Something about the way this thread was written was kind of confusing,
Ok, sorry about that. I’m more than happy to update it if you want to give me some constructive feedback on what was confusing about it. Note though that this was the 3rd part in the series, and maybe you didn’t go back and read the previous 2 parts? They start here
Is it just that the terminology is wrong? Or am I missing something?
Yes and yes. :-) The 2 actual rules of Maths that apply here are Terms and The Distributive Law. These are 2 different rules of Maths - neither of which is “multiplication” - and so when lumping them together as “implicit multiplication” you end up with unpredictable, and usually wrong, answers. The only way to always get the right answer is to follow the actual rules of Maths.
a x b, a*b, ab, and a(b) are all acceptable notations to describe the operation “multiply a and b.”
No, they’re not. The first two are multiplications, the second two are Terms. Note that a Term is a product, the result of a multiplication. In the mnemonics, “Multiplication”" refers literally to multiplication signs, and nothing else.
Ok, sorry about that. I’m more than happy to update it if you want to give me some constructive feedback on what was confusing about it. Note though that this was the 3rd part in the series, and maybe you didn’t go back and read the previous 2 parts? They start here
NP. I’m not really great at giving writing advice, so can’t really help there. Something about it just didn’t click when I read it. The extra context you linked did help a bit.
As far as the issue: After reading it I think it does just seem to be a matter of terminology mixed with problems that arise with when you need to write math expressions inline in text. If you can write things out on paper or use a markdown language, it’s really easy to see how a fractional expression is structured.
8
2(1+3)
is a lot easier to read than 8/2(1+3) even if they technically are meant to be evaluated the same. There’s no room for confusion.
And as for distributive law vs multiplication, maybe this is just taking for granted a thing that I learned a long time ago, but to me they’re just the same thing in practice. When I see a(x+1) I know that in order to multiply these I need to distribute. And if we fill in the algebraic symbols for numbers, you don’t even need to distribute to get the answer since you can just evaluate the parentheses then use the result to multiply by the outside.
Conversely, if I was factoring something, I would need to do division.
ax + a
a
= x+1, thus: a(x+1)
I think we’re basically talking about the same thing, I’m just being a bit lose with the terminology.
And while we’re at it, the best way to make sure there’s no misunderstanding is to just use parenthesis for EVERYTHING! I’m mostly kidding, this can rapidly get unreadable once you nest more than a few parens, although for these toy expressions, it would have the desired effect.
(8)/(2(1+3)) is obviously different than (8/2)(1+3)
8/2(1+3) even if they technically are meant to be evaluated the same
But 8/2(1+3) isn’t a fraction. The / - the computing equivalent of ÷ (which can only be written using Unicode on a computer, so a bit of a pain to use compared to / )- is an operator, which means they’re 2 separate terms. A fraction bar is a grouping symbol, which means it’s 1 term. In this particular case it doesn’t matter, but if it appeared in a bigger expression then it absolutely does matter. The way to write 8/2(1+3) as a fraction inline is to add extra brackets. i.e. (8/2(1+3)) - because brackets are also a grouping symbol.
And as for distributive law vs multiplication, maybe this is just taking for granted a thing that I learned a long time ago, but to me they’re just the same thing in practice
Bu they’re not, for the same reason. Firstly, the Distributive Law isn’t multiplication at all - which only applies literally to multiplication symbols - it applies to bracketed terms (i.e. is a single term which needs to be distributed) - and secondly it applies to a single term, whereas multiplication applies to 2 terms (one before and one after). Anyone who talks about 2(1+3) needing to be “multiplied” has already made the mistake that is going to lead to a wrong answer (unless they just happen to “multiply” before they divide, which is an accidental way to get the right answer).
if I was factoring something
Indeed, that is the precise reason the Distributive Law exists - they are the opposite operation to each other! Anyone who adds a multiplication symbol has broken up the factorised term, again leading to the wrong answer.
I’m just being a bit lose with the terminology
Yeah, and that’s all I was pointing out in the first place - please don’t use “implicit multiplication”. The term itself - i.e. it includes “multiplication” - leads people to do it wrong (because they treat it as multiplication, not brackets, then argue about the precedence of “multiplication”!). It needs to die!
this can rapidly get unreadable once you nest more than a few parens,
Well that’s why the rules of Terms ab=(axb) and The Distributive Law a(b+c)=(a*(b+c))=(ab+ac) exist to begin with - less brackets! :-) Imagine having to write a fraction as (1/(axb)) all the time!
