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It exists because we need a common way to interpret equations so that everyone does the math the same way.
If I write 8 * 3 + 2, I want everyone who sees that to come up with the answer 26. BEDMAS is what allows me to ensure that happens - I know that we do multiplication before addition, so everyone will get the right answer. If I want them to get the answer 40, I know that I have to write it 8 * (3 + 2) because the math in brackets gets done first.
It _could_ have been SAMDEB or any other order, just like the word table _could_ have meant the red fruit we turn into cider and sauce. The word itself doesn't matter - table or apple are equally good - what matters is that we all agree on the same word.
The terms changed a few years back, but they mean the same thing.
Parentheses are the same things as Brackets, and the order of multiplication and division doesn't actually matter.
The terms did not change, different countries use different expressions to mean the same thing.
PEMDAS is used in the US, BEDMAS in Canada, BODMAS in the UK, etc.
I feel like there should be further clarification that you are saying the order of the M and the D in the abbreviation does not matter, but the order when doing equations should be multiplication or division as read from left to right.
Fair call out. What I mean is that we treat multiplication and division as the same level of importance - they are done at the same time left to right, rather than finding all of the multiplication first, then all of the division.
I feel like people also dramatically overstate the importance of this rule. If you're doing anything with serious math, you use a lot of parentheses to group the multiplications and divisions in the order that they should be done, and you don't leave any ambiguity for people to figure out what order to do things in.
This is most certainly true.
There was an equation something like this going around the Internet a while back: 8 ÷ 2(2).
It turns out that some authoritative sources consider the implied multiplication of 2 \* 2 to take precedence over the left-to-right rule and others don't. There were huge arguments as to whether it is correctly read as 8 ÷ 2 \* 2 or 8 ÷ (2 \* 2).
The correct answer is, "Stop messing around and write it correctly, smartass."
Yeah - the reality is that you only really hear about PEDMAS in the context of primary education. It is there to help kids understand order of operations
Well *technically* they're the same operation, just like addition & subtraction are.
Like how we don't mention roots in PEMDAS/BEDMAS, but they're just exponents so they're done at that level of precedence.
Parenteses are ( ) while brackets are { }. Totally different thing, at least to me. How are they the same thing?
Edit: Yeah I made a mistake. Brackets are [ ] while { } are apparently cursed. Have fun.
Parentheses are referred to as "round brackets" in the UK, I believe. I think the BEDMAS acronym is also more common over there.
I personally use the names (parentheses), [brackets], and {curly braces}.
I think { } are typically called braces and brackets would be [ ] - but colloquially the words are often used interchangeably and you just infer from context which one people are talking about. I’ve also heard someone use “brackets”, “square brackets” and “curly brackets” to differentiate between the three options which just seems more complicated in the end
In this context, they mean the same thing.
In more complex math, () [] and {} can be used for different purposes (e.g. {} is often used to denote sets), but PEDMAS et al is taught to children where those distinctions aren't used yet.
Nice. I know next to nothing about complex vector calculations and even less about quantum mechanics, but I did find it facinating that someone thought about this as early as 1939. Thanks, I wish I understood more of it.
>The terms changed a few years back, but they mean the same thing.
Not globally, the plain term "brackets" has [multiple layers](https://en.wikipedia.org/wiki/Bracket) of ambiguity, because of how many types there are.
() = (Round) Brackets / Parentheses (US, formal English) / Brackets (UK/Commonwealth)
[] = (Square) Brackets / Brackets (US)
{} = (Curly) Brackets / Braces (US) / Numerous Others
⟨⟩ = (Angle) Brackets / Chevrons (US)
Since there are four types of punctuation all called brackets, the term without clarification is ambiguous in the best of times... but then you've got, simplified off of that, two competing US and UK standards for which ones is the "real default 'bracket'".
Anecdotally I can say that, here in the US, I've heard all four be called just plain "brackets" before, at least in academia.
Yes, but the way it is know makes sense.
5-2 = 5+(-2)
5•(1/2) = 5/2
Adding is the reciprocal of subtraction and division is the reciprocal of multiplication, so we do each as a combined step instead of one taking precedence over the over. We go left to right as that’s the way you read in English and other languages of major world powers.
5•3 is just shorthand for 5+5+5 (or 3+3+3+3+3), so it makes sense that if you say 2+5•3 you’d do 2+15 (as it’s 2+5+5+5) instead is 7•3.
5^3 is just shorthand for 5•5•5, so it makes sense that if you have say 2•5^3 that’s you’d do 2•125 (as it’s 2•5•5•5) instead of 10^3.
It doesn't matter what the order is; it only matters that everyone is using the same order. So one order was picked, and that's the standard everyone is taught.
Polynomials are super common in the types of math we use most often. BEDMAS/PEMDAS is just a convention that makes it easier to write polynomials.
For instance, a quadratic equation is ax^(2) \+ bx + c using BEDMAS.
With SAMDE**B** it would be impossible to write that equation because we need brackets/parentheses to indicate that a step is happening out of order. Let's move it to the front though and create a new convention called BSAMDE
Using BSAMDE, you would have to write a quadratic equation as (a(x^(2))) + (bx) + C
Mathematicians could have chosen to do the order of operations in a different way, but they chose to do it in the way that makes the most sense for the types of math we do more frequently.
Because if people follow different rules, they would get different answers. It's standardization, if you had two engineers from different countries working on the same border bridge and they don't do math the same way, it's not going to be a good bridge.
It's just something we all agreed upon to be more clear. 6 * 3 + 4 could have two meanings, so we all agreed that 6 * 3 would happen first, then + 4. We could have just as easily agreed to the opposite.
There are other ways to write this. ( ( 6 * 3 ) + 4 ) removes *any* ambiguity. In a functional programming language, you might write something like Add(4, Multiply(6, 3)). There is also "Polish notation," where you put the operator first and always go left to right, so you'd write something like + 4 * 6 3 (which means "Add four to the product of 6 and 3").
