I think you misunderstand my argument
No, you demonstrably didnât understand mine, which is, what you are saying is impossible, but youâre still saying itâs possible.
I could use still math to solve a real-world problem with an altered order of operations
No, you canât. You already tried to do addition first in 2+3x4 and found out why it doesnât work. Ever since then youâve been ignoring that result and pretending that thereâs some other way to make it work. No, there isnât. As long as multiplication is defined in terms of addition (i.e. 3x4=3+3+3+3) then itâs impossible to get a right answer unless you do multiplication before addition.
You could still do anything you can do with regular math, if you had a different order of operations
No, you canât. Again, you already proved you canât.
Do you need me to calculate something, to prove it to you?
Go ahead - Iâm not holding my breath. I already told you why it literally canât work. But note that adding brackets isnât changing the order of operations - brackets are already part of the order of operations. Writing 2+3x4 as 2+(3x4) is exactly the same thing.
BTW just to FURTHER prove your âaddition firstâ doesnât work, look at this exampleâŠ
3x4+2=3x6=18. But earlier you did 2+3x4=5x4=20 - not even the same answer in an âaddition firstâ world! Welcome to why itâs impossible to make addition-first work. But knock yourself out - youâre welcome to try! đ
The order of operations is just part of a system of notation
No, it isnât. Itâs part of the rules of Maths. Notation is how you write it - underlying that is how Maths actually works. This is embodied in the rules of Maths.
is inherently arbitrary
Completely fixed, and a result of the way the operators are defined - that was the only âarbitraryâ bit, deciding what the operators were and what they were going to mean, but once you did that then the order of operations rules were already written for you (having already been determined as soon as you made the definitions of the operators in the first place).
number 5 has no inherent meaning behind it other than the convention of how we interpret it
Again, not a convention, a rule of how to interpret it. You canât just decide to interpret 5 as four, or again, you end up with wrong answers. The rules of Maths prevent you from getting wrong answers. You found that out yourself when you tried to do addition first in 2+3x4.
the number 5 has no inherent meaning behind it other than the convention of how we interpret it
Again, not a convention, a rule of how to interpret it. You canât just decide to interpret 5 as four, or again, you end up with wrong answers. The rules of Maths prevent you from getting wrong answers. You found that out yourself when you tried to do addition first in 2+3x4.
Itâs only a wrong answer if you use the same expression you would with the standard order of operations. And Iâm not saying we can randomly start interpreting 5 as four, just that there is no law of the universe that makes 5 look like that, and we could theoretically (not practically ofc) switch the definitions of the symbols 5 and 4 if we did it all at once and revised old math expressions to match the new standard. Just as there is no reason the letters âbikeâ mean what they do other than thatâs what someone decided to call it, there is no reason the order of operations is what it is other than that is how someone decided to write it.
Scratch doesnât even have an order of operations. You can still do math in it.
Iâm not saying you can take any expression and get the same answer by doing addition before multiplication. Iâm saying you can take any problem and get the correct answer by doing addition before multiplication. In your milk example, that means I would use the expression 2+(3x4) because 2+3x4 is no longer the correct expression needed to solve the problem.
(For an example of my distinction of the words âexpressionâ and âproblemâ, â(4x)+2â is an expression, and âI start with 2 litres of milk. For every dollar I spend, I get 4 more liters of milk. How much milk do I have?â is a problem.)
My argument also relies on a distinction between the language of modern math and the concept of doing math, defining math as the dictionary definition of âThe study of the measurement, properties, and relationships of quantities and sets, using numbers and symbolsâ. As you can see, this makes no mention of the notation commonly used in math. All I am saying is that you can still use numbers to solve problems with an altered order of operations, or by altering any part of the system of notation.
Perhaps seeing how I could solve a problem with a different order of operations will help illustrate my argument:
Problem: 2 cars approach an interchange at a 90 degree angle to each other. Car A approaches the station from 15 meters away at 30 meters/second and Car B approaches the station from 50 meters away at 20 meters/second. How fast is the distance between the cars decreasing?
Answer: the rate of change of the distance between the cars is approximately -27.777 meters per second.
As you can see, I used my altered math notation to find the correct answer. I can still solve a real-world problem with this notation, but the same expressions you would use before may not work now.
