I don’t think I ever used a divide symbol like that beyond elementary school. In practice always use fraction style notation for division because it’s not ambiguous or a gotcha.
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?
I believe they do mean the fact that subtraction is just adding the negative and division is just multiplying by the inverse. You can look up field axioms to see how real arithmetic is really defined. It’s much more convenient to have two operations instead of four.
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.
Yup, I found an old comment of mine but unfortunately that post was deleted. The numbers are different but its the same riddle
I think the confusion is in the way it’s displayed. The notation in the comic is ambiguous, where the division is shown as a symbol, while the multiplication is implied with the brackets, so some people see the question as 8/(2*(2+2))=1, while others see it as 8/2*(2+2).
For the later, my understanding is that multiplication and division actually have equal priority and are solved left to right (rather than an explicit order as PEDMAS and BEDMAS seem to suggest). So the second interpretation would give 8/2*(2+2)=8/2*(4)=4*4=16
The reason this isn’t a problem more often is because
math questions should be written unambiguously, using symbols everywhere and fraction bars
in real life problems, there is a certain order in which you manipulate the numbers, and we can use correct notation (with an excessive number of brackets if needed) to keep it crystal clear
I don’t think I ever used a divide symbol like that beyond elementary school. In practice always use fraction style notation for division because it’s not ambiguous or a gotcha.
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?
I believe they do mean the fact that subtraction is just adding the negative and division is just multiplying by the inverse. You can look up field axioms to see how real arithmetic is really defined. It’s much more convenient to have two operations instead of four.
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.
Yup, I found an old comment of mine but unfortunately that post was deleted. The numbers are different but its the same riddle