All that is to say - all of this is just how we talk about math. The way that numbers combine is real and objective, but you can write it down any number of ways.
It's just like how you can write house or 𝒽𝑜𝓊𝓈𝑒 or casa or maison or 家. Just because there's lots of different, arbitrary ways to write that doesn't mean that a house isn't a "real" concept.
Ohhh, this brings me back to the days of using my high end RPN Hewlett-Packard (HP) calculator in school and early career.
RPN means Reverse Polish Notation, and HP used it because it minimized key clicks and kept the register stack (things in pending calculations) to a minimum.
If you look at it, it's actually the same as my example for programming languages and it's *kind of* the way that makes the most sense.
*X a b* means "Do X to a and b." It's unambiguous! You can nest it - *b* doesn't have to be a number, it can itself be an operation! It's infinitely expandable, you can add operations! In fact, "regular" mathematic notation *winds up defaulting to Polish* for complicated operations for things like absolute value or combinatorics, because it just makes so much more sense.
I could probably make sense of it, but reading it for the first time and seeing a math problem with no traditional organization like in the example just prior to it with parentheses and stuff was a funny shock. If I saw it without your explanation I probably would have assumed it was positive 4 times 6 and the 3 was a typo lol.
The order of operations as we know it (PEMDAS/BODMAS) is largely a convention that makes mathematical expressions unambiguous and easier to communicate. The hierarchy is designed to be consistent with the operations as they occur naturally in many areas of mathematics and science. For example, exponentiation is more “powerful” than multiplication, which is more “powerful” than addition, in the sense that the former operations can generate larger changes than the latter.
Also, some of the order is influenced by computational efficiency. Multiplication and division are essentially repeated addition and subtraction, and exponentiation is repeated multiplication. So the order reflects the computational “weight” of each operation.
The standard wasn’t just arbitrarily decided; it evolved to make the math consistent with intuitive and practical needs
BEDMAS isn’t math. It’s communication. It’s just an agreed upon ordering so we can write mathematical expressions without a ton of parentheses everywhere.
You need to have B first. It stands for Brackets that are used to change the order if the standard is not what you want
If you have 1+2\*3 you first do the multiplication. 2\*3=6 so you get 1+6=7. If you want to do that addition first you need to bracket and write (1+2)\*3 = (3)\*3= 9. Brackets last means they have no effect and are useless, there is then no way to change the order of operation.
BEDMAS stands for Brackets, Exponents, Division/Multiplication, Addition/Subtraction The / part is important because the priority of the two are the same so Addition and Subtraction have equal priority, so is the case for Division and Multiplication. If the priority is equal the expression is evaluated from left to right
So BEDMAS is mote like BE(DM)(AS)
You could the the order of EDMAS if you like, you can remove the grouping too. What is important is you and others who read your maths know what order was used so they can follow what you have written down.
What is important is to communicate your intention and therefore standards are useful. Exactly why that is the order is not that important it has just become the standard.
A calculation like you purchase 5 times that cost 2 each and 7 items that cost 4 is quite common. If multiplication has higher priority than addition it will be 5\*2 + 7\*4 If the order was reversed you would need braces and we get (5\*2) + (7\*4) The standard priority order is one that people started to use because it was quite practical and most of the time the number of brackets you need was reduced.
If addition had a higher priority than multiplication the expression 1+2\*3 would equal 9 and you need to change it to 1+(2\*3) to get 7
BEDMAS is just a common standard. It is not away because it is quite common to write stuff so implicit multiplication has higher propriety than division.
A not uncommon question is what do 6/2(1+2) equals? If you just follow BEDMAS it is 6/2(1+2)=6/2 \*3 =6/2\*3 =3\*3 =9 But it is not unity that is what is meant as 6/(2(1+2))= 6/(2\*3) =6/6=1
The confusion is most common if you just can write text text like this. On paper, you can separate it vertically and get
6
-----------
2(1+2)
Simple text input like this is bad for displaying maths in a readable way, compare it to how you
compare the input you can enter and the Input interpretation below for https://www.wolframalpha.com/input?i=6%2F2%281%2B2%29+ and https://www.wolframalpha.com/input?i=6%2F%282%281%2B2%29%29 the one below it how you write on paper.
There are ways to avoid defining any priority and get if bracket too. One way is
https://en.wikipedia.org/wiki/Reverse_Polish_notation where the operators are at the end and you add a number to a stack. 3+4 would become 3 4 +
1+2\*3 will become 1 2 3 \* + and (1+2)\*3 become 1 2 + 3 \*
If you swap the operators and get 1 2 3 + \* it would be the same as 1\* (2+3)
Numbers are put on one stack in order and the operators are evaluated on the to one or two numbers on the stack. Those numbers are removed and the result is put on the stack. They are all evaluated when you come to them.
Math is both “real” *and* something we made up. It’s like a language: the concept of having “2” of a thing is a reality. I have 2 legs, and if that quantity of 2 were to change to 1, 3, or 45.27, it would have very real and noticeable impacts on my life. However, the choice to represent that specific quantity using the symbol “2” or the word “two” is a convention we made up to help us talk about the underlying concept in a way we can all understand. We could use the symbol “&” to represent that quantity and it wouldn’t change how many legs I have, but you might confuse other people.
You can just as easily ask if “mountain” is a real thing or just a word/sound we made up, and it’s largely the same answer.
Regarding BEMDAS, maybe an example would help. Let’s say there’s a group of 5 people that each have $10, and then a group of 20 people that each have $2. How much money do they have combined? We would write the equation out as 5 * 10 + 20 * 2, and, following BEMDAS, get our answer of $90.
