+9491 The left-to-right rule does work for evaluating these expressions. However, in the case of division, instead of thinking that operations on the same level are done "left to right," I prefer to think that only the "item" immediately following the division symbol goes in the denominator, where an "item" is either a number or an expression enclosed in parenthesees.
The expression \(1\div2+3+4\) is equal to \(\frac12+3+4\) because we place
only the item immediately following the division symbol, 2 , in the denominator.
The expression \(1\div(2+3)+4\) is equal to \(\frac{1}{(2+3)}+4\) because we place
only the item immediately following the division symbol, (2 + 3) , in the denominator.
Following this rule also means the expression \(12/6/3\) is equal to \(\dfrac{\frac{12}{6}}{3}\) which is \(\dfrac23\)
The alternative rule would be that everything after the division symbol goes the denominator.
I personally do not like to say we must follow "left to right" because
for expressions like this: 1 + 2 + 3 + 4 + 5
we don't have to follow left to right. We could evaluate it like this:
1 + 2 + 3 + 4 + 5 = 1 + 2 + 3 + 9 = 1 + 5 + 9 = 10 + 5 = 15
It doesn't hurt to follow left-to-right to evaluate that expression; it just isn't necessary.
Another reason I prefer the rule "only use the item immediatley adjacent to the operator as the operand" is that we can use a similar rule for exponents. In the case of exponents, the rule would be "only the item immediately preceding the caret is the base."
4^2^3
Should 4^2^3 be 4^(2^3) or should it be (4^2)^3 ?
If we just stick with the left-to-right rule, we would get
4^2^3 = 16^3 = 4096
If we use the rule "only the item immediately preceding the caret is the base," we would get
4^2^3 = 4^8 = 65536
Here, WolframAlpha says the answer is 65536:
https://www.wolframalpha.com/input/?i=4%5E2%5E3
The only operations that need such a rule are division, exponentiation, and subtraction.
It is unnecessary to follow left-to-right for multiplication and addition.
+9491 Thanks EP for that link, but I'm pretty sure the information in that article is definitely incorrect.
The question is:
\(8\div2(2+2)\ =\ ?\)
And the ONLY correct answer is:
\(8\div2(2+2)\ =\ 8\div2(4)\ =\ \dfrac82(4)\ =\ 4(4)\ =\ 16\)
See here that WolframAlpha says the answer is 16:
https://www.wolframalpha.com/input/?i=8%C3%B72(2%2B2)
There is unfortunately much confusion about this question, but I do not think it is ambiguous.
PEMDAS and BODMAS mean the same thing, and they both say it's 16.
Parenthesses and Brackets are the same thing. Exponents and Orders are the same thing.
Multiplication and Division are on the same level and so can be written either as MD or DM.
We are also taught that when two operations are on the same level, we do them "left to right".
See this Khan Academy video:
https://www.khanacademy.org/math/pre-algebra/. . ./v/introduction-to-order-of-operations
Notice that he writes Mult/Div on one line and Add/Sub on one line.
And especially watch the example that starts around the time 4:10
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