An algebraic expression written as a product or quotient of numerals or variables or both is called a term
So b * c, which is a product of the variables b and c, is a term, according to this textbook.
You seem to be getting confused because none of the examples on this particular page feature the multiplication symbol ×. But that is because on the previous page, the author writes:
When a product involves a variable it is customary to omit the symbol × of multiplication.
That means that the expression bc is just another way of writing b×c; it is treated the same other than requiring fewer strokes of the pen or presses on a keyboard, because this is just a *custom. That should clear up your confusion in interpreting this textbook (though really, the language is clear: you don’t dispute that b×c - or bc - are products, do you.)
Elsewhere in this thread you are clearly confused about what brackets mean. They are explained on page 20 of your textbook, where it says that you evaluate the expression inside the (innermost) brackets before doing anything else. Notice that, in its elucidation of several examples, involving addition and multiplication, the “distributive law” is not mentioned, because the distributive law has nothing to do with brackets and is not an operation.
Thus the expression 3 × (2 + 4) can be evaluated by first performing the summation inside the brackets to get 3 × 6 and then the product to get 18. The textbook then says that it is customary to omit the multiplication symbol and instead write 3(2+4), again indicating that these expressions are merely different ways of writing the same thing.
The exact same process of course must be followed whether numbers are represented by numerals or by letters designating a variable. You cannot do algebra if you don’t follow the same procedure in both cases. So consider the expression 2(a+b)². You have suggested that you must evaluate this as (2a+2b)² because you must “do brackets first”, but this is not what “doing brackets” means. You haven’t produced any authority to suggest that it is, and your own textbook makes it clear that “doing brackets” means do what is inside the brackets first. Not what is outside the brackets. Distributing 2 over a+b is not “doing brackets”; it is multiplication and comes afterwards.
If 2(a+b)² were equal to (2a+2b)² let us try with a=b=2. Let us first evaluate (2a+2b)²: it is equal to (2×2+2×2)² = (4+4)² = 8² = 64. Now let us evaluate 2(a+b)²: it is 2(2+2)² and now, following your textbook’s instruction to do what is inside the brackets first, this is equal to 2(4)². The next highest-priority operation is the exponent, giving us 2×16 (we now must write the × because it is an expression purely in numerals, with no brackets or variables) which is 32.
The fact that these two answers are different is because your understandings of what it means to “do brackets” and the distributive law are wrong.
Since I’m working off the textbook you gave, and I referred liberally to things in that textbook, I’m sure if you still disagree you will be able to back up your interpretations with reference to it.
By the way, I noticed this statement on page 23, regarding the order of operations:
However, mathematicians have agreed on a rule to fall back on if someone omits punctuation marks.
it does rather seem like this rule is one established not by the fundamental laws of mathematics but by agreement as they say, does it not? Care to comment?
SmartmanApps@programming.dev 2 days ago
Nope. bc is the product of b and c. bxc is Multiplication of the 2 Terms b and c.
Says person who clearly didn’t read more than 2 sentences out of it 🙄
and why do you think that is? Do explain. We’re all waiting 😂 Spoiler alert: if you had read more than 2 sentences you would know why
No it doesn’t. it means bxc is Multiplication, and bc is the product 🙄 Again you would’ve already known this is you had read more than 2 sentences of the book.
No it isn’t, and again you would already know this if you had read more than 2 sentences. If a=2 and b=3, then…
1/ab=1/(2x3)=1/6
1/axb=1/2x3=3/2
Nope, an actual rule of Maths. If you meant 1/axb, but wrote 1/ab, you’ve gonna get a different answer 🙄
says person who only read 2 sentences out of it 🙄
It sure is when the read the rest of the page 🙄
What don’t you understand about only ab is the product of a and b?
Not me, must be you! 😂
Until all brackets have been removed. on the very next page. 🙄 See what happens when you read more than 2 sentences out of a textbook? Who would’ve thought you need to read more than 2 sentences! 😂
Image
And yet, right there on Page 21, they Distribute in the last step of removing Brackets, 🙄 5(17)=85, and throughout the whole rest of the book they write Products in that form, a(b) (or just ab as the case may be).
Brackets aren’t an operator, they are grouping symbols, and solving grouping symbols is done in the first 2 steps of order of operations, then we solve the operators.
3x6 isn’t a Product, it’s a Multiplication, done in the Multiplication step of order of operations.
It says you omit the multiplication sign if it’s a Product, and 3x6 is not a Product. I’m not sure how many times you need to be told that 🙄
Nope, completely different giving different answers
1/3x(2+4)=1/3x6=6/3=2
1/3(2+4)=1/3(6)=1/18
Yep
Yes it is! 😂
Image
Yes it is! 😂 Until all Brackets have been removed, which they can’t be if you haven’t Distributed yet. Again, last step of the working out…
Image
Yes it is! 😂 Until all Brackets have been removed
Nope, it’s Distribution, done in the Brackets step, before doing anything else, as per Page 21
Which, when you finish doing the brackets, is 8²
After you have finished the Brackets 🙄
Nope. Giving us 8²=64
Nope! If you write it at all, which you don’t actually need to (the textbook never does), then you write (2x4)², per The Distributive Law, where you cannot remove the brackets if you haven’t Distributed yet. There’s no such rule as the one you just made up
You disobeyed The Distributive Law in the second case, and the fact that you got a different answer should’ve been a clue to you that you did it wrong 🙄
No, that would be your understanding is wrong, the person who only read 2 sentences 🙄 I’m not sure what you think the rest of the chapter is about.