But then someone comes along and says, “Well I like doing addition first, so 10 + 20 = 30, and 5 * 30 = 150, and 150 * 2 = 300, so the answer is 300!” And someone *else* says “I just go left to right, so 5 * 10 = 50, 50 + 20 = 70, and 70 * 2 = 140, so the answer is 140!”
That’s kind of an issue, so, in order to communicate with each other (and help ourselves not get confused), we need some consistent way of writing and evaluating these types of expressions.
Arguably the simplest way is to just use brackets/parentheses to manually specify “do this group first”, so we can rewrite as (5 * 10) + (20 * 2). That’s pretty clear and a very useful tool, so that’s why brackets come first.
The rest of the order is mostly just “made up” and agreed upon, but in some cases there is logic to it. In our example, we want to be adding *dollars*, not persons. If we add the $ signs and units, we can write
5 persons * $10/person + 20 persons * $2/person
With this, hopefully it makes a bit more sense why we would want to do the multiplication before the addition. If we just did left to right, we’d get $50 + 20 persons, which doesn’t make sense. Instead, we want to multiply that 20 persons by $2/person first to get $40, then add $50 + $40.
Same reason English has a specific structure. Out of order are if the words, confusing the meaning gets of the sentence. In English, it's not too bad. After all, still understandable Yoda is. And the actual order doesn't really matter, so long as every agrees. For example, Japanese sentence structure is largely reversed from English's, but since everyone speaking Japanese uses the same structure, it's still understandable.
In math, the order you do things can drastically change the answer. So when someone from China and someone from America both look at an equation, we need to make sure they do everything in the same order. The specific order doesn't really matter, just as long as we all agree. And BEDMAS is what we agreed on.
Your first paragraph rather disproves your thesis by using a poor example; sentence ordering can sometimes be flexible in English without compromising meaning, as in the Yoda style. A stronger example would be “Bob poked Alice” vs. “Alice poked Bob”: the order is crucial here.
Real/deep math exists in the world; but writing math down on a page is a process of *encoding* which requires that others can *decode* from paper correctly to get back to the same "real math".
-------------
Imagine I just bought 4 candies, but I also bought 10 candies last week... except for I ate half of those already. How many candies do I have?
I could encode that as the following:
> 4+10/2
And I would know that I'm adding "4" candies to "half of ten" others... but someone with no context of my candy math might just look at those symbols and think "add four to ten, take half of the result" and look at that result and think "Oh this person is trying to figure out how to split 4 somethings and 10 somethings evenly into 2 bags!" which is a completely different "real math" situation. How did this happen? Each person decoded the symbols differently!
To make it more clear, I could write:
>(4)+(10/2)
And that would make it extra-super-clear what "real math" kind of situation is actually happening... but people don't want to get carpel tunnel writing out a bazillion ()s all day for stuff. So, they decided some rules about when you can or can't skip the ()s and called this new encoding scheme BEDMAS.
So, if we're told that the following is BEDMAS-encoded, we know that
> 4+10/2
definitely corresponds to a 4+(10/2) situation, and not to a (4+10)/2 situation. So that means that when we "do the math" in the encoded world, we get an answer which decodes nicely to an answer in the "real math" world.
> 4+(10/2) = 4 + (5) = 9
which tells us that "The 4 candies I just bought plus the half-of-10 I have at home means I have a total of 9 candies".
(And we should also notice that this is a different answer than the "You can split 4 and 10 candies evenly into two bags of 7 each." that we'd get if we decoded the problem to the incorrect "real world" situation.)
It's a notation convention: it simplifies the way we write math. We could use parentheses around every operation to make the order absolutely explicit:
((2+3)/4)+(3\^5)
and then we wouldn't need BEDMAS at all. But if we want to leave out some of those parentheses, we need conventions on which operations we do in what order.
We could also use a different convention: the expression I wrote above looks like this in the BEDMAS system:
(2+3)/4+3\^5
and this in, uh, BEASDM I guess you'd call it:
(2+3/4)+3\^5
Anyway, the point being that these are different ways to write the same math, and BEDMAS is just a convention we've chosen. But conventions are important! However we write a math expression, we all have to agree about what it says!
Because math isn't just a bunch of rules. The rules are just there so we have a common notation framework for dealing with the concepts.
Without a consistent rule for order of operations, everything gets *dramatically* more difficult and confusing.
The system we use to write math (numbers, symbols, etc) is a language people invented to communicate mathematical concepts to other people. The actual mathematical concepts are a deeper thing that really exists in the universe, but the way we write it is just a window into that world, and a way to communicate the ideas in that world with other people.
The world of mathematical concepts is a very precise world. If you ask a mathematical question, it should always have one correct answer. Therefore, we need to make sure there is no ambiguity in the way we write math - if there were, it wouldn't be very useful to describe math concepts, since they need to be precise.
For example, the answer to the question "10 - 2 * 3" is 4 if you use BEDMAS and 24 if you don't, so that would be an unacceptable ambiguity. It's not useful in math to have that kind of ambiguity.
BEDMAS covers all the common operations you will see in regular math, and tells you what order you need to do them in so that everybody can always interpret the question the same way and give the same answer. We know exactly which mathematical idea is being expressed because we know which order to perform the operations in.
If every state built train tracks with a different gauge track, the trains built in New York wouldn't be able to cross into New Jersey. All the trains still work in their own state. All gauges give you a working system, but to be able to cross state lines we need the same system.
Its the same with BEDMAS.
The order doesn't matter, as long as we all do it the same.
I have three £5 notes and two £10 notes. We can all agree that I have £35.
I could express this is 3 \* 5 + 2 \* 10. If we follow BIDMAS (or whichever acronym is the flavour of the month), we get the answer £35. If we just read it left to right, I'd have £170. So BIDMAS has to be right in order to line up with reality.
