Why are the division rules for surds the way they are?

[paulb203]

TL;DR Summary

Why are the division rules regards surds the way they are?

Why does, 4√64÷2√4
=2√16 ?

I can see the method (4÷2=2, √64÷√4=√16)

But why is it not;

4 times √64 divided by 2 times √4 (as per BODMAS)
i.e, 4 times 8 divided by 2 times 2, which = 32

Nb. When I put it into my calculator that way it gives 32 as the answer
(4*√64÷2√4)

 

[fresh_42]

You cannot use division in a linear notation without parentheses. It will always be ambiguous. So either you use a nonlinear notation $\dfrac{4\sqrt{64}}{2\sqrt{4}}$ or parentheses $\left(4\sqrt{64}\right)/\left(2\sqrt{4}\right)$ or - and that is my favorite notation - you do not use division at all $4\cdot \sqrt{64} \cdot 2^{-1} \cdot \sqrt{4}^{-1}.$

Every other notation is ambiguous and only suited for playing around with people on fb having endless and senseless discussions.

 

[gmax137]

Every other notation is ambiguous and only suited for playing around with people on fb having endless and senseless discussions.

I detest everything about those FB things.

 

[PeroK]

I detest everything about those FB things.

A few years ago there was a thread asking for "pet peeves". Mine was all this order of operations nonsense.

 

[DaveC426913]

I detest everything about those FB things.

They do have an upside of providing something less brain-rotting than celeb clickbait and Kardashian gossip.

 

[paulb203]

Thanks, everybody.

I share your frustrations, to a lesser degree (given the level I am at, which is preparing for GCSE, in the UK. Google tells me GCSE is equivalent to pre-high school diploma in the U.S.).

The problem for me is the resources given out, and recommended by, my college is full of linear notation with the division symbol. It’s not often problematic as we just follow the (much-despised-by-some) BODMAS ‘rules’ and there rarely seems to be any ambiguity, at least in that context.

Incidentally, I came across the surds example on Maths Genie, online, which I’ve found really helpful, but I take the point about this particular example.

Who sets the agenda in this regard? I Googled it once I’d read your replies and saw the BODMAS thing being explained (happily) by a university as if it wasn't an issue, and I read elsewhere that it goes back centuries.

How do you guys know the order of operations when it comes to tackling a maths problem. Or are the problems you are presented with at degree and beyond level (and in the workplace) devoid of any ambiguity?

 

[paulb203]

You cannot use division in a linear notation without parentheses. It will always be ambiguous. So either you use a nonlinear notation $\dfrac{4\sqrt{64}}{2\sqrt{4}}$ or parentheses $\left(4\sqrt{64}\right)/\left(2\sqrt{4}\right)$ or - and that is my favorite notation - you do not use division at all $4\cdot \sqrt{64} \cdot 2^{-1} \cdot \sqrt{4}^{-1}.$

Every other notation is ambiguous and only suited for playing around with people on fb having endless and senseless discussions.

Thanks, fresh_42.

Is this example ambiguous;
√30 ÷ √6 = √5 ?

This one didn’t confuse me like the other one did.

 

[gmax137]

Or are the problems you are presented with at degree and beyond level (and in the workplace) devoid of any ambiguity?

If something appears like this:
A x B ÷ C x D
the only response should be "ambiguous."

I have not seen this "÷ " used since grade school.

 

[PeroK]

How do you guys know the order of operations when it comes to tackling a maths problem. Or are the problems you are presented with at degree and beyond level (and in the workplace) devoid of any ambiguity?

If done properly mathematics should be unambiguous. I believe that brackets should be used wherever there is potential doubt. And that if brackets are not used where they are needed, then the resulting expression is ambiguous, hence meaningless.

There is, however, a school of thought that every sequence of symbols has a single correct meaning. The problem is that the people who hold this belief often disagree among themselves about the precise order of operations!

I wouldn't be surprised if those who set the GCSE syllabus fall into the second camp!

 

[fresh_42]

How do you guys know the order of operations when it comes to tackling a maths problem.

