I wasn't. I quoted Maths textbooks, and if you read further you'll find I also quoted historical Maths documents, as well as showed some proofs.
I didn't say the distributive property, I said The Distributive Law. The Distributive Law isn't ax(b+c)=ab+ac (2 terms), it's a(b+c)=(ab+ac) (1 term), but inaccuracies are to be expected, given that's a wikipedia article and not a Maths textbook.
I did read the answers, try doing that yourself
I see people explaining how it's not ambiguous. Other people continuing to insist it is ambiguous doesn't mean it is.
If you read the wikipedia article, you would find it also stating the distributive law, literally in the first sentence, which is just that the distributive property holds for elemental algebra. This is something you learn in elementary school, I don't think you'd need any qualification besides that, but be assured that I am sufficiently qualified :)
By the way, Wikipedia is not intrinsically less accurate than maths textbooks. Wikipedia has mistakes, sure, but I've found enough mistakes (and had them corrected for further editions) in textbooks.
Your textbooks are correct, but you are misunderstanding them. As previously mentioned, the distributive law is about an algebraic substitution, not a notational convention. Whether you write it as a(b+c) = ab + ac or as a*(b+c) = a*b + a*c is insubstantial.
If you read the wikipedia article
...which isn't a Maths textbook!
also stating the distributive law, literally in the first sentence
Except what it states is the Distributive property, not The Distributive Law. If I call a Koala a Koala Bear, that doesn't mean it's a bear - it just means I used the wrong name. And again, not a Maths textbook - whoever wrote that demonstrably doesn't know the difference between the property and the law.
This is something you learn in elementary school
No it isn't. This is a year 7 topic. In Primary School they are only given bracketed terms without a coefficient (thus don't need to know The Distributive Law).
be assured that I am sufficiently qualified
No, I'm not assured of that when you're quoting wikipedia instead of Maths textbooks, and don't know the difference between The Distributive Property and The Distributive Law, nor know which grade this is taught to.
Wikipedia is not intrinsically less accurate than maths textbooks
BWAHAHAHAHA! You know how many wrong things I've seen in there? And I'm not even talking about Maths! Ever heard of edit wars? Whatever ends up on the page is whatever the admin believes. Wikipedia is "like an encyclopedia" in the same way that Madonna is like a virgin.
but you are misunderstanding them
And yet you have failed to point out how/why/where. In all of your comments here, you haven't even addressed The Distributive Law at all.
Whether you write it as a(b+c) = ab + ac or as a*(b+c) = ab + ac is insubstantial
And neither of those examples is about The Distributive Law - they are both to do with The Distributive Property (and you wrote the first one wrong anyway - it's a(b+c)=(ab+ac). Premature removal of brackets is how many people end up with the wrong answer).
Let me quote from the article:
"In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x*(y+z) = x*y + x*z is always true in elementary algebra."
This is the first sentence of the article, which clearly states that the distributive property is a generalization of the distributive law, which is then stated.
Make sure you can comprehend that before reading on.
To make your misunderstanding clear: You seem to be under the impression that the distributive law and distributive property are completely different statements, where the only difference in reality is that the distributive property is a property that some fields (or other structures with a pair of operations) may have, and the distributive law is the statement that common algebraic structures like the integers and the reals adhere to the distributive property.
I don't know which school you went to or teach at, but this certainly is not 7th year material.
which clearly states that the distributive property is a generalization of the distributive law
Let me say again, people calling a Koala a Koala bear doesn't mean it actually is a bear. Stop reading wikipedia and pick up a Maths textbook.
You seem to be under the impression that the distributive law and distributive property are completely different statements
It's not an impression, it's in Year 7 Maths textbooks.
this certainly is not 7th year material
And yet it appears in every Year 7 textbook I've ever seen.
Looks like we're done here.
If you don't want to see why you're wrong that's your thing, but I tried. I can just say, try to re-read the math textbook you took pictures of, and try to understand it.
About the ambiguity: If I write f^{-1}(x), without context, you have literally no way of knowing whether I am talking about a multiplicative or a functional inverse, which means that it is ambiguous. It's correct notation in both cases, used since forever, but you need to explicitly disambiguate if you want to use it.
I hope this helps you more than the stackexchange post?
If I write f^{-1}(x), without context, you have literally no way of knowing whether I am talking about a multiplicative or a functional inverse, which means that it is ambiguous
The inverse of the function is f(x)^-1. i.e. the negative exponent applies to the whole function, not just the x (since f(x) is a single term).
You can define your notation that way if youlike to, doesn't change the fact that commonly f^{-1}(x) is and has been used that way forever.
If I read this somewhere, without knowing the conventions the author uses, it's ambiguous
You can define your notation that way if you like
Nothing to do with me - it's in Maths textbooks.
without knowing the conventions the author uses, itβs ambiguous
Well they should all be following the rules of Maths, without needing to have that stated.
Exactly! It's in math textbooks, in both ways! Ambiguous notation, one might say.
Exactly! Itβs in math textbooks, in both ways!
And both ways are explained, so not ambiguous which is which.
Yeah, doesn't mean that you know what an author is talking about when you encounter it doing actual math
The notation is not intrinsically clear, as any human writing. Ambiguous, one may say.
The notation is not intrinsically clear
It is to me, I actually teach how to write it.
