's nothing to group: they don't "wrap around" anything. (Parentheses and the like explicitly create a group by delimiting a beginning and an end.) They are just a postfix operator.
A "term" is just a name given to a maximal sequence of symbols that contains no ungrouped operators with precedence equivalent to or lower than addition. (Yes, there are operators with precedence _lower_ than addition.) It's just a shortcut used to explain the most common cases of precedence. But when you have more than a few tiers of precedence involved, the shortcut falls short.
@@smartmanapps5588 That would be if you expanded them. By the same token, 3 × 4 groups (3 + 3 + 3 + 3). And also, what about non-integer powers or factorials? What multiplication is 0.5! grouping? (And yes, that's well defined.)
@@smartmanapps5588 But again, what about operations that don't expand to anything?
Consider the volume of an n-dimensional ball (so for n = 2 you get the area of a circle, for n = 3 you get the volume of a sphere, etc.):
V = (R √π)ⁿ / (n/2)!
For even values of n, you can expand that factorial the way you showed. But for odd values of n (such as the sphere case, n = 3), the argument to that factorial isn't an integer and the recursive definition doesn't terminate: you can't expand it that way. (And yet, 1.5! = 0.75√π is well defined.) What is that factorial grouping? It would be tempting to say that it's grouping the (n/2) before it, but it's not - that's the reason I needed to add parentheses there in the first place!