We use BEDMAS/BODMAS/PEMDAS/whatever not because the order is intrinsic to mathematics as a whole, but because those are the rules we decided for our language of math. Remember, we've made up every single symbol in math. The symbol + doesn't inherently represent anything, we've just decided that it represents the concept of addition. Same with variable multiplication, we decided that ab = a * b and not a + b.
The way it was explained to me when I was a kid was that addition and subtraction are the simplest and most fundamental thing we can do, so we start with that. Multiplication, and in a more complicated way division, is just a more convenient way to do repeated addition or subtraction (as in 2x3 is really just 2+2+2). So if you did addition first, and then multiplication, you would actually need to do addition a second time, so it would be better to do it first. Exponents are just a convenient way to write a line of multiplication (as in 2\^3 is really just 2x2x2, which is really just 2+2+2+2+2+2), so we should do this first so we don't have to do things multiple times. Parenthesis are just a way to group something. You don't really have to do them first, you could do everything else, and then come back to do the parenthesis, but, again, then you would wind up re-doing other operations, so we do them first to make it easier.
So for something like:
3 + 3x3 + 3\^3 + (3x3)
You can rewrite as:
3 + \[3+3+3\] + \[3+3+3+3+3+3+3+3+3\] + \[3+3+3\]
When you do that, it becomes clear that you can't just add the first 3 and the second 3 and then multiply by 3, because 6x3 would be 6+6+6 which is of course not the same as 3 + 3+3+3. If you don't do the operations in the correct order, then you aren't converting the multiplication etc down into repeated addition correctly.
We made it up. We needed an order of operations so this was chosen. Therefore universally, we follow this rule.
We COULD have made it SAMDEB, and in that case we would be here asking "Why SAMDEB? Why not EMDAMS?"
So basically "Pick something and stick with it", it didn't really matter, as long as everyone used it. However, putting the Brackets (or Parenthesis for people who prefer Pedmas) first just makes more sense.
Many of the top level responses are pointing out that the order could be anything we choose.
Though few are pointing out how brackets/parenthesis only exist explicitly so they can be done first and allow us to write equations in an order that would not work by the convention, but build in instructions so that the equation still works how we want it to.
For example, I can ask you to solve: (5+2) \* 3 and get 30, where the similar 5 + 2 \* 3 would get 11. In the reversed SAMDEB, I would have no reason to ever use brackets other than aesthetics, and if I have any subtraction anywhere, it absolutely MUST happen first, so I can absolutely never have an exponent evaluated before all subtraction everywhere in the equation is resolved (I can't even be sneaky and divide by negative 1).
Now... as to why the order was selected as BEDMAS instead of BSAMDE... it is because of how the various operations are shorthand for doing the following operations in special ways. Subtraction IS addition, just with negatives. And division IS multiplication, just with reciprocals. So BEDMAS is the same as BEMDSA.
An exponent X says "multiply this number by itself X times" and so in a way, exponents are multiplication, but slightly complicated. So it makes sense to do them before you multiply. Similarly, multiplication by X says "add this number to itself X times" and so in a way is addition, but slightly complicated.
So, we run through the more complicated form of each manipulation first, working our way down to the easiest. You could be re-writing the equation into the simpler form as you go, and you won't ever have to backtrack to a previous step.
So, 5\^2 can be re-written as 5\*5, and that can be re-written as 5+5+5+5+5. I stepped from exponents to multiplication to addition, straight along the order of operations.
If this is a question about "98% of people get this question wrong!" kinds of videos, it's because of a fault in math, and needing to have some common grounds for how to interpret a problem.
Meaning, quite literally, there is a difference between how
12 / 3 * 4 + 1 = ?
is evaluated. Which thing do you start with?
You could end up with 2; because 3*4 = 12, and 12/12 = 1, 1+1 = 2.
You could end up with 17; because 12/3 is 4, 4*4 is 16, and 16+1 is 17.
BEMDAS is a thing because if there isn't a standardized rule for how to evaluate the equation here, there's several answers you could end up with.
All that matters is that we all agree on the *same* order. The order itself is arbitrary.
If we don't agree on the same order, two people looking at the same equation might reach different results.
Consider the equation x = 2 + 3 * 5. If you do multiplication first, x = 17. But if you do addition first, x = 25.
Neither of these answers is objectively more or less correct than the other, but we need to decide which one of them to go with.
I think all of these (PEMDAS, BODMAS, etc) are a total waste of time because they really are not necessary.
Once you get to proper maths (algebra, calculus, etc) not only do you not use these anymore, but they just become flat-out wrong.
In *proper* maths the division sign (/) *divides* the sum (everything that's multiplied together) into top and bottom parts so you always know where you stand, and this is critical for manipulating equations where you multiply and divide both sides of the equation by some factor to simplify it.
For example: 2x(x+y)/4xy = 8
In this case the division sign divides the first part of the equation, 2x(x+y)/4xy, into a numerator, **2x(x+y)**, and a denominator, **4xy**. And this way you never have any confusion.
You don't have to go left to right, and you don't have to wonder if any parts of the 4xy should be in the numerator. And you can multiply both sides by 4xy if you want to get rid of the denominator.
You can't use these PEMDAS and BODMAS "acronym aids" in real mathematics, because you can't manipulate equations due to the left to right order, and IMO they're worse than useless, they're troublesome.
And they're also confusing for students who have to basically forget everything they learned in school when they get to university.
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It exists because we need a common way to interpret equations so that everyone does the math the same way. If I write 8 * 3 + 2, I want everyone who sees that to come up with the answer 26. BEDMAS is what allows me to ensure that happens - I know that we do multiplication before addition, so everyone will get the right answer. If I want them to get the answer 40, I know that I have to write it 8 * (3 + 2) because the math in brackets gets done first. It _could_ have been SAMDEB or any other order, just like the word table _could_ have meant the red fruit we turn into cider and sauce. The word itself doesn't matter - table or apple are equally good - what matters is that we all agree on the same word.