(8)/(2(1+3)) is obviously different than (8/2)(1+3)
Correct, though a lot of people treat it as the latter (yet another way to do it wrong - doing division before brackets) because they figure the 8/2 is “outside the brackets”, but in fact only the 2 is, because the slash separates them as being 2 terms.
Ok, sorry about that. I’m more than happy to update it if you want to give me some constructive feedback on what was confusing about it. Note though that this was the 3rd part in the series, and maybe you didn’t go back and read the previous 2 parts? They start here
Yes and yes. :-) The 2 actual rules of Maths that apply here are Terms and The Distributive Law. These are 2 different rules of Maths - neither of which is “multiplication” - and so when lumping them together as “implicit multiplication” you end up with unpredictable, and usually wrong, answers. The only way to always get the right answer is to follow the actual rules of Maths.
No, they’re not. The first two are multiplications, the second two are Terms. Note that a Term is a product, the result of a multiplication. In the mnemonics, “Multiplication”" refers literally to multiplication signs, and nothing else.
NP. I’m not really great at giving writing advice, so can’t really help there. Something about it just didn’t click when I read it. The extra context you linked did help a bit.
As far as the issue: After reading it I think it does just seem to be a matter of terminology mixed with problems that arise with when you need to write math expressions inline in text. If you can write things out on paper or use a markdown language, it’s really easy to see how a fractional expression is structured.
8
2(1+3)
is a lot easier to read than 8/2(1+3) even if they technically are meant to be evaluated the same. There’s no room for confusion.
And as for distributive law vs multiplication, maybe this is just taking for granted a thing that I learned a long time ago, but to me they’re just the same thing in practice. When I see a(x+1) I know that in order to multiply these I need to distribute. And if we fill in the algebraic symbols for numbers, you don’t even need to distribute to get the answer since you can just evaluate the parentheses then use the result to multiply by the outside.
Conversely, if I was factoring something, I would need to do division.
ax + a
a
= x+1, thus: a(x+1)
I think we’re basically talking about the same thing, I’m just being a bit lose with the terminology.
And while we’re at it, the best way to make sure there’s no misunderstanding is to just use parenthesis for EVERYTHING! I’m mostly kidding, this can rapidly get unreadable once you nest more than a few parens, although for these toy expressions, it would have the desired effect.
(8)/(2(1+3)) is obviously different than (8/2)(1+3)
But 8/2(1+3) isn’t a fraction. The / - the computing equivalent of ÷ (which can only be written using Unicode on a computer, so a bit of a pain to use compared to / )- is an operator, which means they’re 2 separate terms. A fraction bar is a grouping symbol, which means it’s 1 term. In this particular case it doesn’t matter, but if it appeared in a bigger expression then it absolutely does matter. The way to write 8/2(1+3) as a fraction inline is to add extra brackets. i.e. (8/2(1+3)) - because brackets are also a grouping symbol.
Bu they’re not, for the same reason. Firstly, the Distributive Law isn’t multiplication at all - which only applies literally to multiplication symbols - it applies to bracketed terms (i.e. is a single term which needs to be distributed) - and secondly it applies to a single term, whereas multiplication applies to 2 terms (one before and one after). Anyone who talks about 2(1+3) needing to be “multiplied” has already made the mistake that is going to lead to a wrong answer (unless they just happen to “multiply” before they divide, which is an accidental way to get the right answer).
Indeed, that is the precise reason the Distributive Law exists - they are the opposite operation to each other! Anyone who adds a multiplication symbol has broken up the factorised term, again leading to the wrong answer.
Yeah, and that’s all I was pointing out in the first place - please don’t use “implicit multiplication”. The term itself - i.e. it includes “multiplication” - leads people to do it wrong (because they treat it as multiplication, not brackets, then argue about the precedence of “multiplication”!). It needs to die!
Well that’s why the rules of Terms ab=(axb) and The Distributive Law a(b+c)=(a*(b+c))=(ab+ac) exist to begin with - less brackets! :-) Imagine having to write a fraction as (1/(axb)) all the time!
Correct, though a lot of people treat it as the latter (yet another way to do it wrong - doing division before brackets) because they figure the 8/2 is “outside the brackets”, but in fact only the 2 is, because the slash separates them as being 2 terms.