I keep seeing questions that say 9÷3(4-1) or something similar of the form a÷b(c-d). Everyone seems to assume that there is an answer to this question, but the use of deliberately ambiguous notation means there is not!
From the example above I could mean:
\[\frac{9}{3(4-1)} = 1\]
or
\[\Big(\frac{9}{3}\Big)(4-1) = 9\]
All of you who have done GCSE maths should be familiar with: BODMAS, BIDMAS, BEDMAS, take your pick there are a lot of mnemonics. These are the orders of operation, which are:
- Brackets first
- Indices (powers and roots) second
- Multiplication and division third
- Addition and subtraction last
But what people seem to forget is that all operations of each order are performed simultaneously, not from left to right. So questions like this have no answer, without more explicit notation they are meaningless.
What your calculator says is wrong...
What Google says is wrong...
Each of these will have had to interpret your ambiguous notation and will have either implicitly but brackets around one expression or have worked from left to right as it was probably programmed to do when notation was not explicit. This does not mean the answer it returned was right!
To summarise: the whole purpose of notation is make communication clear and explicit. Writing mathematical expressions such as the above is like removing all verbs from a sentence, you can attempt to guess at what it means. But different people will interpret the sentence differently, as with the above expression.
Try interpreting the following sentence, I doubt everyone will come up with the same answer. I've even been kind enough to show you where the verbs that I removed were.
"Paradoxically ____ may occur with an immediate increase in ____ after ____."
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