Says person who only read 2 sentences out of it 🙄
Yep, ignoring all the parts that prove you are wrong 🙄
Exact same reference! 😂
You know Mathematicians tend to agree when something has been proven, right? 😂
Yep, read the whole chapter 🙄
FishFace@piefed.social 2 days ago
Do you teach classes like this? “That’s not a product, it’s a multiplication” – those are the same thing. Shouldn’t you, as a teacher, be explaining the difference? I’m starting to believe you don’t think they’re is one, but are just using words to be annoying. Or maybe you don’t explain because you don’t know.
You could argue that “product” refers to the result of the multiplication rather than the operation, but there’s no sense in which the formula “a × b” does not refer to the result of multiplying a and b.
Of course, you don’t bother to even make such an argument because either that would make it easier to see your trolling for what it is, or you’re not actuality smart enough to understand the words you’re using.
It’s interesting, isn’t it, that you never provide any reference to your textbooks to back up these strange interpretations. Where in your textbook does it say explicitly that ab is not a multiplication, or that a multiplication is different from a product in any substantive sense, eh? It doesn’t, does it? You’re keen to cite textbooks any time you can, but here you can’t. You complain that people don’t read enough of the textbook, yet they read more than you ever refer to.
In the other thread I said I wouldn’t continue unless you demonstrated your good faith by admitting to a simple verifiable fact that you got wrong. Here’s another option: provide an actual textbook example where any of the disputed claims you make are explicitly made. For example, there should be some textbook somewhere which says that mathematics would not work with different orders of operations - you’ve never found a textbook which says anything like this, only things like “mathematicians have agreed” (and by the way, hilarious that you commit the logical fallacy of affirming the consequent on that one).
Likewise with your idea of what constitutes a term, where’s your textbook which says that “a × b is not a term”? Where is the textbook that says 5(17) requires distribution? (All references you have given are that distribution relates multiplication and addition, but there’s no addition) Where’s your textbook which says “ab is a product, not multiplication”? Where’s a citation saying “product is not the same as multiplication and here’s how”? Because there is a textbook reference saying “ab means the same as a × b”, so your mental contortions are not more authoritative.
Find any one of these - explicitly, not implicitly, (because your ability to interpret maths textbooks is poor) and we can have a productive discussion, otherwise we cannot.
My prediction: you’ll present some implicit references and try to argue they mean what you want. In that case, my reply is already prepared 😁
SmartmanApps@programming.dev 1 day ago
Yep! And if you read more than 2 sentences out of the textbook you would know why 🙄
Says person who only read 2 sentences out of a whole chapter 🙄
Yep, and it’s right there in the textbook! 🙄
So you think if a=2 and b=3, then…
1/ab=1/(2x3)=1/6
1/axb=1/2x3=3/2
Are somehow the same answer?? 😂 Which one is it then? 1/6 or 3/2?? 😂
Yep by definition!
There’s no sense in which it does refer to the result you mean. The result of axb is ab. If a=2, b=3, axb=ab. 2x3=6, axb=2x3, ab=6
Says someone revealing that they haven’t read a word I’ve said 🙄
says someone who has just proven they haven’t been reading them 🙄
Yes I did, and you only read 2 sentences out of it 😂
Read on dude, read on, like I have been telling you the whole time. Oh wait, that would prove you were wrong. Oh, I wonder why you haven’t read it… 🙄
The page that you only read one sentence from 🙄
I already did and you only read 2 sentences out of it 🙄
says person who has repeatedly proven they’ve only read 2 sentences 🙄
And I pointed out that in fact you got it wrong, and Mr. Hypocrite has failed to admit it 🙄
Same one I already told you and you only read 2 sentences out of a whole chapter
It’s easy enough to prove yourself, like I did. Go ahead and try it out and let me know how you go.
No, I was able to prove it myself 🙄
Because it was proven 🙄
Same textbook that you only read 2 sentences from
It tells you tight there on the same page that you must remove all brackets, 🙄 which you also haven’t admitted to being wrong about yet, surprise, surprise, surprise
Image
Image
Image
Image
Image
Image
Image
Image
Image
Image
Same one you only read 2 sentences from
And you stopped reading at that point didn’t even finish the page, never mind the chapter 🙄 Just started making false claims (contradicted by same textbook) that “means” means “equals”, instead of realising they have explicitly not said equals 🙄
Says person who made the mental contortion that “means” means “equals” instead of reading the rest of the page
says person who only read 2 sentences out of a whole chapter 🙄
when you decide to read more than 2 sentences 🙄
Wrong, as usual
says person trying to argue that “means” means “equals” 🙄
FishFace@piefed.social 1 day ago
It’s amazing that you think these are explicit references. Notice how the text never says “you MUST use the distributive law”? It always says some variation of “*in order to simplify*, you must…”?
No, you don’t notice, because you’re blind, and don’t understand what distributivity actually is.
You also got me confused with someone else trying to explain in short words how you’re wrong, but that won’t be a problem now you demonstrated such abject failure to hold a productive discussion - bye.
mindbleach@sh.itjust.works 1 day ago
“a X b is written ab.” Modern Algebra: Structure And Method, page 36.
mindbleach@sh.itjust.works 2 days ago
“When a product involves a variable, it is customary to omit the symbol X of multiplication. Thus, 3 X n is written 3n and means three times n, and a X b is written ab and means a times b.”
Illiterate fraud.
SmartmanApps@programming.dev 2 days ago
says person who thinks “means” and “equals” mean the same thing 😂
mindbleach@sh.itjust.works 2 days ago
By all means, humiliate yourself by splitting that hair.