Itâs only a wrong answer
Really? You want to do that again? Ok, fine⊠If I have 1 2 litre bottle of milk, and 4 3 litre bottles of milk - i.e. 2+3x4 - how many litres of milk do I have?
you would with the standard order of operations
The definition of 5 as being 1+1+1+1+1 has nothing to do with order of operations.
there is no law of the universe that makes 5 look like that
No, but there is a rule of Maths which defines it.
switch the definitions of the symbols 5 and 4 if we did it all at once and revised old math expressions to match the new standard
In other words everything would be the same as now but we just switched the notation around. I already said that to you a while back. Now youâre getting it.
there is no reason the order of operations is what it is other than that is how someone decided to write it
Got nothing to do with how itâs written - Maths is written differently in many different countries, and yet the underlying order of operations rules are universal.
Iâm not saying you can take any expression and get the same answer by doing addition before multiplication
And if itâs not the same answer then itâs wrong. Youâre nearly had it.
Iâm saying you can take any problem and get the correct answer by doing addition before multiplication
And I told you you canât. Waiting on a proof from you. Start with 2+3x4 - show me how you can get the correct answer by doing addition first - itâs a nice simple one. :-)
that means I would use the expression 2+(3x4) because 2+3x4
Theyâre literally the same thing.
All I am saying is that you can still use numbers to solve problems with an altered order of operations, or by altering any part of the system of notation
And I told you that itâs impossible. Changing the notation doesnât change the Maths.
As you can see, I used my altered math notation to find the correct answer
BWAHAHAHAHA! Nope! I see you putting brackets around the multiplication to make sure it gets done first - same as if you hadnât used brackets at all! Itâs the exact same notation we use now, just with some redundant brackets added to it! And, predictably, you left the addition for last.
Ok, letâs take your example and do addition first (like you claimed can be done)âŠ
15ÂČ+50ÂČ=15x15+50x50=15x65x50=48,750. But 15ÂČ+50ÂČ is 2,725 according to my calculator. Ooooh, different answers - I wonder which one is right⊠I wonder which one is rightâŠ???
Thanks for proving it can only be done by following the order of operations rules (just like Iâve been saying to you all along). Bye now.
I"m beginning to wonder if you are willfully misunderstanding my point. Or perhaps you have sunk so much time into this argument you assume I must be wrong. Take another look at my third and fifth paragraphs. I promise, I am not trying to say what you think Iâm trying to say.
I see you putting brackets around the multiplication to make sure it gets done first - same as if you hadnât used brackets at all! Itâs the exact same notation we use now, just with some redundant brackets added to it! And, predictably, you left the addition for last.
All I did was use the expression necessary to evaluate correctly with the altered order of operations. There are, in fact, times when you can remove brackets that you would otherwise need, for example (x+4)(x-2) would no longer need brackets. The fact that âoldâ expressions often have to be written with new brackets to evaluate correctly with an altered order of operations is something I fully understand. The presence of brackets where there would be none otherwise does not invalidate my point.
15ÂČ+50ÂČ=15x15+50x50=15x65x50=48,750. But 15ÂČ+50ÂČ is 2,725 according to my calculator. Ooooh, different answers - I wonder which one is right⊠I wonder which one is rightâŠ???
What? I never wrote 15ÂČ+50ÂČ. That is an expression you copied incorrectly. Your incorrectly copied expression has little relevance to the problem at hand.
Ok, fine⊠If I have 1 2 litre bottle of milk, and 4 3 litre bottles of milk - i.e. 2+3x4 - how many litres of milk do I have?
If we were doing math with an altered order of operations, the expression 2+3x4 is just simply wrong. 2+(3x4) is the expression you need. If you try to do math the same as it is with the regular order of operations, it will not work. But that does not mean math with an altered order of operations is useless. It is still math. It can still be used to âstudy of the measurement, properties, and relationships of quantities and sets using numbers and symbolsâ.
I fully understand that to correctly evaluate an expression written with a certain order of operations in mind, you need to use that order of operations. If someone wrote an expression with a different order of operations in mind, you could solve it with a different order of operations and still get what the author of the expression intended. For example, I write the equation a+2xa-2 with my order of operations, expecting you to use the same order of operations, and tell you to simplify. If you get 3a-2, that is wrong, because you used an order of operations different than the one I intended to be used to solve the problem. Imagine, for a moment, an alternate universe where everyone uses a different order of operations and a+2xa-2 simplifies to a^2-4. All I am trying to say is that that their math, with a different order of operations, would be no less useful then our math.
In summary, my only claim is that you can still use a different order of operations to manipulate numbers and solve real world problems.
Waiting on a proof from you.
I wrote and evaluated all of those expressions in my last comment with a different order of operations in mind, and was still able to come to the correct answer.
