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I just watched this video (click here) on YouTube, where this guy tries to show that we take the rules of algebra for granted. He does this by showing examples of him using BODMAS backwards, and claims that it doesn't work. This completely messed up my mind.

The response to the video seem mixed, with mathematicians claiming that it proves nothing, while several maths teachers remarking that it was an amazing demonstration. I sort of got the gist of what he was saying, but is BODMAS the only system that can be used to solve equations, or is it sinply a form of notation? I have several questions:

1) Does this prove that BODMAS/PEDMAS is the only system which will help us solve equations in algebra? (Is the order of operations something discovered or agreed upon)

2) Or will changing the order of operations simply change the meaning of the mathematical statement, for example 11 in base 2 vs 11 in base 10? (In the video he says $2+2=22$, but would the + just carry the meaning of $* 10+$?

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    As long as B / P is first, anything can be made unambiguous by adding brackets. The whole reason to have a standardised order of operations is to make it so that we don't need as many brackets. See, for instance, this MinutePhysics video. The standardised order of operations is something we humans as a whole have adapted to make reading mathematics easy to read and write. There is no real, objective, deep mathematical reason it should be the way that it is, it's just what corresponds best to how we think about quantities. – Arthur Sep 06 '17 at 11:37
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    To illustrate what I mean by "it's just what corresponds best to how we think about quantities", imagine you have boxes with peas in them. You have three boxes with five peas in each, and two boxes with ten in each. In conventional English this means that what we have are "five and five and five and ten and ten". If you were to write down what that sentence says about the number of peas using algebra, you would get $3\cdot 5 + 2\cdot 10$. The order of operations simply says that we interpret that quantity the same way the original sentence is interpreted: $5+5+5+10+10$. – Arthur Sep 06 '17 at 11:44
  • IMO: The whole mathematical theory is based on axioms and conventions. If you change them, which you can, you need to build the whole construct again. A small change is for example $0\notinℕ$ vs $0∈ℕ$. In this context: You would first have to define what "equation" and "solving an equation" means. – P. Siehr Sep 06 '17 at 11:44
  • ". If you were to write down what that sentence says about the number of peas using algebra, $3\cdot 5 + 2\cdot 10$ [...]". As I mentioned in the comment above. You only get this, because you have a mathematical theory that allows you to do this abstraction. – P. Siehr Sep 06 '17 at 11:47
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    These are not the „rules for algebra“, these are notational conventions. Theorems like the distributive law for the natural numbers etc. hold independently. The rules are only there to avoid ambiguity and clearly communicate what you mean. One can always replace the current situation by an unambiguous one: Consider not only putting brackets around everything, but even not using operators inbetween numbers: Instead of $x\cdot y$ you would write $\cdot (x, y)$. It would be clear what you mean, but immensely „complicated“ because it would not be easy to read anymore. – Lukas Juhrich Sep 06 '17 at 11:48
  • @Luke Your suggestion is pretty much Polish Notation, isn't it? It has some attractions but I think that polynomials might look rather ugly. – badjohn Sep 06 '17 at 12:19

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Many other rules are possible. The most important aspect is that the writer and the reader agree. For example: on which side of the road should we drive? The left or the right will work provided that we agree.

As Arthur says, as long as brackets / parentheses are regarded as the highest priority, the other rules could be pretty much anything. One possibility is that there are no other rules and the parentheses are mandatory. In other words, $2 + 3 \times 4$ would not be a valid expression. You would need to write either: $(2 + 3) \times 4$ or $2 + (3 \times 4)$.

Or, you could devise a very different system which does not even need parentheses. This has been done, see Polish Notation or Reverse Polish Notation which was once popular on calculators.

Yet more schemes could be easily devised.

However, most of us have agreed on one system so if you want to switch then you might be rather lonely and you will need to carefully warns others of your system.

Is the accepted system the best? That is hard to answer precisely and I think that most would agree that it is not perfect. It does have some advantages, for example polynomials can be represented quite compactly which would not be the case in some other schemes.

badjohn
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  • Great answer, I would also like to add from Luke's comment above that theorems such as the distributive law property hold independently. –  Sep 06 '17 at 12:02
  • @imagentab Yes, this question seems to be about notation and not the rules of algebra. For example, the associativity of addition and multiplication make the usual system more workable. The lack of associativity of division or exponentiation sometimes causes confusion with $a / b / c$ and $a^{b^c}$. – badjohn Sep 06 '17 at 12:23