With how many people on Twitter argue from a third grade understanding of sex and gender I can’t say I’m surprised See also climate deniers and Trump supporters
It is not really about lack of math knowledge, for people who are used to working with calculators that show only one number at a time it acts unpredictable
Hold up. without parenthesis that says: here's 50 of something here's 50 more of something now double it. It is worked left to right. In which case, the addition would go first. Without parenthesis, it is a different statement. It says this: 50+50×2=a ÷50 from the left is 50x2=a/50 Which is 100=a/50 Which is a=200 Or you can ÷ 100 from the left Which is 2 = a/100 Which is a=200 Without parenthesis, it is already broken down.
@@dogbreaththe3rd851 Cute. No. However, seriously, write the equation out to prove 200 is incorrect. The flow chart the video presents is not mathematics.
Ive been trying to explain to people that multiplication and division are the exact same thing. Multiplying by 2 is the same as dividing by 1/2, they are the same operation and therefore have the same order of operation. This also applies with addition/subtraction (+2=-[-2]) and exponentiation/roots (x^[1/2]=[sqrt x])
Make that 1950. The fifties were a great time for futurism. Just look at the Atomium in Brussels, which was built in 1958. A great example of the sixties is 2001: a space odyssey, where HAL9000 is an AI computer built in 1992.
@@EaglePicking You've just reminded that I have been to the Atomium. I've also been to the Dali exhibition in the Pompidou centre in Paris. HAL. In each case one letter before IBM. Back in the day, IBM meant "It's Being Mended", ICL meant "It's Coming Later", and DEC meant "Don't Even Consider". Yes, I am actually that old!
you can imagine what other things that individual is getting all sorts of wrong in his life. Including voting. You can only fear his choices if they impacted your daily life as well.
People did Maths for hundreds of years without the order of operations we mostly use now, they used different ones, or none - it's not a fundamental part of maths just an arbitrary set of rules that are relatively recent, and are not needed
Except it isn't really "math"...it's busy work. PEMDAS crap is why kids think they "hate math" or that "they're no good at math"...it is the math equivalent of diagramming sentences.
The problem here is that Apple calculator app is absolute garbage because it doesn't show you your whole input. Some people then think that it actually evaluates the 50+50 part - exactly like those cheapest single-line calculators, which evaluate every single operation as soon as possible. Apple's calculator app looks like it does the same thing, but doesn't - thus the confusion.
@@Duke_of_Lorraine I wouldn’t. I guess I’ve been using too many cheap calculators. I would use a lot of parenthesis on scientific calculators just to be sure.
She is not that wrong... Apple's calculator does not show the expression, compared to other calculators, and old calculators used to work like..... calculators! and not parsers + calculators
This is the just the first step in dealing with the order of operations. There are other very common videos out there that give an expression like 6÷2(2+1) and the debate rages over whether that should be 1 or 9. Those who say it should be 9 tend to scream “PEMDAS!!1!!111!!” That is, you do the Parentheses and Exponents first, then Multiplication and Division, and then Addition and Subtraction. Thus, you do the “(2+1)” first since that is the part that is in parentheses and you get 6÷2(3). They then see that “2(3)” is a multiplication and thus, they do the division and multiplication left-to-right: 6÷2 gives you 3 and you then multiply it by (3) to get 9. But those who say it should be 1 point out that PEMDAS is only the beginning of the order of operations, not the end. Yes, “2(3)” is a multiplication. But how do we know that? There is no mathematical operator there. That’s because it is “implicit” multiplication. Terms that are placed next to each other are “implicitly” understood to be multiplied together rather than added or divided or some other operation. And for those who say that it is 1, implicit multiplication takes precedence over division and explicit multiplication. Thus, you still do the (2+1) first and get 6÷2(3), but you then do the implicit multiplication first to get 6÷6 which is 1. As a side note, the concept that PEMDAS is only the beginning of order of operations is trivially shown to be true. For example, what is -5^2? Is it equivalent to (-5)^2 and thus the answer is 25? Or, is it equivalent to -(5^2) and thus the answer is -25? PEMDAS won’t tell you because this is unary negation and PEMDAS doesn’t indicate where unary negation falls in the order of operations. Or, what is 2^3^4? Is it equivalent to (2^3)^4 which is 8^4? Or, is it equivalent to 2^(3^4) which is 2^81? And this is just exponents. And yet, PEMDAS