We've been at this point, I'm not going to explain this again. But you weren't able to read a single sentence of a wikipedia article without me handfeeding it to you, so I guess I shouldn't be surprised. I'm sorry for your students.
a single sentence of a wikipedia article without me handfeeding it to you
And I told you why it was wrong, which is why I read Maths textbooks and not wikipedia.
No, The Distributive Law states that a(b+c)=(ab+ac), and that you must expand before you simplify.
Examples by people who simply don't remember all the rules of Maths. Did you read the answers?
Please learn some math before making more blatantly incorrect statements. Quoting yourself as a source is... an interesting thing to do.
https://en.m.wikipedia.org/wiki/Distributive_property
I did read the answers, try doing that yourself.
I'm a Maths teacher - how about you?
I wasn't. I quoted Maths textbooks, and if you read further you'll find I also quoted historical Maths documents, as well as showed some proofs.
I didn't say the distributive property, I said The Distributive Law. The Distributive Law isn't ax(b+c)=ab+ac (2 terms), it's a(b+c)=(ab+ac) (1 term), but inaccuracies are to be expected, given that's a wikipedia article and not a Maths textbook.
I see people explaining how it's not ambiguous. Other people continuing to insist it is ambiguous doesn't mean it is.
If you read the wikipedia article, you would find it also stating the distributive law, literally in the first sentence, which is just that the distributive property holds for elemental algebra. This is something you learn in elementary school, I don't think you'd need any qualification besides that, but be assured that I am sufficiently qualified :)
By the way, Wikipedia is not intrinsically less accurate than maths textbooks. Wikipedia has mistakes, sure, but I've found enough mistakes (and had them corrected for further editions) in textbooks. Your textbooks are correct, but you are misunderstanding them. As previously mentioned, the distributive law is about an algebraic substitution, not a notational convention. Whether you write it as
a(b+c) = ab + acor asa*(b+c) = a*b + a*cis insubstantial....which isn't a Maths textbook!
Except what it states is the Distributive property, not The Distributive Law. If I call a Koala a Koala Bear, that doesn't mean it's a bear - it just means I used the wrong name. And again, not a Maths textbook - whoever wrote that demonstrably doesn't know the difference between the property and the law.
No it isn't. This is a year 7 topic. In Primary School they are only given bracketed terms without a coefficient (thus don't need to know The Distributive Law).
No, I'm not assured of that when you're quoting wikipedia instead of Maths textbooks, and don't know the difference between The Distributive Property and The Distributive Law, nor know which grade this is taught to.
BWAHAHAHAHA! You know how many wrong things I've seen in there? And I'm not even talking about Maths! Ever heard of edit wars? Whatever ends up on the page is whatever the admin believes. Wikipedia is "like an encyclopedia" in the same way that Madonna is like a virgin.
And yet you have failed to point out how/why/where. In all of your comments here, you haven't even addressed The Distributive Law at all.
And neither of those examples is about The Distributive Law - they are both to do with The Distributive Property (and you wrote the first one wrong anyway - it's a(b+c)=(ab+ac). Premature removal of brackets is how many people end up with the wrong answer).
Let me quote from the article:
"In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality
x*(y+z) = x*y + x*zis always true in elementary algebra."This is the first sentence of the article, which clearly states that the distributive property is a generalization of the distributive law, which is then stated.
Make sure you can comprehend that before reading on.
To make your misunderstanding clear: You seem to be under the impression that the distributive law and distributive property are completely different statements, where the only difference in reality is that the distributive property is a property that some fields (or other structures with a pair of operations) may have, and the distributive law is the statement that common algebraic structures like the integers and the reals adhere to the distributive property.
I don't know which school you went to or teach at, but this certainly is not 7th year material.
Let me say again, people calling a Koala a Koala bear doesn't mean it actually is a bear. Stop reading wikipedia and pick up a Maths textbook.
It's not an impression, it's in Year 7 Maths textbooks.
And yet it appears in every Year 7 textbook I've ever seen.
Looks like we're done here.
If you don't want to see why you're wrong that's your thing, but I tried. I can just say, try to re-read the math textbook you took pictures of, and try to understand it.
About the ambiguity: If I write
f^{-1}(x), without context, you have literally no way of knowing whether I am talking about a multiplicative or a functional inverse, which means that it is ambiguous. It's correct notation in both cases, used since forever, but you need to explicitly disambiguate if you want to use it.I hope this helps you more than the stackexchange post?
The inverse of the function is f(x)^-1. i.e. the negative exponent applies to the whole function, not just the x (since f(x) is a single term).
You can define your notation that way if youlike to, doesn't change the fact that commonly
f^{-1}(x)is and has been used that way forever.If I read this somewhere, without knowing the conventions the author uses, it's ambiguous
Nothing to do with me - it's in Maths textbooks.
Well they should all be following the rules of Maths, without needing to have that stated.
Exactly! It's in math textbooks, in both ways! Ambiguous notation, one might say.
And both ways are explained, so not ambiguous which is which.
Yeah, doesn't mean that you know what an author is talking about when you encounter it doing actual math
The notation is not intrinsically clear, as any human writing. Ambiguous, one may say.
It is to me, I actually teach how to write it.
We've been at this point, I'm not going to explain this again. But you weren't able to read a single sentence of a wikipedia article without me handfeeding it to you, so I guess I shouldn't be surprised. I'm sorry for your students.
And I told you why it was wrong, which is why I read Maths textbooks and not wikipedia.
My students are doing good thanks