To add two fractions (let's say, 4/7 and 2/3), instead of using LCM, why I can't simply add numerators first (4+2), and divide it by sum of denominators (7+3) ? Well, I know the division has more precedence than addition, so we can't perform 4+2 first. BUT my question is if precedence is a MAN-DEFINED rule and there's no logic behind performing division before addition, why the precedence rule always holds true for every day problems? I mean, there must be a reason behind this rule!
2026-03-26 12:35:56.1774528556
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Does there lie a logic/reason behind precedence rule of mathematical operators?
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You seem to be thinking of the denominator as a number in the sense of "how many I have" whereas it is a descriptor telling you what type of thing you have. It tells you how many parts you need to make a whole one. So, for instance, if you have $\frac{3}{7}$ it means you have three things and that if you had seven you would make a whole. So it makes no sense to add the denominators.
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Let's see how we would add $1\over2$ and $1\over4$. Just like adding one thousand and one we write $1000+1$ using units, we need to use the same unit of measurement for $1\over2$ and $1\over4$ This unit can be picked in many ways but the simpliest one is $1\over4$. Now, $1\over2$ contains how many of these units? That's right, two. So together we have $2+1=3$ units and each unit is $1\over4$. Thus, $\large{\frac{1}{2} + \frac{1}{4} = 2\cdot \frac{1}{4}+\frac{1}{4}=\frac{3}{4}}$. We could've picked $1\over8$ as a unit of measurement and the result would be $6\over8$ which is the same. Hopefully you see now where LCM is coming into picture as $1\over LCM$ is the simpliest unit of measurement we can use while adding or subtracting two fractions.
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Your question seems to be “I think $+$ is defined arbitrarily, with no connection to the world. Given that assumption, explain why $+$ describes the world.” The answer is simple: the first sentence is false.
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Another problem, suppose you accept that $$1=\frac{1}{1}=\frac{2}{2}=\frac{3}{3}=\dots$$
Then adding any of these equal values to $\frac{1}{2}$ should give the same answer, but $$\frac{1}{1}+\frac{1}{2}=\frac{2}{3}\\\frac{2}{2}+\frac{1}{2}=\frac{3}{4}\\\frac{3}{3}+\frac{1}{2}=\frac{4}{5}\\\dots$$ So this would mean that $$\frac{2}{3}=\frac{3}{4}=\frac{4}{5}=\frac{5}{6}=\dots$$ Indeed if $$\frac{2}{3}=\frac{3}{4}$$ then $$\frac{2}{3}-\frac{3}{4}=0\\\frac{2-3}{3-4}=0\\\frac{-1}{-1}=0\\1=0$$ So for your system to work, you need $1=0$, which means that every number you work with is equal to $0$. This is because any number $n$ can be written $n=n\cdot 1=n\cdot 0=0$.
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Yes, precedence is man-defined. For example, there is nothing per se saying that $1 / 2 + 1 / 3$ should be interpreted as $(1/2) + (1/3).$ It could as well be interpreted as $((1/2)+1)/3,$ which we might call left-to-right precedence. We could also have said that addition has higher precedence than multiplication, and that operators of equal precedence should be evaluated from right to left, resulting in $1 / ((2+1) / 3).$
The addition rule for fractions, however, has nothing to do with operator precedence. It's a consequence of how we want them to behave.
Because it doesn't get you the correct result. For example, if you try adding two halves, you should get a whole, but your convention would give $$ \frac12 + \frac12 = \frac{1+1}{2+2} = \frac{2}{4} = \frac{1}{2}. $$ Of course, you are free to redefine what the symbol $+$ means -- this is where your comment about man-defined rules come in -- but then you have to give up any expectations about how the symbol $+$ works. In fact, the definition you suggest has another problem: since $\frac13 = \frac26$, you would get the result $$ \frac25 = \frac12 + \frac13 = \frac12 + \frac26 = \frac39 = \frac13, $$ but $\frac25$ and $\frac13$ are different numbers.