As far as I am concerned, I ignore the existence of a division. Division is in my world the multiplication by an inverse element. I only distinguish between the properties of the multiplicative domain, i.e. the numbers which do not multiply to zero and have an inverse. It can be a proper subgroup like $\{\pm 1\}$ in the case of integers, or on the clock with hands, where only $\{1,5,7,11\}\subseteq \{0,1,2,3,4,5,6,7,8,9,10,11\}$ have inverses, or all numbers ($\neq 0$) in the case of rational numbers. Division in my world is at most the Euclidean algorithm
$$
n=q\cdot m + r \, , \,|r|<m
$$
But I have learned it like everybody else at school. It took me a while to recognize that division is not necessary in mathematics. However, I'm the only one here who sees it as such. All others love their divisions. It creates senseless discussions about my opinion, or why $0$ cannot be divided by, or how linear notations should be read. All things that are impossible in my world. I do not even have such questions: $0$ isn't even an element of the multiplicative sets as long as we demand $1\neq 0,$ and linear notations are no problem if you only multiply. Btw, the same is true for subtraction; only the addition with an inverse.

Or are the problems you are presented with at degree and beyond level (and in the workplace) devoid of any ambiguity?

There is simply no reason to save parentheses if a linear notation has to be. All these BODMAS or whatever artificial abbreviation is used depend on one single fact: multiplication and addition only meet each other in the distributive laws $a\cdot (b+c)=a\cdot b+ a\cdot c$ and $(b+c)\cdot a= b\cdot a+ c\cdot a.$

That's it.

Everything else are abbreviations: $\div x$ for $\cdot x^{-1}$ and $ab$ for $a\cdot b.$ Exponentiation is in its basic version also an abbreviation, $x^2$ for $x\cdot x,$ or a function in case of $x^\alpha$ where $\alpha$ is something nastier than integers. In that case, write $f(x)$ instead of $x^\alpha$and you won't be tempted to multiply the $x$ by something. All you have to memorize is
$$
a(b+c)=ab+ac\neq ab+c .
$$

In all other cases: be generous with parentheses and brackets.

 

[Mark44]

Is this example ambiguous;
√30 ÷ √6 = √5 ?

No.

This one didn’t confuse me like the other one did.

The left side of the expression above consists of a single binary operation whose operands are $\sqrt{30}$ and $\sqrt 6$. This is different from your example in post #1, 4√64÷2√4. The ambiguity comes from not being able to determine whether the left operand of the division operation should be divided only by 2 or instead by $2\sqrt 4$.

As I've said in a similar thread, the concept of order of operations/operator precedence isn't well developed in the mathematics community, unlike in the field of computer science. Every programming language that I know about specifies a very precise order of operations together with the associativity of the operators; i.e., whether operators at the same precedence level operate left-to-right or right-to-left.

For example, in Python, the multiplication (*) and integer division (//) operators are at the same precedence level, and these operators associate left to right. The expression 12 // 2 * 3 evaluates to 18 because the first operation yields 6, and the second operation is to multiply that result by 3. The expression 12 // 2 * 3 is equivalent to (12 // 2) * 3.

 

[gmax137]

For example, in Python ...

This is interesting. The difference between this approach and, say, BODMAS or PEDMAS or whatever is, the programming language actually does have a manual and the coding actually does work as stated, so there is a definite correct answer not subject to opinion or argument.

 

[Mark44]

This is interesting. The difference between this approach and, say, BODMAS or PEDMAS or whatever is, the programming language actually does have a manual and the coding actually does work as stated, so there is a definite correct answer not subject to opinion or argument.

The main difference between these programming languages such as Python, C, C++, Java, etc. and BODMAS/PEDMAS is the associativity: how operators at the same precedence level group, either left-to-right or right-to-left.

 

[DaveC426913]

I have lived on this planet the better part of six decades, most of it surrounded by science and math, and this is the first time I have ever heard the term "surd". I assumed it was maybe an unfamiliar acronym for a term I am familiar with, but no, it's a straight-up word, from Latin, that has simply eluded me all this time.