doesn’t tell you because it just says you do the exponents first. It doesn’t tell you how to do serialized exponentiation. It turns out that unary negation happens after exponentiation. Thus, -5^2 = -(5^2) = -25. And, serial exponentiation is done right-to-left (or top-down, depending on how you look at it). Thus, 2^3^4 = 2^81. And it turns out that if you look at how actual mathematicians write their symbology, implicit multiplication is done before division or explicit multiplication. If you look at the style guides for the journals of the American Mathematical Society and the American Physical Society, they tell you that multiplication is done first, period, and that if you’re going to put division in there, you should use the vinculum (the horizontal bar) in order to make it clear what is in the numerator and what is in the denominator. Thus, you would never write “6÷2(2+1).” That’s the obelus. Instead, you’d write it as either: 6 - (2+1) 2 Or 6 --- 2(2+1) Depending upon what you were trying to communicate. Now, while I appreciate the nod to more sophisticatedly written calculators like Wolfram Alpha, you need to be careful. A calculator is only as good as the programmers and if they programmed it incorrectly with regard to the order of operations, then the answers you get will be wrong. It’s why Wolfram Alpha says that -5^2 = -25 while Excel says that it’s 25. Excel doesn’t follow the rules regarding unary negation. And sadly, both WA and Excel don’t follow the rules regarding implicit multiplication. Both of them say that 6÷2(2+1) = 9. Now, the only reason that we need to have an “order of operations” is due to the way that we write expressions. We use a system known as “infix.” That is, the operators are placed between (“infixed”) the operands. The “+” in “1 + 2” is put between the two numbers that are to be added together. Thus, when you string operations together, you’ll have operand operator operand operator operand operator operand. Thus, we need to determine which operands are being associated with which operator and when do those operators kick off. There is a way to get around this: Separate the operators from the operands. One common way to do this is known as “postfix” or “Reverse Polish Notation.” Instead of putting the operator between the operands, you put it after them (“postfixed”). Thus, instead of 1 + 2, you’d write 1 2 +. One of the nice things about this is that it eliminates the need for parentheses. The way you calculate is to take the left-most operator and apply it to the operands to the left on the stack. The result is then pushed on the stack at that point and you repeat. Thus, “6÷2(2+1)” would be written as: 6 2 2 1 + × ÷ You take the left-most operand, +, and the two operands to the left of it, 2 and 1. 2 + 1 = 3. This leaves: 6 2 3 × ÷ So you take the left-most operator, ×, and the two operands to the left of it, 2 and 3. 2 × 3 = 6. This leaves: 6 6 ÷ So you take the left-most operator, ÷, and the two operands to the left of it, 6 and 6. 6 ÷ 6 = 1 When you write the notation this way, you build the order of operations into operand/operator stacks. There is no convention to use. It’s whatever you write down. If you wanted it to be 6÷2×(2+1) (note the explicit multiplication), you’d write: 6 2 ÷ 2 1 + × Thus, you start with the ÷ and apply it to the 6 and 2: 6 ÷ 2 = 3. Push that into the stack at that point: 3 2 1 + × So you then take the + and apply it to the 2 and 1: 2 + 1 = 3. Push that into the stack at that point: 3 3 × So you then take the × and apply it to the 3 and 3: 3 × 3 = 9. And thus, you never have to worry about doing the operations out of order: The notation is simply carried out left-to-right. The problem with this sort of notation is that it can be a bit difficult to read for a complicated expression. The operators can be physically separated from the operands. In the expression 6 2 ÷ 2 1 + ×, what is that “×” referring to? What is being multiplied with what? If the expression is longer and you have a whole bunch of things that are being manipulated before you finally get to that last multiplication, it isn’t obvious what is being operated upon. By placing the operators between the operands, it is more obvious as to what the operator is operating upon. But the tradeoff is that you can have operators on both sides of an operand and that means you need a convention to determine when to do which operator to that operand. Remember: The order of operations isn’t there to trick you. It is actually there to make things easier to understand. By coming up with a convention that everybody agrees to, it means that every expression will be understood in the exact same way by everybody. There is no ambiguity so long as the convention completely covers all possible ways of combining the operators. Instead, the problem is that the convention isn’t sufficiently described and taught. Again: PEMDAS is the beginning of the order of operations, not the end.
