By Clive Newstead on 12th May 2011
I recently witnessed a discussion about what the true value of 6÷2(1+2) is, with some arguing that it is equal to 1 and some arguing that it is equal to 9. My goal with this article is to convince the reader that either could be true but neither is true without more context.
Before we start, we need to know how to differentiate between mathematics and notation. Mathematics is the study of quantity, structure, space and change; notation is what we use to make maths understandable. Mathematical rules are ones like "an even number plus an odd number is an odd number," and notational rules are ones like BIDMAS, an acronym standing for 'brackets, indices, division & multiplication, addition & subtraction'. That is, mathematical rules are logical consequences of the axioms we use to define abstract objects such as numbers; whereas notational rules are rules that we impose to have a consistent and clear way of communicating what happens in the mathematical world.
My claim is this: 6÷2(1+2) is ambiguously written and cannot be evaluated without further context. The rest of the article shall be my justification of this point.
BIDMAS is a rule taught in secondary school to pupils learning how to evaluate expressions like 12×5+3-4×(3+2)2. To clarify a point, it gives division and multiplication equal preference, and addition and subtraction equal preference (in fact, some people call it PEMDAS or other such abbreviations). According to these rules, the bracket 1+2 must be evaluated first, meaning we have 6÷2(3). This is where the problem lies; do we evaluate 2×3 next, or 6÷2?
This is an ambiguity that BIDMAS cannot clear up, because we do not know what 3 is multiplying. Is it multiplying the 2 on the denominator of the fraction, or is it multiplying 6/2 as a single entity? This is where most people begin to argue. Some say that in the face of 6÷2×3 we must proceed from left to right, meaning we divide 6 by 2 first, and then multiply what we get by 3. This group of people form the '9 camp', as it tells us that 6÷2(1+2)=9. Others say that the notation clearly indicates that the (1+2) bracket is on the denominator of the fraction, so we must multiply the 3 by 2 and then divide 6 by this. This group of people form the '1 camp', as it tells us that 6÷2(1+2)=1.
Sadly, neither of the reasons used to get either of these answers are justified. The left-to-right rule is merely a (non-standard) notational rule used to make 13 year olds ask less questions when presented with something like 2×3×5 to evaluate, and those who claim that (1+2) is 'obviously on the denominator' are mistaken -- if it were obvious, this debate wouldn't be happening. Claiming that if it were on the numerator there would be a × symbol there, as many do claim, is an arbitrary rule and is certainly non-standard; we simply can't know this without more context.
The truth of the matter is that we are missing a set of brackets. Specifically, either \[ 6 \div 2(1+2) = (6 \div 2)(1+2) = \frac{6}{2}(1+2) = 9 \]
Computers (including calculators, Google, Wolfram Alpha, etc) operate in the way they were programmed. Computers can't usually deal with ambiguity, and so the options are either to present "ERROR" or to interpret it in one way or the other. Usually, they do the latter, especially because most computers aren't programmed thoroughly enough to detect ambiguous statements. If a computer interprets an ambiguous statement in one particular way, it doesn't mean that way is right, it just means that it defaults to a particular interpretation of the statement as dictated by its programmer. This is why some calculators say 1 and some calculators say 9.
Well no it doesn't. If write "2×6+÷4-" it means nothing; and indeed, just because something is written using mathsy things like + symbols, numbers and letters, doesn't mean that it evaluates to anything. In the 2×6+÷4- example it is obvious that the expression is meaningless, but just because 6÷2(1+2) appears coherent does not mean that it is coherent. (Don't even get me started on what happens when you divide zero by zero!) This has happened because the evaluation of 6÷2(1+2) is a two-stage problem. The first stage is notational: to convert the notation into a sequence of mathematical operations; and the second step is mathematical: to perform that sequence of operations. The notational stage is where the problem lies, and because it is notational rather than mathematical, it does not require a definite answer.
I often hear people say "maths is a language" or even that it is the "universal language". I disagree; mathematics is a study, and it is conveyed using mathematical notation. In some sense mathematical notation is a language, but it is only universal because it has the rare property of transcending international borders. Languages are normally used to make your thoughts clear and understandable and to understand clearly another person's thoughts. This is often what mathematical notation does: for example, because we have numbers, it is suddenly clear that if you have three apples in a bag and you put two more apples in, and if you have three oranges in another bag and you put two more oranges in, then each bag has the same number of pieces of fruit in it. That is, the problem of counting apples and the problem of counting oranges are equivalent problems: counting! Without maths, such connections are not obvious -- or at least, they aren't always. For example, without maths we would never have discovered that gravity and electric charge are governed by (almost) exactly the same equations, so properties of one can be transferred to the other.
However, in many cases, mathematical notation blurs the reality and, as discovered time and time again in history, the invention of an ingenious new notation (e.g. representing linear transformations by matrices) can make the study of abstract objects much more easily accessible -- what this means is that many incredibly simple problems are still intractable because we have not yet discovered the right notation under which to study them. (I would argue that this may well be the case in knot theory.)
My point is that we need notational clarity in order to understand maths. Notation serves to make our lives easier and tells us how we can convey what we mean mathematically in a clear way. But writing 6÷2(1+2) does not do this. Yes, we can sit around and invent 134 new notational rules to interpret ambiguous statements, but it boils down to this: what is written is not always what is meant. For example, if I were to write 1+x / 3+x I would expect it to be interpreted as $\frac{1+x}{3+x}$, despite BODMAS dictating otherwise. What I wrote is ambiguous, but it would probably be clear from the context what I meant, in which case notational clarity is achieved despite rigid rules being disobeyed. By contrast, writing 6÷2(1+2) with no further context is just bad maths, for all of the above reasons.
If you have any comments, corrections or queries, feel free to send me an email.