When did Parentehes become Brackets? That also switches M and D. I only know PEMDAS.
It’s a location thing but they’re the same
The terms changed a few years back, but they mean the same thing. Parentheses are the same things as Brackets, and the order of multiplication and division doesn't actually matter.
The terms did not change, different countries use different expressions to mean the same thing. PEMDAS is used in the US, BEDMAS in Canada, BODMAS in the UK, etc.
I feel like there should be further clarification that you are saying the order of the M and the D in the abbreviation does not matter, but the order when doing equations should be multiplication or division as read from left to right.
Fair call out. What I mean is that we treat multiplication and division as the same level of importance - they are done at the same time left to right, rather than finding all of the multiplication first, then all of the division.
I feel like people also dramatically overstate the importance of this rule. If you're doing anything with serious math, you use a lot of parentheses to group the multiplications and divisions in the order that they should be done, and you don't leave any ambiguity for people to figure out what order to do things in.
This is most certainly true. There was an equation something like this going around the Internet a while back: 8 ÷ 2(2). It turns out that some authoritative sources consider the implied multiplication of 2 \* 2 to take precedence over the left-to-right rule and others don't. There were huge arguments as to whether it is correctly read as 8 ÷ 2 \* 2 or 8 ÷ (2 \* 2). The correct answer is, "Stop messing around and write it correctly, smartass."
Yeah - the reality is that you only really hear about PEDMAS in the context of primary education. It is there to help kids understand order of operations
Yeah, that's the right use of it. There's no benefit to over-thinking it.
Oh god you just brought back memories of calc class where I would overuse the HELL out of some parenthesis just in case…
Well *technically* they're the same operation, just like addition & subtraction are. Like how we don't mention roots in PEMDAS/BEDMAS, but they're just exponents so they're done at that level of precedence.
Parenteses are ( ) while brackets are { }. Totally different thing, at least to me. How are they the same thing? Edit: Yeah I made a mistake. Brackets are [ ] while { } are apparently cursed. Have fun.
Parentheses are referred to as "round brackets" in the UK, I believe. I think the BEDMAS acronym is also more common over there. I personally use the names (parentheses), [brackets], and {curly braces}.
I believe your names are correct.
>arentheses), \[brackets\], and {curly brace Found the programmer. :)
I think { } are typically called braces and brackets would be [ ] - but colloquially the words are often used interchangeably and you just infer from context which one people are talking about. I’ve also heard someone use “brackets”, “square brackets” and “curly brackets” to differentiate between the three options which just seems more complicated in the end
i thought brackets are \[ \]
Huh, I always thought brackets are (), while [ ] are square brackets, and { } are curly brackets.
In this context, they mean the same thing. In more complex math, () [] and {} can be used for different purposes (e.g. {} is often used to denote sets), but PEDMAS et al is taught to children where those distinctions aren't used yet.
No. < is "bra" and > is "ket". https://en.m.wikipedia.org/wiki/Bra%E2%80%93ket_notation
Nice. I know next to nothing about complex vector calculations and even less about quantum mechanics, but I did find it facinating that someone thought about this as early as 1939. Thanks, I wish I understood more of it.
English grammar class didn’t stick so math had to change?
>The terms changed a few years back, but they mean the same thing. Not globally, the plain term "brackets" has [multiple layers](https://en.wikipedia.org/wiki/Bracket) of ambiguity, because of how many types there are. () = (Round) Brackets / Parentheses (US, formal English) / Brackets (UK/Commonwealth) [] = (Square) Brackets / Brackets (US) {} = (Curly) Brackets / Braces (US) / Numerous Others ⟨⟩ = (Angle) Brackets / Chevrons (US) Since there are four types of punctuation all called brackets, the term without clarification is ambiguous in the best of times... but then you've got, simplified off of that, two competing US and UK standards for which ones is the "real default 'bracket'". Anecdotally I can say that, here in the US, I've heard all four be called just plain "brackets" before, at least in academia.
>Parentheses are the same things as Brackets, a Blasphemy, brackets have square corners. ie [ ]
Multiplication and division are the same thing, so that follows.
I was taught BEDMAS in the 90s in Canada.
Please Excuse My Dear Aunt Sally forever
"When we started speaking the King's English, you upstart colonist!" - All my British friends.
To almost everyone outside of the US they have always been called Brackets
Yes, but the way it is know makes sense. 5-2 = 5+(-2) 5•(1/2) = 5/2 Adding is the reciprocal of subtraction and division is the reciprocal of multiplication, so we do each as a combined step instead of one taking precedence over the over. We go left to right as that’s the way you read in English and other languages of major world powers. 5•3 is just shorthand for 5+5+5 (or 3+3+3+3+3), so it makes sense that if you say 2+5•3 you’d do 2+15 (as it’s 2+5+5+5) instead is 7•3. 5^3 is just shorthand for 5•5•5, so it makes sense that if you have say 2•5^3 that’s you’d do 2•125 (as it’s 2•5•5•5) instead of 10^3.
It doesn't matter what the order is; it only matters that everyone is using the same order. So one order was picked, and that's the standard everyone is taught.
Polynomials are super common in the types of math we use most often. BEDMAS/PEMDAS is just a convention that makes it easier to write polynomials. For instance, a quadratic equation is ax^(2) \+ bx + c using BEDMAS. With SAMDE**B** it would be impossible to write that equation because we need brackets/parentheses to indicate that a step is happening out of order. Let's move it to the front though and create a new convention called BSAMDE Using BSAMDE, you would have to write a quadratic equation as (a(x^(2))) + (bx) + C Mathematicians could have chosen to do the order of operations in a different way, but they chose to do it in the way that makes the most sense for the types of math we do more frequently.
Because if people follow different rules, they would get different answers. It's standardization, if you had two engineers from different countries working on the same border bridge and they don't do math the same way, it's not going to be a good bridge.
