My thought process: if I had 8÷(4+4) = 1 and factored out a 2, I get 8÷2(2+2) = 1. However, if I say 2(2+2) = 2*(2+2), then 8÷2(2+2) = 8÷2*(2+2) = 1, BUT 8÷2*(2+2) = 16.
Please help I’m feeling dumber by the second
In: Mathematics
First up, mathematics notation is a language and it’s most helpful to think of it as such. To make this point clear, take an example from a different language such as English. Say you tell someone, “I saw a man with a telescope.” You might mean that you saw a man carrying a telescope. Or you might mean that you looked through a telescope and saw a man. The fact that your actual meaning is unclear to your listener is not a sign that you are a super smart English genius. It’s a sign that you are a barely literate moron.
Same here. The fact that the meaning of the expression 8÷2(2+2) is unclear to you doesn’t mean that you are stupid. But it does tell you a lot about the people asking the question.
So, why is this expression unclear? A small part of it is that the obelus is ambiguous. Most of the time, a-b÷c-d means a-(b÷c)-d. However, occasionally a-b÷c-d means (a-b)÷(c-d). The second meaning is much less common but not entirely unheard of.
By far though, a much greater source of ambiguity is in the implicit multiplication. This is because most of us use implicit multiplication in an ambiguous way most of the time. Consider the expressions 3/2a and the expression 3a/2. Are they equivalent? Most people would say no. Most people would say that 3/2a evaluates to 1.5/a, while 3a/2 evaluates to 1.5×a. But if you apply strict BODMAS, you’d conclude that 3/2a and 3a/2 both evaluate to 1.5×a, ie they are both the same.
The problem is that *most* people sometimes use implicit multiplication to mean high priority multiplication (ie multiplication that must be done before all other multiplication/division) and sometimes use implicit multiplication to mean regular multiplication that is done in the normal left to right order. What we mean by implicit multiplication is usually clear from the context, but in an instance like this question, there is no context.
As a result, it’s not unreasonable for you to conclude that 8÷2(2+2) means 8÷8. This assumes that the implicit multiplication is being used to mean high priority multiplication which is a reasonable assumption because almost everyone uses implicit multiplication this way *some of the time*.
Equally, it’s not unreasonable for you to conclude that 8÷2(2+2) means 4×4. This assumes that the implicit multiplication is being used to mean regular multiplication which is a reasonable assumption because almost everyone also uses implicit multiplication this way *some of the time*.
It really depends on what the person asking the question thinks the answer is.
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