And as for implicit multiplication, a hundred years ago it was taught to take precedence over explicit multiplication. Then it was reversed and taught that it was taken equally left to right with explicit multiplication. Now it's coming round again that implicit is the higher priority ; and calculators are split about 50/50 as to which way they do implicit multiplication. My calculator app actually has a setting where I can choose which way to do implicit multiplication. But to get it to do unary negation, I have to physically enter the multiplying by - 1.
@@Fountainofyouth007 When was implicit multiplication ever given equal precedence to explicit multiplication in the past 100 years? Certainly not in the past 40 years if my textbooks have anything to say about it. From my H&R to my SZ&Y; my calculus, probability, and statistics texts; my chemistry texts, they all do implicit multiplication before division. It's why the reduced Planck constant is commonly written as h/2π and nobody thinks that means the same thing as hπ/2. It's why Richard Feynman had his lectures using implicit multiplication first. So when was it ever on equal precedence as explicit multiplication or division?
@@dhwyllThat is what was presented in an article I found on the subject the first time this issue came up months ago. It may have also been mentioned in the video I saw about calculators and the fact that some are giving it higher precedence now while some do not.
the fact that it has to be memorized, and cannot be worked out from first principals, should tell you it is an arbitrary set of rules, and nothing really to do with maths at all
Calculators used to (and some still do) calculate the result each time you pressed an operator, not just equal, so if you tiped 50 + 50 then X, it will calculate to 100 (as if you pressed =), then 2 = 200
Why do you continue to post these same type of videos repeatedly ? In recent days, I think you've posted more bodmas problems than actual maths questions
The problem is not about math but rather about interface: Apple calculator doesn't display the full expression being evaluated, just like about any simple calculator, which in turn presumes left-to-right evaluation order (pressing an action button is performing current calculation and going to input of the next argument using selected action). In Android example the expression is displayed fully, and there is no confusion.
Exactly this! Apple's calculator violates good Usability principles, specifically 'visibility of system status'. People's confusion is justified here, as the confusion is not about arithmetics.
Windows default calculator interprets both the "addition" key and the "multiplication" key as also the "equals" key. But if you switch to the scientific view then you get a different result.
Yeah, this has been an issue for many years. But I'm pretty sure they used to have a calculator that did it correctly. But that was the version before they added the "Standard", "Scientific" and other modes. So, maybe 15-20 years ago. I was shocked, the first time I discovered the error and I only discovered it, because I'm good at calculating in my head. And yes, it is an error.
the difference is between calculators that use single-step evaluation (as pretty much all old calculators did), and newer calculators that use expression (or formula) evaluation. the Windows calculator lets you choose between single-step (Standard) and expression (Scientific/Programmer), which give the answers 200 and 150, respectively. to say that old calculators evaluate the "expression incorrectly" is not a valid statement - they don't evaluate mathematical expressions at all, they evaluate their input step-by-step (not expressions), by-design.
Bingo. Apple's looks like the simple type but acts like the advanced type. It's ridiculous to think that someone would put in a complex expression totally blind, and imagine that it's storing it all internally and then evaluating it using order of operations, with no evidence of this.
The issue is not expecting calculators to follow the formal order of operations, but to follow as typed. I grew up as we transitioned from the latter, which is why I use RPN calculators - they behaved in a deterministic manner.
@@mrsoisauce9017 Extensively praising Elon Musk, racism (including that against his own nationality), supporting war crimes, and arguing 16-year-olds should be allowed to get married. That's all I can remember now.
It’s a political Twitter account, so “worst” posts are going to be a highly subjective category. This one is blatantly a false post, but he can be excused as it is very much not in his area of expertise.
This isn't even an order of operations problem. The first example explicitly uses the "=" key, which means "evaluate the tree so far, display the result, and treat the result as the first operand".