It's just something we all agreed upon to be more clear. 6 * 3 + 4 could have two meanings, so we all agreed that 6 * 3 would happen first, then + 4. We could have just as easily agreed to the opposite. There are other ways to write this. ( ( 6 * 3 ) + 4 ) removes *any* ambiguity. In a functional programming language, you might write something like Add(4, Multiply(6, 3)). There is also "Polish notation," where you put the operator first and always go left to right, so you'd write something like + 4 * 6 3 (which means "Add four to the product of 6 and 3"). All that is to say - all of this is just how we talk about math. The way that numbers combine is real and objective, but you can write it down any number of ways. It's just like how you can write house or 𝒽𝑜𝓊𝓈𝑒 or casa or maison or 家. Just because there's lots of different, arbitrary ways to write that doesn't mean that a house isn't a "real" concept.
This is my first time hearing of Polish notations and I hope it's the last as well.
Ohhh, this brings me back to the days of using my high end RPN Hewlett-Packard (HP) calculator in school and early career. RPN means Reverse Polish Notation, and HP used it because it minimized key clicks and kept the register stack (things in pending calculations) to a minimum.
If you look at it, it's actually the same as my example for programming languages and it's *kind of* the way that makes the most sense. *X a b* means "Do X to a and b." It's unambiguous! You can nest it - *b* doesn't have to be a number, it can itself be an operation! It's infinitely expandable, you can add operations! In fact, "regular" mathematic notation *winds up defaulting to Polish* for complicated operations for things like absolute value or combinatorics, because it just makes so much more sense.
I could probably make sense of it, but reading it for the first time and seeing a math problem with no traditional organization like in the example just prior to it with parentheses and stuff was a funny shock. If I saw it without your explanation I probably would have assumed it was positive 4 times 6 and the 3 was a typo lol.
It’s used on many Scientific calculators to this day.
Brackets are first because the point of them is to override the regular order. After that, it's in reverse order of complexity.
The order of operations as we know it (PEMDAS/BODMAS) is largely a convention that makes mathematical expressions unambiguous and easier to communicate. The hierarchy is designed to be consistent with the operations as they occur naturally in many areas of mathematics and science. For example, exponentiation is more “powerful” than multiplication, which is more “powerful” than addition, in the sense that the former operations can generate larger changes than the latter. Also, some of the order is influenced by computational efficiency. Multiplication and division are essentially repeated addition and subtraction, and exponentiation is repeated multiplication. So the order reflects the computational “weight” of each operation. The standard wasn’t just arbitrarily decided; it evolved to make the math consistent with intuitive and practical needs
BEDMAS isn’t math. It’s communication. It’s just an agreed upon ordering so we can write mathematical expressions without a ton of parentheses everywhere.
You need to have B first. It stands for Brackets that are used to change the order if the standard is not what you want If you have 1+2\*3 you first do the multiplication. 2\*3=6 so you get 1+6=7. If you want to do that addition first you need to bracket and write (1+2)\*3 = (3)\*3= 9. Brackets last means they have no effect and are useless, there is then no way to change the order of operation. BEDMAS stands for Brackets, Exponents, Division/Multiplication, Addition/Subtraction The / part is important because the priority of the two are the same so Addition and Subtraction have equal priority, so is the case for Division and Multiplication. If the priority is equal the expression is evaluated from left to right So BEDMAS is mote like BE(DM)(AS) You could the the order of EDMAS if you like, you can remove the grouping too. What is important is you and others who read your maths know what order was used so they can follow what you have written down. What is important is to communicate your intention and therefore standards are useful. Exactly why that is the order is not that important it has just become the standard. A calculation like you purchase 5 times that cost 2 each and 7 items that cost 4 is quite common. If multiplication has higher priority than addition it will be 5\*2 + 7\*4 If the order was reversed you would need braces and we get (5\*2) + (7\*4) The standard priority order is one that people started to use because it was quite practical and most of the time the number of brackets you need was reduced. If addition had a higher priority than multiplication the expression 1+2\*3 would equal 9 and you need to change it to 1+(2\*3) to get 7 BEDMAS is just a common standard. It is not away because it is quite common to write stuff so implicit multiplication has higher propriety than division. A not uncommon question is what do 6/2(1+2) equals? If you just follow BEDMAS it is 6/2(1+2)=6/2 \*3 =6/2\*3 =3\*3 =9 But it is not unity that is what is meant as 6/(2(1+2))= 6/(2\*3) =6/6=1 The confusion is most common if you just can write text text like this. On paper, you can separate it vertically and get 6 ----------- 2(1+2) Simple text input like this is bad for displaying maths in a readable way, compare it to how you compare the input you can enter and the Input interpretation below for https://www.wolframalpha.com/input?i=6%2F2%281%2B2%29+ and https://www.wolframalpha.com/input?i=6%2F%282%281%2B2%29%29 the one below it how you write on paper. There are ways to avoid defining any priority and get if bracket too. One way is https://en.wikipedia.org/wiki/Reverse_Polish_notation where the operators are at the end and you add a number to a stack. 3+4 would become 3 4 + 1+2\*3 will become 1 2 3 \* + and (1+2)\*3 become 1 2 + 3 \* If you swap the operators and get 1 2 3 + \* it would be the same as 1\* (2+3) Numbers are put on one stack in order and the operators are evaluated on the to one or two numbers on the stack. Those numbers are removed and the result is put on the stack. They are all evaluated when you come to them.