My life is a lie.

 

[PeroK]

The main difference between these programming languages such as Python, C, C++, Java, etc. and BODMAS/PEDMAS is the associativity: how operators at the same precedence level group, either left-to-right or right-to-left.

The difference between a computer language and mathematics is that each computer language has its own well defined syntax and will happily process anything that is well formed.

Mathematics, however, is not a precise set of syntactical rules. At least not one that is universally agreed on. And, there are subtlies such as powers being raised lettering and division being split over two lines. Moreover the following would not be acceptable:
$$f(x) = a \ \ \ x \ \ ^2+b$$That may be valid as far as PEMDAS is concerned, but it fails the more general mathematical conventions.

 

[pbuk]

Incidentally, I came across the surds example on Maths Genie, online, which I’ve found really helpful

Really? Are you sure it wasn't $\frac{4\sqrt{64}}{2\sqrt{4}}$? Do you have a link?

Independent revision sites aren't perfect (some are downright awful), but apart from BBC Bitesize I'd say that Maths Genie is one of the most reliable for GCSE and A level so I'd be surprised if they had an ambiguous example like this.

...are the problems you are presented with at degree and beyond level ... devoid of any ambiguity?

Yes, and so are the questions you are presented with by an exam board at GCSE level so don't worry about it.

...are the problems you are presented with ... in the workplace devoid of any ambiguity?

In some workplaces you encounter all sorts of nonsense, but you are only likely to encounter surds in an engineering or science workplace where things are stated without ambiguity.

 

[pasmith]

In some workplaces you encounter all sorts of nonsense, but you are only likely to encounter surds in an engineering or science workplace where things are stated without ambiguity.

And it is always acceptable to ask for clarification if something is ambiguous.

 

[Mark44]

Mathematics, however, is not a precise set of syntactical rules.

Which was pretty much my point. The mathematics community has agreed on a convention for operator precedence as defined by the acronyms BEDMAS and PEMDAS. It would be beneficial if the community would also agree on how operators at the same precedence level group so as to clarify expressions such as this one:
$${2^3}^2$$
Is this $(2^3)^2$ or is it $2^{(3^2)}$?

 

[WWGD]

All you need to remember:
Square of opposite side + Square of adjacent side= Square of Hippopotamus.

 

[paulb203]

If something appears like this:
A x B ÷ C x D
the only response should be "ambiguous."

I have not seen this "÷ " used since grade school.

Thanks, gmax. I hear you, but I'm wondering if things like that can be unambiguous in certain contexts. For example if a text book explains BODMAS (or their particular version of it) and then presents you with that problem, in the same book, presumably the correct procedure would be to follow BODMAS/whatever? So if it was say, 2x6÷3x4 the answer, in that context, would have to be 16, yeah? They've said, "Here's the rules. Now solve this." But I do get from all the comments here that it's problematic generally.

 

[paulb203]

Really? Are you sure it wasn't $\frac{4\sqrt{64}}{2\sqrt{4}}$? Do you have a link?

Independent revision sites aren't perfect (some are downright awful), but apart from BBC Bitesize I'd say that Maths Genie is one of the most reliable for GCSE and A level so I'd be surprised if they had an ambiguous example like this.Yes, and so are the questions you are presented with by an exam board at GCSE level so don't worry about it.In some workplaces you encounter all sorts of nonsense, but you are only likely to encounter surds in an engineering or science workplace where things are stated without ambiguity.

Thanks, pbuk. Yeah they used the "÷" symbol, rather than a slash. Here's the link; https://www.mathsgenie.co.uk/surds.html
I'm not complaining; the site has been really helpful. And thanks for the reassurance about the exam board. I'm not really worried about it though, I'm more curious about it.

 

[pbuk]

Thanks, pbuk. Yeah they used the "÷" symbol, rather than a slash. Here's the link; https://www.mathsgenie.co.uk/surds.html

The errors are in the middle of the videos, here is a direct link to an example:

I've sent them a message.

