While working on a compiler's grammar to follow this, It occurred to me that although it's something we are all taught to and simply follow, I don't know the origin or the reason we do it in the first place. I've done some research, but feel an unfamiliarity with the subject is limiting results / scope of my search terms.
Is there some naturally-occurring basis? For example, if I think of the fundamentals of addition, taking say a fruit and another fruit you can observe two fruits, even negative integers make sense in terms of debt but is there any naturally occurring instance we base these precedence rules off of? Is it just an arbitrary but helpful omission tool as the answers here essentially conclude?
I understand and accept that it is a convention, but what I'm curious about is the motivation behind that being the decided convention? I don't see a definitive answer: There's some contention that it just makes 'sense' but I'm not convinced that doesn't just simply seem the case because it's such a socially ingrained concept and taught really early on.
Best Answer
Imagine a world in which BEDMAS does not exist and where it is necessary to denote exactly in which order the operations in an expression are evaluated, using parentheses. A valid expression looks something like this:
$$((a + b) \times (c \div (d - e)))$$
Look at this expression. It has four operations and four sets of parentheses. Every set of parentheses belongs to exactly one operation and vice versa.
$$\color{red}(\color{orange}(3 \color{orange}+ 5\color{orange}) \color{red}\times \color{blue}(6 \color{blue}\div \color{#00bf00}(9 \color{#00bf00}- 7\color{#00bf00})\color{blue})\color{red})$$
With these parentheses, we know exactly how to evaluate this expression. The result of this expression is $24$.
Now consider a large expression. What if you don't want to write all these parentheses all the time? What if you want large expressions to be easier to read? Wouldn't you think of ways to reduce the amount of parentheses and general symbols you need in such a way that the resulting expression is still unambiguous? Well, mathematicians would. And that's why there is BEDMAS.
First, let's consider expressions that use only $+$. Consider the expression $((a + b) + ((c + d) + e))$. This expression has enough parentheses for the order of operations to be uniquely determined. But there's a nice property of $+$: For any numbers $a, b, c$, it always is the case that $(a+b)+c = a+(b+c)$. This is called the law of associativity. Now if we have a bunch of sums of sums, we can just leave out the parentheses between them, because no matter how you put them back in, you always get the same result. Our expression simplifies to $a + b + c + d + e$. Great, we've already got rid of a lot of parentheses.
Next, let's add $\times$ to the fray. The law of associativity also holds for $\times$, so we can write $a \times b \times c \times d \times e$ instead of $((a \times b) \times ((c \times d) \times e))$. Now let's look at what happens when you combine $+$ and $\times$. We know that $a + (b \times c)$ and $(a + b) \times c$ are different expressions, so we can't remove the parentheses in both cases. But here's a new law: We can rewrite $(a + b) \times c$ as $(a \times c) + (b \times c)$. This is the law of distributivity. This means that we can write any expression using $+$ and $\times$ as an expression where you always do all multiplications first, only then all additions.
As a consequence, we can invent a new way to hide parentheses. In a term where multiplication and addition are next to each other without parentheses, we agree always do the multiplication first. This way is great, because when we use this, we can actually write any term using $+$ and $\times$ without any parentheses at all!
Example:
Using the law of distributivity, we can rewrite
$$(a \times (b + ((c + d) \times (e + f))))$$ to $$((a \times b) + (((a \times (c \times e)) + (a \times (c \times f))) + ((a \times (d \times e)) + (a \times (d \times f)))))$$
There's a lot of operators here, but if we agree to do multiplication first and remember the law of associativity, we get
$$a \times b + a \times c \times e + a \times c \times f + a \times d \times e + a \times d \times f$$
which does not contain any parentheses at all! This would not always be possible if we evaluated from left to right or addition before multiplication.
