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

[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.

 

[gmax137]

The Wiki
https://en.wikipedia.org/wiki/Order_of_operations
provides some interesting info.
It states that

BODMAS meaning Brackets, Operations, Division/Multiplication, Addition/Subtraction. Sometimes the O is expanded as "Of" or "Order" (i.e. powers/exponents or roots).

 

[pbuk]

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.

But again this convention is not part of any UK syllabus: if there was any need for stacked exponentials the order would be made explicit ($2^{(3^2)} = 262144$ or $(2^3)^2 = 64$).

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.

Order. BODMAS was the standard term used in the UK until [a few years ago] when it was replaced with BIDMAS in the national curriculum, although you will still see a lot of materials using BODMAS. We have never used E for exponentiation (or P for parentheses) here.

 

[pbuk]

BODMAS meaning Brackets, Operations, Division/Multiplication, Addition/Subtraction.

That is wrong - note that none of the references say this. I will correct some time.

 

[gmax137]

Well, I was just pointing to Wiki. I do agree with you, "operations" doesn't make much sense.

As always, Wiki is... wiki.

 

[Mark44]

But again this convention is not part of any UK syllabus: if there was any need for stacked exponentials the order would be made explicit ($2^{(3^2)}$ or $(2^3)^2 = 64$).But

That's all well and good about the UK syllabus, but stacked exponentials without parentheses do occur in the wild; e.g., $e^{x^2}$ and similar. Some readers would recognize that x should be squared before the exponentiation, but not all would. If the mathematics community would take a leaf from computer science about precedence and associativity, it's my view that this would be a good thing, possibly eliminating most of the stupid, click-bait Youtube videos like the one in the OP of this thread.

 

[paulb203]

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.

When you assumed it might be an acronym did you consider, Simplify Ur Roots, Dude?