I'd like to delve deeper to clarify this concept. If we're talking PURELY numbers, PURELY mathematics, there IS NO order of operations. The math just *IS*.

For example, 1+2+3 = 6 doesn't happen left to right, right to left, or whatever order at all. 1+2+3 *IS* 6 and it *IS* 12-6 and it *IS* 18/3. All of these terms are just different ways of representing the number 6. They are equivalent, one in the same. Another way to think of it is that ALL the operations are done *at the same time*.

However, human beings aren't instant and we lose track of things all the time. Therefore, we devised the order of operations as an INTERPRETATION of how we observe the math actually works.

But I do want to clarify, this order of operations isn't just an "agreed upon standard" arbitrarily. It's following observed rules. The reason why multiplication comes before addition is because it is already a form of repeated addition.

7 + 3 x 4 is the same as 7+(4+4+4).

If we try it purely left to right, it doesn't make sense.

7+3x4 is not the same as 10x4 because the 3 is *modifying* the 4.

The 3 and the 4 are not separate terms. They are telling us that 4 happens 3 times (or that 3 happens 4 times). The 7 is just another term being added to the evaluation of 3 times 4. So we MUST evaluate 3x4 first.

Here's an illustration:

7 apples + 3 baskets of 4 apples. You don't add 7 apples to 3 baskets. You'd get what, 10 apple/baskets? That doesn't make sense. You add the baskets first (12 apples total), then add the extra 7 apples for a total of 19 apples.

you're trying to be clever using those decimals, but the concept is still the same.

First let's convert those decimals into symbology that's easier to work with (fractions):

0.75 = 3/4

0.44 = 11/25

0.75 x 0.44 = 3/4 x 11/25

So therefore using ADDITION to evaluate the multiplication of the fractions,

11+11+11 = 33 for the numerator

25+25+25+25 = 100 for the denominator

The fraction 33/100 simplifies to 0.33 in decimal form.

No, but it IS still equivalent to repeated addition. It just gets more complicated in how that's represented symbolically. The fundamental operation doesn't change just because we use less clean symbology of rational and irrational numbers. The concept is still the same. Multiplication is an operation upon addition.

It's simply a fraction of pi. You don't need to think of the parenthesis as multiplication per se, just that it's a modifier that represents the leftover.

You can say "half of 4 is 2", "a quarter of 12 is 3". In a similar vein, .1415...th of pi is 0.44...

No, the rules apply through the real numbers. It's just easier to visualize in integers.

Let's take 1.4 × 2.5. Both are non-integers.

You can still interpret this as taking 1.4 and adding it to itself 2.5 times: 1.4 + 1.4 + 0.7 = 3.5

Another example, take pi, an irrational number. pi × pi = pi + pi + pi + (.1415...)pi = 9.8696...