The P in PEMDAS just means resolve what’s inside the parentheses first. After that, it’s just simple multiplication with adjacent terms, and multiplication and division happen together left to right.
This is actually a generational thing. Millennials were taught “PEMDAS”:
Parenthesis
Exponent
Multiplication
Division
Addition
Subtraction
But younger generations have been taught “BEDMAS” instead:
Brackets
Exponent
Division
Multiplication
Addition
Subtraction
Notably, Division and Multiplication are swapped on PEMDAS and BEDMAS, to make this “both happen at the same time” more straightforward. But that only applies if the entire operation can happen at the same time.
For instance, let’s say 6/2(3) compared to 6÷2(3). At first glance, they both appear to be the same operation. But in the former, the 6 dividend would be over the entire2(3) divisor. Which means you would need to simplify the divisor (by resolving the multiplication of 2•3) before you divide. So the former would simplify to 6/6=1, while the latter would divide first and become 3(3)=9.
Technically, if you wanted to be completely clear, you would write it using multiple parenthesis as needed. For instance, you would write it as either: (6÷2)(3)=9 or 6÷(2(3))=1 to avoid the ambiguity. Then it wouldn’t matter if you’re using PEMDAS or BEDMAS.
But in the former, the 6 dividend would be over the entire2(3) divisor.
I have never heard of or seen an example of anyone using / and ÷ in different ways. If you want multiple terms in your divisor, either write it as a large fraction with all relevant terms in the dividend or divisor, or use parentheses. This just seems like sloppy notation to me.
The slash was just because MarkDown doesn’t really make mathematical notation easy. The point is that with a slash, the 6 is over the entire 2(3) divisor. It’s the difference between these:
You can even see that the automatic solution (in yellow) parses the two differently. In the first example, it correctly resolves the 2(3) first, because you should always simplify both the top and the bottom as much as possible before you resolve the division. But in the second, it parses the 6÷2 first, because it is left ambiguous. The slash is literally the horizontal bar, putting the dividend above the entire divisor. Except it’s in a single line, instead of taking up three lines of text for a single operation.
would you say the same thing if the division was written out like a line under 2(3) and under that 6
idk how this’ll come out but something like this:
2(1+2)
-----------
6
edit : wow i did a formatting thing
edit2: i got it (ish)
In that case, I’d say the answer is 1. Top and bottom are each resolved to the fullest extent possible before dividing top by bottom. It’s equivalent to (top)÷(bottom), but it’s clearer and preferable if you can easily format that way in my opinion, just harder on a computer.
I think that’s why people are complaining about the division sign.
It’s been decades since I took a math class so I am definitely not the right person to explain things, but I am using technology to confirm my understanding of the various notations:
So yeah, if you put 6 over a denominator of 2(1+2), the answer is different (1) because the equation is different. But if you write it out literally, it would be 6 over 2 times (1+2).
What you wrote swapped the denominator to make it 2(1+2)÷6, which will always be 1.
The P in PEMDAS just means resolve what’s inside the parentheses first. After that, it’s just simple multiplication with adjacent terms, and multiplication and division happen together left to right.
6÷2(1+2)
6÷2(3)
3(3)
9
This is actually a generational thing. Millennials were taught “PEMDAS”:
But younger generations have been taught “BEDMAS” instead:
Notably, Division and Multiplication are swapped on PEMDAS and BEDMAS, to make this “both happen at the same time” more straightforward. But that only applies if the entire operation can happen at the same time.
For instance, let’s say
6/2(3)compared to6÷2(3). At first glance, they both appear to be the same operation. But in the former, the6dividend would be over the entire2(3)divisor. Which means you would need to simplify the divisor (by resolving the multiplication of2•3) before you divide. So the former would simplify to6/6=1, while the latter would divide first and become3(3)=9.Technically, if you wanted to be completely clear, you would write it using multiple parenthesis as needed. For instance, you would write it as either:
(6÷2)(3)=9or6÷(2(3))=1to avoid the ambiguity. Then it wouldn’t matter if you’re using PEMDAS or BEDMAS.I have never heard of or seen an example of anyone using / and ÷ in different ways. If you want multiple terms in your divisor, either write it as a large fraction with all relevant terms in the dividend or divisor, or use parentheses. This just seems like sloppy notation to me.
The slash was just because MarkDown doesn’t really make mathematical notation easy. The point is that with a slash, the 6 is over the entire
2(3)divisor. It’s the difference between these:You can even see that the automatic solution (in yellow) parses the two differently. In the first example, it correctly resolves the
2(3)first, because you should always simplify both the top and the bottom as much as possible before you resolve the division. But in the second, it parses the6÷2first, because it is left ambiguous. The slash is literally the horizontal bar, putting the dividend above the entire divisor. Except it’s in a single line, instead of taking up three lines of text for a single operation.Usually, no sign before the bracket means juxtaposition. Scientific calculators do account for it (not all, tho), while regular ones may not.
So 2(1+2) is really (2+4)
Compare 2/2x and 2/2×X where x is (1+2)
The first is 2/(2+4)=1/3, the second is (2/2)×(1+2)=3
Basically, either 1 or 9 can be considered correct. And yes, it’s ambiguous.
Also, there’s no real rule about solving left to right due to associative and commutative properties: 1×2×3 = 1×(2×3) = (1×2)×3 = 3×1×2 = 2×1×3 = 6
would you say the same thing if the division was written out like a line under 2(3) and under that 6
idk how this’ll come out but something like this:
2(1+2)
-----------
6
edit : wow i did a formatting thing
edit2: i got it (ish)
In that case, I’d say the answer is 1. Top and bottom are each resolved to the fullest extent possible before dividing top by bottom. It’s equivalent to (top)÷(bottom), but it’s clearer and preferable if you can easily format that way in my opinion, just harder on a computer.
I think that’s why people are complaining about the division sign.
It’s been decades since I took a math class so I am definitely not the right person to explain things, but I am using technology to confirm my understanding of the various notations:
So yeah, if you put 6 over a denominator of 2(1+2), the answer is different (1) because the equation is different. But if you write it out literally, it would be 6 over 2 times (1+2).
What you wrote swapped the denominator to make it 2(1+2)÷6, which will always be 1.