Yes, he completely missed the point the Twitter post was angry about. It's that clicking the "equals" button implicitly surrounds your previous expression with parentheses.
Re. the laws, isn't that mostly a matter of notation? If you change the notation conventions you'll also have to change the notation of the laws: (a) + (b) = (b) + (a) a(b + c) = (ab) + (ac) Edit: I certainly don't mean to say that would be better, just that I wonder if it's about the laws not holding vs. having to express the laws differently when conventions change
Yeah, he forgot that a left-to-right convention regardless of operation, means that a + b × c means (a + b) × c Applying the commutative law of addition as he did, he should've started with a + (b × c) which equals b × c + a And the distributive law should've been notated as a × (b + c) = a × b + (a × c) You're right. If you use a different convention, the properties still hold, they just look different. Another example, if using a convention of addition having a higher priority than multiplication, then the distributive law would be written as a × b + c = (a × b) + (a × c) or, using implied multiplication, ab + c = (ab) + (ac) I agree, even though the properties hold under different conventions, it doesn't mean they're useful.
Commutative law part if you have (10 + 50) x 2 and change the order of the addition, you get (50 + 10) x 2. Your way of treating it as 10 + (50 x 2) = (50 x 2) + 10 is a result of the order of operation you already chosen 6:48
Distributive law: a * (b + c) = (a * b) + (a * c) in the normal-order of operation you can remove the parentheses, in the reverse you can't. If you write a law in one systems format, of course it's not going to look the same in the other system.
It's a valid point that if the entire expression isn't shown on screen while you type the rest, then it's updating the total as you go along, ergo not following the order of operations.
He types 50+50 and hit equals and after that times 2 and hit equals which is 100 and later it's 200 But when he types 50+50*2 that's just 1 and not 2 different equations And the other thing which is 2nd or 3rd grade Maths: multiplications and divisions are sooner than addition or substraction
Order of operations is just grammar for math. Yes we could just decide to do it differently but then we would have a harder time understanding each other and wouldn’t know how to write equations to best be interpreted the way we intended
You want to start a flame war? Multiplication by juxtaposition. There is no universal standard so sometimes there are 2 different and valid answers. Most of the world including Europe, academia, and engineering use it. Some of the world, notably North American grade schools do not. Even calculators don’t agree- some use it, some don’t, some have changed back and forth between models. Multiplication by juxtaposition, sometimes called implied multiplication, is where you multiply adjacent terms that do not have a symbol before regular addition, subtraction, multiplication, and division. People construct problems such as the examples you had at the end of the video often with text saying “most get this wrong”, that because there two answers are designed nothing more than to start an argument between two groups that cannot understand why the other group says the other answer. As multiplication by juxtaposition is more common, I say use it. But if your equation could be ambiguous, use parentheses to be explicit.
Love how you explained why the order of expressions is valid past the dad answer of ‘because’. Not using the order of expression will violate the Commutative and the Distributive Laws. I will be referencing this video the next time my family asks these types of math questions. Thanks for sharing.
You highlight the statement in William Oughtred's Key of the Mathematics that shows him evaluating 12x3 first before adding 4, but the very next step (23) doesn't seem to follow: he has the expression 40+3x13=572. What's going on there?
It confused me as well. What he ment to say ist the rule vor the sum of an arithmetical progression. Compare to en.wikipedia.org/wiki/Arithmetic_progression. 4 (first term); 7; 10; ....;37; 40 (thirteenth term), sum of terms (Z) = 4+7+10....+37+40 = 572/2 =286 (4+40)*13 = 572 = 2*286 Maybe the "3" in the equation was a typo and he forgot brackets.
The example of swapping with commutative addition isn't accurate. a + b = b + a would not move the multiplication left if you were using left to right evaluation. It would only swap the 50s, so does not prove your point.
besides tech, you can put the blame for at least 2 generations not knowing the precedence rules on the degeneracy and corruption of the educational and academic institutes due to the decline of western civilization. like or not, the decline is occurring. there is a rise in home schooling.
The vinculum is *not* the horizontal bar used to represent division. The vinculum is the under or over bar used to group digits or to group operations or show conjugates or negation or various other things but *not* division for which the division bar is used.