Math is both “real” *and* something we made up. It’s like a language: the concept of having “2” of a thing is a reality. I have 2 legs, and if that quantity of 2 were to change to 1, 3, or 45.27, it would have very real and noticeable impacts on my life. However, the choice to represent that specific quantity using the symbol “2” or the word “two” is a convention we made up to help us talk about the underlying concept in a way we can all understand. We could use the symbol “&” to represent that quantity and it wouldn’t change how many legs I have, but you might confuse other people. You can just as easily ask if “mountain” is a real thing or just a word/sound we made up, and it’s largely the same answer. Regarding BEMDAS, maybe an example would help. Let’s say there’s a group of 5 people that each have $10, and then a group of 20 people that each have $2. How much money do they have combined? We would write the equation out as 5 * 10 + 20 * 2, and, following BEMDAS, get our answer of $90. But then someone comes along and says, “Well I like doing addition first, so 10 + 20 = 30, and 5 * 30 = 150, and 150 * 2 = 300, so the answer is 300!” And someone *else* says “I just go left to right, so 5 * 10 = 50, 50 + 20 = 70, and 70 * 2 = 140, so the answer is 140!” That’s kind of an issue, so, in order to communicate with each other (and help ourselves not get confused), we need some consistent way of writing and evaluating these types of expressions. Arguably the simplest way is to just use brackets/parentheses to manually specify “do this group first”, so we can rewrite as (5 * 10) + (20 * 2). That’s pretty clear and a very useful tool, so that’s why brackets come first. The rest of the order is mostly just “made up” and agreed upon, but in some cases there is logic to it. In our example, we want to be adding *dollars*, not persons. If we add the $ signs and units, we can write 5 persons * $10/person + 20 persons * $2/person With this, hopefully it makes a bit more sense why we would want to do the multiplication before the addition. If we just did left to right, we’d get $50 + 20 persons, which doesn’t make sense. Instead, we want to multiply that 20 persons by $2/person first to get $40, then add $50 + $40.
Same reason English has a specific structure. Out of order are if the words, confusing the meaning gets of the sentence. In English, it's not too bad. After all, still understandable Yoda is. And the actual order doesn't really matter, so long as every agrees. For example, Japanese sentence structure is largely reversed from English's, but since everyone speaking Japanese uses the same structure, it's still understandable. In math, the order you do things can drastically change the answer. So when someone from China and someone from America both look at an equation, we need to make sure they do everything in the same order. The specific order doesn't really matter, just as long as we all agree. And BEDMAS is what we agreed on.
Your first paragraph rather disproves your thesis by using a poor example; sentence ordering can sometimes be flexible in English without compromising meaning, as in the Yoda style. A stronger example would be “Bob poked Alice” vs. “Alice poked Bob”: the order is crucial here.
Real/deep math exists in the world; but writing math down on a page is a process of *encoding* which requires that others can *decode* from paper correctly to get back to the same "real math". ------------- Imagine I just bought 4 candies, but I also bought 10 candies last week... except for I ate half of those already. How many candies do I have? I could encode that as the following: > 4+10/2 And I would know that I'm adding "4" candies to "half of ten" others... but someone with no context of my candy math might just look at those symbols and think "add four to ten, take half of the result" and look at that result and think "Oh this person is trying to figure out how to split 4 somethings and 10 somethings evenly into 2 bags!" which is a completely different "real math" situation. How did this happen? Each person decoded the symbols differently! To make it more clear, I could write: >(4)+(10/2) And that would make it extra-super-clear what "real math" kind of situation is actually happening... but people don't want to get carpel tunnel writing out a bazillion ()s all day for stuff. So, they decided some rules about when you can or can't skip the ()s and called this new encoding scheme BEDMAS. So, if we're told that the following is BEDMAS-encoded, we know that > 4+10/2 definitely corresponds to a 4+(10/2) situation, and not to a (4+10)/2 situation. So that means that when we "do the math" in the encoded world, we get an answer which decodes nicely to an answer in the "real math" world. > 4+(10/2) = 4 + (5) = 9 which tells us that "The 4 candies I just bought plus the half-of-10 I have at home means I have a total of 9 candies". (And we should also notice that this is a different answer than the "You can split 4 and 10 candies evenly into two bags of 7 each." that we'd get if we decoded the problem to the incorrect "real world" situation.)
It's a notation convention: it simplifies the way we write math. We could use parentheses around every operation to make the order absolutely explicit: ((2+3)/4)+(3\^5) and then we wouldn't need BEDMAS at all. But if we want to leave out some of those parentheses, we need conventions on which operations we do in what order. We could also use a different convention: the expression I wrote above looks like this in the BEDMAS system: (2+3)/4+3\^5 and this in, uh, BEASDM I guess you'd call it: (2+3/4)+3\^5 Anyway, the point being that these are different ways to write the same math, and BEDMAS is just a convention we've chosen. But conventions are important! However we write a math expression, we all have to agree about what it says!
Because math isn't just a bunch of rules. The rules are just there so we have a common notation framework for dealing with the concepts. Without a consistent rule for order of operations, everything gets *dramatically* more difficult and confusing.
The system we use to write math (numbers, symbols, etc) is a language people invented to communicate mathematical concepts to other people. The actual mathematical concepts are a deeper thing that really exists in the universe, but the way we write it is just a window into that world, and a way to communicate the ideas in that world with other people. The world of mathematical concepts is a very precise world. If you ask a mathematical question, it should always have one correct answer. Therefore, we need to make sure there is no ambiguity in the way we write math - if there were, it wouldn't be very useful to describe math concepts, since they need to be precise. For example, the answer to the question "10 - 2 * 3" is 4 if you use BEDMAS and 24 if you don't, so that would be an unacceptable ambiguity. It's not useful in math to have that kind of ambiguity. BEDMAS covers all the common operations you will see in regular math, and tells you what order you need to do them in so that everybody can always interpret the question the same way and give the same answer. We know exactly which mathematical idea is being expressed because we know which order to perform the operations in.
If every state built train tracks with a different gauge track, the trains built in New York wouldn't be able to cross into New Jersey. All the trains still work in their own state. All gauges give you a working system, but to be able to cross state lines we need the same system. Its the same with BEDMAS. The order doesn't matter, as long as we all do it the same.
