Those rules are based on axioms which are used to create statements which are used within proofs. As far as I know statements are pretty common and are a foundational part of all math.
Defining math as a language though is also going to be pointless here. It’s not really a yes or no thing. I’ll say it is a language but sure it’s arguable.
And again laws are created using statements. I have plenty of textbooks that contain “statements”
Nope! The order of operations rules come from the proof of the definitions in the first place. 3x4=3+3+3+3 by definition, therefore if you don’t do the multiplication first in 2+3x4 you get a wrong answer (having changed the multiplicand).
As far as I know statements are pretty common
And yet you’ve not been able to quote a Maths textbook using that word.
are a foundational part of all math
Expressions are.
It’s not really a yes or no thing
It’s really a no thing.
And again laws are created using statements
Not the Laws of Maths. e.g. The Distributive Law is expressed with the identity a(b+c)=(ab+ac). An identity is a special type of equation. We have…
Numerals
Pronumerals
Expressions
Equations (or Formula)
Identities
No statements. Everything is precisely defined in Maths, everything has one meaning only.
Left to right is a convention. Left Associativity is a hard rule. Left to right is a convention which obeys the rule of Left Associativity.
It’s something agreed upon
It’s something that is a natural consequence of the definitions of the operators in the first place. As soon as Multiplication was defined in terms of Addition, that guaranteed we would always have to do Multiplication before Addition to get right answers.
is it not something that is universally true
Yes it is! All of Maths is universally true! 😂
Solve for X X^2=4
You know that’s no longer an order of operations problem, right?
What proof do you have that using a left to right rule is universally true?
From my understanding It’s an agreed convention that is followed which doesn’t make it a universal truth. If we’re all doing it just to make things easier to understand, that implies we could have a right to left rule. It’s also true that not all cultures right in the same way.
But here is an interesting quote from Florian Cajori in his book a history of mathematical notations.
Lastly here is an article that also highlights the issue.
Some of you are already insisting in your head that 6 ÷ 2(1+2) has only one right answer, but hear me out. The problem isn’t the mathematical operations. It’s knowing what operations the author of the problem wants you to do, and in what order. Simple, right? We use an “order of operations” rule we memorized in childhood: “Please excuse my dear Aunt Sally,” or PEMDAS, which stands for Parentheses Exponents Multiplication Division Addition Subtraction.* This handy acronym should settle any debate—except it doesn’t, because it’s not a rule at all. It’s a convention, a customary way of doing things we’ve developed only recently, and like other customs, it has evolved over time. (And even math teachers argue over order of operations.)
What proof do you have that using a left to right rule is universally true?
From my understanding It’s an agreed convention that is followed
Read what I wrote again. I already said that left to right is a convention, and that Left Associativity is a rule. As long as you obey the rule - Left Associativity - you can follow whatever convention you want (but we teach students to do left to right, because they often make mistakes with signs when they try doing it in a different order, as have several people in this thread).
that implies we could have a right to left rule
You can have a right to left convention if the rule is Right Associativity.
It’s also true that not all cultures right in the same way
Yeah, I don’t know how they do Maths - if they do it the same as us or if they just flip everything back-to-front (or top to bottom - I guess they would). In either case all the rules on top stay the same once the direction is established (like I guess exponents would now be to the top left not the top right? but in any case the evaluation of an exponent would stay the same).
But here is an interesting quote from Florian Cajori in his book a history of mathematical notations
Yeah, he’s referring to the conventions - such as left to right - not the rule of Left Associativity, which all the conventions must obey. For a while Lennes was doing something different - because he didn’t understand Terms - and was disobeying Left Associativity, (which meant his rules were at odds with everyone else), but his rule died out within a generation of his death,. Absolutely all textbooks now obey Left Associativity, same as before Lennes came along.
Lastly here is an article that also highlights the issue
Not really. Just another person who has forgotten the rules.
“as it happens, the accepted convention says the second one is correct”
No it isn’t. The Distributive Law says the first is correct (amongst 4 other rules of Maths which also say the answer is only 1). The second way they did it disobeys The Distributive Law (and 4 other rules) and is absolutely wrong.
Those rules are based on axioms which are used to create statements which are used within proofs. As far as I know statements are pretty common and are a foundational part of all math.
Defining math as a language though is also going to be pointless here. It’s not really a yes or no thing. I’ll say it is a language but sure it’s arguable.
And again laws are created using statements. I have plenty of textbooks that contain “statements”
Nope! The order of operations rules come from the proof of the definitions in the first place. 3x4=3+3+3+3 by definition, therefore if you don’t do the multiplication first in 2+3x4 you get a wrong answer (having changed the multiplicand).
And yet you’ve not been able to quote a Maths textbook using that word.
Expressions are.
It’s really a no thing.
Not the Laws of Maths. e.g. The Distributive Law is expressed with the identity a(b+c)=(ab+ac). An identity is a special type of equation. We have…
Numerals
Pronumerals
Expressions
Equations (or Formula)
Identities
No statements. Everything is precisely defined in Maths, everything has one meaning only.
Order of operations is not a hard rule. It is a convention. It’s something agreed upon but is it not something that is universally true.
Solve for X
X^2=4
Yes it is.
Left to right is a convention. Left Associativity is a hard rule. Left to right is a convention which obeys the rule of Left Associativity.
It’s something that is a natural consequence of the definitions of the operators in the first place. As soon as Multiplication was defined in terms of Addition, that guaranteed we would always have to do Multiplication before Addition to get right answers.
Yes it is! All of Maths is universally true! 😂
You know that’s no longer an order of operations problem, right?
What proof do you have that using a left to right rule is universally true?
From my understanding It’s an agreed convention that is followed which doesn’t make it a universal truth. If we’re all doing it just to make things easier to understand, that implies we could have a right to left rule. It’s also true that not all cultures right in the same way.
But here is an interesting quote from Florian Cajori in his book a history of mathematical notations.
Lastly here is an article that also highlights the issue.
https://scienceblogs.com/evolutionblog/2013/03/15/the-horror-of-pemdas
Read what I wrote again. I already said that left to right is a convention, and that Left Associativity is a rule. As long as you obey the rule - Left Associativity - you can follow whatever convention you want (but we teach students to do left to right, because they often make mistakes with signs when they try doing it in a different order, as have several people in this thread).
You can have a right to left convention if the rule is Right Associativity.
Yeah, I don’t know how they do Maths - if they do it the same as us or if they just flip everything back-to-front (or top to bottom - I guess they would). In either case all the rules on top stay the same once the direction is established (like I guess exponents would now be to the top left not the top right? but in any case the evaluation of an exponent would stay the same).
Yeah, he’s referring to the conventions - such as left to right - not the rule of Left Associativity, which all the conventions must obey. For a while Lennes was doing something different - because he didn’t understand Terms - and was disobeying Left Associativity, (which meant his rules were at odds with everyone else), but his rule died out within a generation of his death,. Absolutely all textbooks now obey Left Associativity, same as before Lennes came along.
Not really. Just another person who has forgotten the rules.
“as it happens, the accepted convention says the second one is correct”
No it isn’t. The Distributive Law says the first is correct (amongst 4 other rules of Maths which also say the answer is only 1). The second way they did it disobeys The Distributive Law (and 4 other rules) and is absolutely wrong.