Maybe also another reason the order of operations is important to follow is because of the following (correct me if i could be wrong). -Multiplication and division are like a simplified version of addition(multiplication) or subtraction(division somewhat). -A parenthesis may be just an unsimplified mini equation. -exponentiation is a simplified version of many multiplications and/or divisions. And wouldn't it be very logical if you started off with the parenthesis since that is an unsimplified mini equation? Then the exponential since it is just a notation for multiplication/division of the same value? Then multiplication or division next since it is another notation for addition/subtraction? Then lastly addition and subtraction? Just wanted to add for your discussion about why the operations of equations are important:))
I think your Proof with the communative law is wrong. The summands are Not 50 and 50*2 but 50 and 50. You are literally assuming in your Proof, that multiplication comes first.
First, follow PEMDAS (Perinthesis, Exponent, Multiplication and Division, ADD and SUBSTRACT) So this evaluation: 50 + 50 * 2 Is something like this: 50 + 50 * 2, 50 + 100, 150 So the answer is 150
Let all be kind here 😂😂, I know we all know it but give some respect for his honest question. He is willing to learn new things although it should be a basic understanding for others. Knowledge is a bliss to those who have it. I hope he got the answer and learned new things.
One ambiguity in the order of operations I personally find is with trigonometry, for instance, seeing something written as: tan αβ Just based on the order of operations of trigonometry coming before multiplication, it should be (tan(α)) * (β). However, I was told it really denoted tan(αβ), because the αβ written together implicitly formed a group, equivalent to that of brackets. Which is the correct order?
the standard is usually, if you're not writing the × or • explicitly then scalars go infront of the term. In your case we would expect (tan(α))*(β) to be written as βtanα. A real annoying issue is when you have multiple functions like tan x√x which can either mean (tan(x))*(√x) or tan(x√x) but it's hopefully clear from context
When I got to college level, the instructor was baffled that this concept was not taught in my secondary school (or I just didn't know it). No problem, I learned it then and there.
The key here is the word "*SCIENTIFIC*". The Windows calculator will give BOTH results, depending on which MODE it is in. In STANDARD mode it adheres to "left-to-right*, but in SCIENTIFIC mode it adheres to *order-of-operations*. *Neither* method is *in-&-of-itself* wrong. It simply depends on what one wants to do. Presh Talwalkar clearly states what the STANDARD mode is typically used for. So any user *MUST* know the rules of & conventions for BOTH *left-to-right* AND *order-of-operations*. AND, if using a calculator, which mode it is set to. (I am still searching for a Help-file for the Windows calculator app...) (PS: My "smart phone" running Android has only the SCIENTIFIC mode, and also NO useful Help-file. But from its 'History' display one can deduce its mode.)
Also, there's nothing wrong with the math in the post. It's clearly, the user expecting 200, and mechanically NOT pressing the equal sign to get 100 and then multiply by 2.
Multiplication is just a special case of addition: repeatedly adding some number, a, some other number, b, amount of times. So, multiplication can just be rewritten in terms of addition. 2 × 5 = 2 + 2 + 2 + 2 + 2. The question then becomes: why should the precedence be what it is? The commutative property of addition being true answers that. 3 + 2 × 5 = 3 + 2 + 2 + 2 + 2 + 2 Since addition is accepted as being commutative, we can rearrange the addends and evaluate some of them. If we simplify the remaining addends into multiplication, we should expect the same result because we only used addition's commutative property: 3 + 2 × 5 = 3 + 2 + 2 + 2 + 2 + 2 = 3 + 4 + 2 + 2 + 2 = 3 + 4 + 2 × 3 Sure enough, the entire expression remains equal to 13 at every step. If a left-to-right evaluation system were to be accepted as true, then either the commutative property of addition would need to be false, multiplication would have to be redefined, or BOTH. More complex equations would even be inconsistent in their rule evaluations with the constant swapping of operator precedence.
In all my years of education and 40+yrs professional experience, that's the first time I've ever heard a justification for the Order of Operations. All I was ever taught was the Order of Operation and, "When in doubt, USE PARENTHESES!". (🤔) Thanks for that, Presh! It was a revelation.