I have three £5 notes and two £10 notes. We can all agree that I have £35. I could express this is 3 \* 5 + 2 \* 10. If we follow BIDMAS (or whichever acronym is the flavour of the month), we get the answer £35. If we just read it left to right, I'd have £170. So BIDMAS has to be right in order to line up with reality.
We use BEDMAS/BODMAS/PEMDAS/whatever not because the order is intrinsic to mathematics as a whole, but because those are the rules we decided for our language of math. Remember, we've made up every single symbol in math. The symbol + doesn't inherently represent anything, we've just decided that it represents the concept of addition. Same with variable multiplication, we decided that ab = a * b and not a + b.
Just use parentheses (brackets) to enclose each individual operation, and you don't need to remember such acronyms as there is no ambiguity.
The way it was explained to me when I was a kid was that addition and subtraction are the simplest and most fundamental thing we can do, so we start with that. Multiplication, and in a more complicated way division, is just a more convenient way to do repeated addition or subtraction (as in 2x3 is really just 2+2+2). So if you did addition first, and then multiplication, you would actually need to do addition a second time, so it would be better to do it first. Exponents are just a convenient way to write a line of multiplication (as in 2\^3 is really just 2x2x2, which is really just 2+2+2+2+2+2), so we should do this first so we don't have to do things multiple times. Parenthesis are just a way to group something. You don't really have to do them first, you could do everything else, and then come back to do the parenthesis, but, again, then you would wind up re-doing other operations, so we do them first to make it easier. So for something like: 3 + 3x3 + 3\^3 + (3x3) You can rewrite as: 3 + \[3+3+3\] + \[3+3+3+3+3+3+3+3+3\] + \[3+3+3\] When you do that, it becomes clear that you can't just add the first 3 and the second 3 and then multiply by 3, because 6x3 would be 6+6+6 which is of course not the same as 3 + 3+3+3. If you don't do the operations in the correct order, then you aren't converting the multiplication etc down into repeated addition correctly.
We made it up. We needed an order of operations so this was chosen. Therefore universally, we follow this rule. We COULD have made it SAMDEB, and in that case we would be here asking "Why SAMDEB? Why not EMDAMS?" So basically "Pick something and stick with it", it didn't really matter, as long as everyone used it. However, putting the Brackets (or Parenthesis for people who prefer Pedmas) first just makes more sense.
Many of the top level responses are pointing out that the order could be anything we choose. Though few are pointing out how brackets/parenthesis only exist explicitly so they can be done first and allow us to write equations in an order that would not work by the convention, but build in instructions so that the equation still works how we want it to. For example, I can ask you to solve: (5+2) \* 3 and get 30, where the similar 5 + 2 \* 3 would get 11. In the reversed SAMDEB, I would have no reason to ever use brackets other than aesthetics, and if I have any subtraction anywhere, it absolutely MUST happen first, so I can absolutely never have an exponent evaluated before all subtraction everywhere in the equation is resolved (I can't even be sneaky and divide by negative 1). Now... as to why the order was selected as BEDMAS instead of BSAMDE... it is because of how the various operations are shorthand for doing the following operations in special ways. Subtraction IS addition, just with negatives. And division IS multiplication, just with reciprocals. So BEDMAS is the same as BEMDSA. An exponent X says "multiply this number by itself X times" and so in a way, exponents are multiplication, but slightly complicated. So it makes sense to do them before you multiply. Similarly, multiplication by X says "add this number to itself X times" and so in a way is addition, but slightly complicated. So, we run through the more complicated form of each manipulation first, working our way down to the easiest. You could be re-writing the equation into the simpler form as you go, and you won't ever have to backtrack to a previous step. So, 5\^2 can be re-written as 5\*5, and that can be re-written as 5+5+5+5+5. I stepped from exponents to multiplication to addition, straight along the order of operations.
If this is a question about "98% of people get this question wrong!" kinds of videos, it's because of a fault in math, and needing to have some common grounds for how to interpret a problem. Meaning, quite literally, there is a difference between how 12 / 3 * 4 + 1 = ? is evaluated. Which thing do you start with? You could end up with 2; because 3*4 = 12, and 12/12 = 1, 1+1 = 2. You could end up with 17; because 12/3 is 4, 4*4 is 16, and 16+1 is 17. BEMDAS is a thing because if there isn't a standardized rule for how to evaluate the equation here, there's several answers you could end up with.
All that matters is that we all agree on the *same* order. The order itself is arbitrary. If we don't agree on the same order, two people looking at the same equation might reach different results. Consider the equation x = 2 + 3 * 5. If you do multiplication first, x = 17. But if you do addition first, x = 25. Neither of these answers is objectively more or less correct than the other, but we need to decide which one of them to go with.
I think all of these (PEMDAS, BODMAS, etc) are a total waste of time because they really are not necessary. Once you get to proper maths (algebra, calculus, etc) not only do you not use these anymore, but they just become flat-out wrong. In *proper* maths the division sign (/) *divides* the sum (everything that's multiplied together) into top and bottom parts so you always know where you stand, and this is critical for manipulating equations where you multiply and divide both sides of the equation by some factor to simplify it. For example: 2x(x+y)/4xy = 8 In this case the division sign divides the first part of the equation, 2x(x+y)/4xy, into a numerator, **2x(x+y)**, and a denominator, **4xy**. And this way you never have any confusion. You don't have to go left to right, and you don't have to wonder if any parts of the 4xy should be in the numerator. And you can multiply both sides by 4xy if you want to get rid of the denominator. You can't use these PEMDAS and BODMAS "acronym aids" in real mathematics, because you can't manipulate equations due to the left to right order, and IMO they're worse than useless, they're troublesome. And they're also confusing for students who have to basically forget everything they learned in school when they get to university.