Now let's add subtraction. First, let's remember that there is a unary minus sign that goes left of an expression, for example $-a$ is shorthand for $(0-a)$. Consider a term of additions and subtractions:
$$(a - (b - (c + (d - e))))$$
Using the laws $a + (b - c) = (a + b) - c$, $~a - (b + c) = (a - b) - c$ and $a - (b - c) = (a-b) + c$, we can actually reorder this sequence of operations so that the parentheses move as far left as possible, so addition and subtraction are executed from left to right:
$$((((a - b) + c) + d) - e)$$
So let's just agree that unparenthesized addition and subtraction are executed left to right. We get
$$a - b + c + d - e$$
This has the advantage that with this notation, you can actually rewrite subtracting as adding negations using the unary $-$:
$$a + (-b) + c + d + (-e)$$
This method therefore makes a term much more easy to read and understand, and this allows us to reorder the summands in this sum much easier.
If we want to mix multiplication and subtraction, we again specify that multiplication is done first.
Now let's come to division. Consider a term mixing multiplication and division. Just as we had for addition and subtraction, we can reorder the parentheses so that the operations are executed from left to right, using perfectly analogous laws $a \times (b \div c) = (a \times b) \div c$, $~a \div (b \times c) = (a \div b) \div c$ and $a \div (b \div c) = (a\div b) \times c$.
We now agree to drop the parentheses when the operations are in fact executed from left to right. So
$$(a \div (b \times (c \div (d \div e))))$$
becomes
$$((((a \div b) \div c) \times d) \div e)$$
which we write as $$a \div b \div c \times d \div e$$
Again, we can interpret division as multiplication with the inverse:
$$a \times (1 \div b) \times (1 \div c) \times d \times (1 \div e)$$
So this description is much easier to work with, for example we can reorder this very easily.
When division is mixed with addition or subtraction, we say that division is always executed first by default.
Now note that terms using $\div$ cannot always be written without parentheses, for example $a \div (b + c)$.
Because of this, mathematicians actually don't use the $\div$ symbol anymore. Instead, they use the fraction notation:
$$a \div b = \frac{a}{b}$$. The nice thing is that we can draw the horizontal line in the middle as long as we want, and that the line clearly separates what's above the line from what's below the line. So we can drop some more parentheses:
$$a \div (b + c) = \frac{a}{b+c}$$
Compare this to
$$(a \div b) + c = \frac{a}{b} + c$$
Clearly we can tell the difference.
Like this, we can actually write any term using $+$, $-$, $\times$, $\div$ completely without parentheses!
Consider the term $((a + b) \times (c \div (d - e)))$ from the top of my answer. It has four sets of parentheses, which we can now technically all avoid by using proper notation and expanding using the distributive law:
$$a \times \frac{c}{d - e} + b \times \frac{c}{d-e}$$
Note that this expression would still usually be written as
$$(a + b) \times \frac{c}{d - e}$$
since this is actually easier to read and understand. We haven't forbidden parentheses after all, we've just made ways to avoid having to write them.
Now be aware that since nobody out of school actually uses $\div$, precedence of $\times$ and $\div$ is not actually known common consensus. This means that we do need to use parentheses when a $\div$ comes before a $\times$, just so people don't get confused.
Now we can finally have a look at the term in the picture of your question:
$$6 - 1 \times 0 + 2 \div 2$$
If we replace the parentheses using the rules we've established, we get
$$((6 - (1 \times 0)) + (2 \div 2)) = 7$$
which should now be perfectly unambiguous.
The rules we've established, as you might have guessed by now, are exactly BEDMAS (or rather BDMAS, since I didn't talk about exponentiation).
Note that if you just punch the given term into a calculator, you won't get the same result, because the calculator evaluates the operations in the order they're punched in, not in BEDMAS order. So a calculator sees
$$((((6 - 1) \times 0) + 2) \div 2) = 1$$
which is clearly a different result. So just remember that calculators can't do BEDMAS and you're good to go.
Here's a different expression:
$$24 \div 4 \times 3$$
What is the result of this expression? By common agreement, we do the division first, but again, technically we should have added parentheses because the agreement is not common enough knowledge. Or we could just use the fraction notation instead of $\div$, in this case we would get two different expressions:
$$\frac{24}{4}\times 3 \neq \frac{24}{4 \times 3}$$
So here we can clearly see the intended order, and all is well again.