Ummm...yeah, no. Both the commutative law of addition and the distributive law hold under a left to right convention regardless of operation. Commutative law of addition: 20 + 50 × 2 = 70 × 2 = 140 50 + 20 × 2 = 70 × 2 = 140 20 + 50 × 2 = 50 + 20 × 2 To do what you did is 50 + (50 × 2) = 50 + 100 = 150 50 × 2 + 50 = 100 + 50 = 150 50 + (50 × 2) = 50 × 2 + 50 Distributive law: 50 × 3 = 150 50 × (1 + 2) = 150 50 × 1 + (50 × 2) = 150 If you include a nonsensical demonstration about the commutative law and distributive law - before you claim they don't hold under left to right - you need to consider your own biases. Next time be consistent using left to right, including parenthesis to group multiplication if you want it done first and it's not the left most operation. In doing so you'll realize two things: 1) The commutative law of addition and the distributive law hold under a left to right convention 2) You can't prove nor disprove a convention. The primary reason for multiplication over addition is it's usefulness in reading and writing polynomials. Other conventions would be considerably more wonkish to work with polynomials. For example if addition has a higher priority than multiplication, then you'd end up writing parenthesis around every monomial term of the polynomial.
Half of the video's MYD posts are really difficult challenges The other half is people on the internet causing a fuss because they forget order of operations
The ones at 10:18 are legitimately ambiguous, though. I had a pair of students try and consult me on one of those a few weeks ago (I am a HS Math Teacher), and I ended up telling them that there are two different problems it very much LOOKS like, but that it is unclear which one it should actually solve to (and thus which of the two popular answers is “right”).
@MindYourDecisions Thank you, Presh, for continuing to (implicitly) hammer on the need for teachers who UNDERSTAND what they are supposed to be teaching.
Calculator to me: Hallo world! Me to calculator: Sorry! I do not know how not use the calculator corrrectly! Paper to me: I am here too! Me to paper: Sorry! I do not know maths basic laws of operation! Paper and calculator to me: ask AI for help.
multiplication is done before addition because of the meaning multiplication holds. if we were to evaluate an expression without the order of operations let's say 20 + 40 * 3 then what is means is 20 + 40 + 40 + 40 = 140. evaluating 20 + 40 * 3 left to right gives 180. but the correct ans is 140 that's why multiplication is done before addition.
It's not even related to order of operations. When you press =, it calculates whatever was entered. Any subsequent operations you do will use that result. The previous operations that gave that result don't matter anymore. If the screen displays "100" after pressing =, then "100" will be used for the next thing you do (if you enter an operator). Whatever happened previously doesn't matter.
This should be apparent to anyone with basic math literacy as soon as they see the tweet. As soon as the enter button is hit, it calculates based on the order of operations from what you've input. The result, if left in for another calculation, is considered its own, separate part of a new equation. So if you enter it as two different equations, you will get two different answers if you don't do them in an order that would match the order of operations had you input it all at once. The only calculator I've ever seen to actually be incorrect and not have the order of operations correctly programmed is Casio brand calculators, which will sometimes give incorrect answers when using more than one type of operation in the same equation, but that's a topic for another video. Simply going from left to right regardless of the order of operations is simply incorrect. The order of operations exists for a reason. It would be like disregarding cooking times and ingredient amounts in a recipe. Doing it the right way will give you a delicious meal but disregarding the system will result in something very different than what the recipe is meant to yield.
The deeper reason is that there is an order to operations. Adding is just counting by numbers other than just 1, multiplying is adding multiple groups. Exponentiation is another order higher yet, multiple multiples if you will. And we can go on, to tetration etc. We do the highest order first unless the expression has been specifically indicated to be done first (parentheses). That being said, while sometimes you may use an operation to figure out the answer an operation isn't always what is going on. 2(1+2) does not mean 2 multiplied by (1 + 2) which would be taking 2 and adding it to itself a total of three times. 2(1+2) means that you have 2 of the value of (1+2). It describes how many of (1+2) you have and so in this case is acting as an adjective. Technically numbers are probably best considered as determiners, but they act like adjectives much of the time. The order of operations is the second thing you do when solving a problem, the first thing you do is figure out what the problem is actually saying. Calculators aren't good at that.