 

[Mark44]

For example if a text book explains BODMAS (or their particular version of it) and then presents you with that problem, in the same book, presumably the correct procedure would be to follow BODMAS/whatever? So if it was say, 2x6÷3x4 the answer, in that context, would have to be 16, yeah? They've said, "Here's the rules. Now solve this." But I do get from all the comments here that it's problematic generally.

No, BEDMAS is no help, because division and multiplication are considered to be at the same level of precedence. What BEDMAS and PEMDAS are missing is how to treat operators at the same precedence level, the point I made earlier in this thread. To make this problem a little bit simpler, consider 12÷3x4. Now the operators are ÷ and x. Do you group them left to right? If so, that's the same as (12÷3) x 4, or 16. If you group them right to left, then the problem is the same as 12 ÷(3x4), or 1.

If it's written as it normally would be in a grade school textbook, it would appear like this: $\frac{12}{3 \times 4}$, and the fraction bar normally would be interpreted as implied parentheses, so the expression would evaluate to 1.

 

[PeroK]

This case exemplifies the point that it's not as simple as some people would like. In this case, the video instructor has taken an expression such as $2\sqrt 3$ to represent a single number. That's just another way to express $\sqrt {12}$. You can argue all day whether this is valid or not.

A similar case would be the decimal point. That's treated as internal to a certain number and not an external operation. So that, for example: $3 \times 2.5 = 7.5 \neq 6.5$.

If you treat mathematics as computer-like parsing of strings, then that's another rule you need to add to PEMDAS etc.

 

[PeroK]

PS this adds a twist to the question of why do we write $\sqrt {12}$ as $2\sqrt 3$ in the first place? It's not a simplification as such. And, if we consider the resulting potential ambiguities involving order of operations, then $\sqrt {12}$ looks like the better option!

And, before anyone says it, of course we would write $4$ instead $\sqrt {16}$. As that is a definite simplification.

 

[Ibix]

PS this adds a twist to the question of why do we write $\sqrt {12}$ as $2\sqrt 3$ in the first place? It's not a simplification as such. And, if we consider the resulting potential ambiguities involving order of operations, then $\sqrt {12}$ looks like the better option!

Isn't it a pre-cheap calculators thing? I don't know $\sqrt{12}$ off the top of my head, but I do know $\sqrt{3}\approx 1.73$ and if I observe that $\sqrt{12}=2\sqrt{3}$ then I can get $\sqrt{12}\approx 3.46$ with mental arithmetic. If your work gives you square roots of smallish integers fairly frequently it may be convenient to be able to parlay knowing the square roots of a few primes into the square roots of a lot more numbers.

Hardly worth the effort when my phone is in my pocket, though. I can get the square root of anything with the same minimal effort, while spotting repeated prime factors and memorising and multiplying out the roots of the unique factors gets more and more complex. On the other hand, it makes a good party trick.

 

[pbuk]

Now the operators are ÷ and x. Do you group them left to right?

Yes, according to the GCSE syllabus (which, unlike in some other countries, is set out nationally). In some places on the Maths Genie site (I have only seen this in the area where they deal with surds, the main BIDMAS videos look OK) they seem to have invented the rule that where multiplication is implied (by the omission of any operator between adjacent terms) it has higher precedence.

This 'rule' does not exist in any material published by any of the exam boards: the ÷ division operator is only used in early stage material wheras multiplication of constants by adjacancy (e.g. $2 \sqrt 3$ instead of $2 \times \sqrt 3$ is only used in later stage material where division is always expressed by a fraction bar $\frac{\text{expression}}{\text{expression}}$.

 

[martinbn]

I agree with @PeroK, when I watched that part of the clip it seemed clear to me that they meant the division of two numbers that have surds, and not the stupid facebook maths expression of the form $A\times B\div C\times D$.

 

[martinbn]

PS this adds a twist to the question of why do we write $\sqrt {12}$ as $2\sqrt 3$ in the first place? It's not a simplification as such. And, if we consider the resulting potential ambiguities involving order of operations, then $\sqrt {12}$ looks like the better option!

