This is the correct answer and it drives me crazy how often this comes up.
As another user commented, division and subtraction are just syntactic flavor for multiplication and addition, respectively. Division is a specific type of multiplication. Subtraction is a specific type of addition.
And so there is a reason mathematicians do not use the division symbol (➗): it is ambiguous as to which of the following terms are in the divisor and which are part of the next non-divisor term.
In other words, the equation as written is a lossy representation of whatever actual equation is being described.
tl;dr: the equation as written provides insufficient information to determine the correct order of operations. It is ambiguous notation and should not be used.
division and subtraction are just syntactic flavor for multiplication and addition
Can you tell me a bit more about how you mean this? I searched a bit but only basic primary school level resources about the relationship between addition and subtraction came up.
Do you mean like subtraction is just adding a negative number, and division is just multiplication by the inverse of a number? In that case I don’t really see how it simplifies things much because negatives and inverses still need as much definition. Or are you talking about bit-wise operations like a computer would use to do these things?
The other commenter is correct, but another way to think or visualize is that any subtraction or division operation can be understood as an addition or multiplication.
X - 5 = X + (-5)
X / 5 = X * (1/5)
You can think of subtraction and division not being distinct or separated from addition and multiplication; instead, they’re just a shortcut notation in mathematics because everyone was tired of having to write extra characters.
Addition asks “What do you get when you combine these two numbers?”
Subtraction asks “What do you need to combine with this number to get this result?”
Multiplication asks “What do you get if you add this number to itself this many times?”
Division asks “How many times do you need to add this number to itself to get this result?”
In many ways, all of these operations are syntactic flavor for addition. Subtraction is addition in reverse. Multiplication is repetitive addition. Division is repetitive addition in reverse. Exponents are recursive repetition of repetitive addition. And so on.
Look into the axiomatic proof of 1+1=2. It will shed some light on how mathematics is just complex notation for very, very simple concepts at scale.
This is the correct answer and it drives me crazy how often this comes up.
As another user commented, division and subtraction are just syntactic flavor for multiplication and addition, respectively. Division is a specific type of multiplication. Subtraction is a specific type of addition.
And so there is a reason mathematicians do not use the division symbol (➗): it is ambiguous as to which of the following terms are in the divisor and which are part of the next non-divisor term.
In other words, the equation as written is a lossy representation of whatever actual equation is being described.
tl;dr: the equation as written provides insufficient information to determine the correct order of operations. It is ambiguous notation and should not be used.
Can you tell me a bit more about how you mean this? I searched a bit but only basic primary school level resources about the relationship between addition and subtraction came up.
Do you mean like subtraction is just adding a negative number, and division is just multiplication by the inverse of a number? In that case I don’t really see how it simplifies things much because negatives and inverses still need as much definition. Or are you talking about bit-wise operations like a computer would use to do these things?
The other commenter is correct, but another way to think or visualize is that any subtraction or division operation can be understood as an addition or multiplication.
X - 5 = X + (-5)
X / 5 = X * (1/5)
You can think of subtraction and division not being distinct or separated from addition and multiplication; instead, they’re just a shortcut notation in mathematics because everyone was tired of having to write extra characters.
Figuratively, at least.
Addition asks “What do you get when you combine these two numbers?”
Subtraction asks “What do you need to combine with this number to get this result?”
Multiplication asks “What do you get if you add this number to itself this many times?”
Division asks “How many times do you need to add this number to itself to get this result?”
In many ways, all of these operations are syntactic flavor for addition. Subtraction is addition in reverse. Multiplication is repetitive addition. Division is repetitive addition in reverse. Exponents are recursive repetition of repetitive addition. And so on.
Look into the axiomatic proof of 1+1=2. It will shed some light on how mathematics is just complex notation for very, very simple concepts at scale.