We need to abolish the ÷ symbol. It's needlessly confusing when we have fraction bars to express exactly where division goes in the equation's order of operations.
6:39 I am not convinced by that as it assumes the conclusion i.e. is circular. Changing the order makes sense only if we take multiplication to be higher precence.
Even without following the mathematical order of operations it should be easy to apply the correct order. 50 + 50 x 2 reads as follows: 50 plus two 50's. That should clarify things a bit. Obviously the answer would be 150.
*Breath *Sigh Bodmas, my guy... Bodmas Edit : Seriously though, as a big fan of math, I'm... a bit disappointed. Edit 2 : And when I say that, I meant the Karen guy.
The issue is not lack of education, or a bug with a calculator - it is simply that Order of Operations is an arbitrary set of rules that are only used to teach people in school, and to put as puzzles for people to discuss and are otherwise not used The fact that people argue over it means it's useless If you want it to be unambiguously interpreted as you wanted - use brackets, like everyone else does If you meant (50 + 50) x 2, then write that and your calculator will show the answer you expected If you meant 50 + (50 x 2), then write that and your calculator will show the answer you expected
The customary order of operations has NOTHING to do with how calculators, spreadsheets and computer programming languages interpret math expressions. They all work a little differently (in the case of RPN, way differently). It's nice when applications follow the customary order of operations, but it is up to the user to know how the application works. As someone who does a LOT of calculations for work...calculations that are written out to be reviewed by others, they MUST be clear. Here are my humble suggestions: 1) Make ample use of brackets and parenthesis. Group operations that have priority. 2) NEVER EVER use obeluses (division signs)...if you have a rational expression, write it out as a fraction. 3) Dividing by fractions/putting fraction in the denominator of a rational expression should be avoided unless you have a damned good reason. (It does come up in algebra and calculus). Mathematic expressions are communication...if you are relying on the reader to strictly interpret your math with PEMDAS/BODMAS, you are, at best, WASTING THEIR TIME, and at worst, CREATING CONFUSION. If people have to ask what you meant, or how you got a particular result, you have expressed yourself poorly. An aside...most of the "math videos" deliberately present expressions that are, to put it nicely, problematic. They have just enough ambiguity to generate lots of clicks and comments. The comments often get very ugly. Another aside, teaching PEMDAS is one of those things I would call "essential, but otherwise unimportant". Teachers are better off teaching students how to express their math ideas clearly.
can someone please excuse my dear aunt Sally and get everyone on board with order of operations already? XD;; seriously though, the explanation of order with properties of addition was really cool!
A large part of the problem is that older calculators would have given you 200 if you entered the calculation as given because they complete each step as you go. When you hit the key for multiplication, older calculators would complete the addition and then apply the multiplication to the result of the addition. The result is that people who are used to only using older, physical calculators likely aren’t even considering that and just see that the result on their phone is different than when they punch it into their calculator manually.
The only possible "excuse" I could think of, why in the 21st century people still don't know the correct order of operation, is that they might be used to very primitive calculators from the electronic "stone age" in their childhood. But on the other hand: every one who has visited school for more than 2 years should know the correct rules.
The problem is that people are told the order of operations but not why it is so. It is not some rule given from above but some to simplify things. It is typical that you for example buy different amounts of different items that each have a different unit price and want to calculate the total. Because multiplication is done before addition one does not need to put every multiplication in brackets. Four function calculators calculate as you go (some say left to right but that incorrect as there is no concept of left or right in the calculator. The entries are separated by time, not by location). On them you must use the memory function. M+ adds the result to the memory. Mr then gives the total.
Actually, if you happen to ever been given an assignment on programming your own calculator in AOS (Algebraic Operating System), you would find that storing the whole expression tree is unnecessary, but would only need to store two variables: one for accumulating addition/subtraction results, and one for memorizing the latest chain of multiplication/division results. For example, if you key in 4 + 4 + 3 + 2 * 2 * 3 = , as you press +, the screen displays the accumulated result(4 -> 8 -> 11), but for the next multiplications it will display the chain of multiplication (2 -> 4 -> 12), then it will be added to the accumulated variable as soon as you press equal or plus/minus.