And, before anyone says it, of course we would write $4$ instead $\sqrt {16}$. As that is a definite simplification.

If you work with the field $\mathbb Q(\sqrt3)$, then you want to write every element in the form $a+b\sqrt3$, so $\sqrt{12}$ is $2\sqrt3$. Of course that is not what is meant here, or what your point is. But to me it seems like a convention that has nor real justification other than because we decided so. Same as why we write $\frac46$ as $\frac23$.

 

[pbuk]

I agree with @PeroK, when I watched that part of the clip it seemed clear to me that they meant the division of two numbers that have surds.

Yes, but again this is against the specification of syllabus which is designed to avoid ambiguities: you should never see 'simplify $6 \sqrt{14} \div 2 \sqrt{7}$' in any GCSE material, it should always be 'simplify $\frac{6 \sqrt{14}}{2 \sqrt{7}}$'.

 

[martinbn]

Yes, but again this is against the specification of syllabus which is designed to avoid ambiguities: you should never see 'simplify $6 \sqrt{14} \div 2 \sqrt{7}$' in any GCSE material, it should always be 'simplify $\frac{6 \sqrt{14}}{2 \sqrt{7}}$'.

I agree, I can only speculate that it was done because they didn't have enough space on the screen.

 

[paulb203]

No, BEDMAS is no help, because division and multiplication are considered to be at the same level of precedence. What BEDMAS and PEMDAS are missing is how to treat operators at the same precedence level, the point I made earlier in this thread. To make this problem a little bit simpler, consider 12÷3x4. Now the operators are ÷ and x. Do you group them left to right? If so, that's the same as (12÷3) x 4, or 16. If you group them right to left, then the problem is the same as 12 ÷(3x4), or 1.

If it's written as it normally would be in a grade school textbook, it would appear like this: $\frac{12}{3 \times 4}$, and the fraction bar normally would be interpreted as implied parentheses, so the expression would evaluate to 1.

I think any BODMAS or similar explanation I've come across has included the instruction to work from left to right regards equal precedence.

 

[paulb203]

I agree, I can only speculate that it was done because they didn't have enough space on the screen.

Thanks, Martin. I doubt that it was due to lack of space; they guy often scrolls and scrolls using loads of space when necessary, for prime factor trees for example, or solving algebraic fractions.

 

[Mark44]

you should never see 'simplify $6 \sqrt{14} \div 2 \sqrt{7}$'

Right. And I don't recall ever seeing the $\div$ symbol in any class beyond primary grade arithmetic classes. The author of the Youtube video linked to in post #22 evidently considered the expression above to be identical to $\frac{6 \sqrt{14}}{2 \sqrt{7}}$.

I think any BODMAS or similar explanation I've come across has included the instruction to work from left to right regards equal precedence.

That's not the case with exponents. $2^{3^2} = 2^9 = 512$ and is evaluated as if written this way: $2^{(3^2)}$, but not as if written as $(2^3)^2 = 64$. With nested exponents, the evaluation goes from the top down -- i.e., right to left. See https://en.wikipedia.org/wiki/Exponentiation#Terminology.

This is another reason why associativity (i.e. grouping rules) should be formally included with PEDMAS/BIDMAS.

BTW I'm not sure that BODMAS is a thing. The E and I parts of the acronyms represent 'exponent' and 'index' respectively. If there's a word that corresponds to O I'm not aware of it.

 

[PeroK]

I think any BODMAS or similar explanation I've come across has included the instruction to work from left to right regards equal precedence.

IMO, this is the problem with the order of operations mandates. The reality is that $2\sqrt 3$ is just another way to express $\sqrt{12}$. It represents a single number that should not be split up. When you convert $\sqrt{12}$ to $2\sqrt 3$, there's an over-riding rule that you must keep those two terms together.

There perhaps ought to be brackets in the original calculation. But, even without them, decomposing $2\sqrt 3$ into $2 \times \sqrt 3$ and then operating on the $2$ separate from the $\sqrt 3$ looks fundamentally wrong to me.