Monday, April 30, 2012

6 ÷ 2 (1+2) = ?

Personal Note: It's been more than a year since I last wrote on this blog, and let's just begin by stating that, YES I AM ALIVE. It's been an interesting fifteen months, where I transformed from a sheepish new intern with a quavering voice while saying "Hi it's Yang one of the... doctors" to a more confident doctor who is still humbled everyday by the nature of the amazing job. Oh and I also got engaged in the process. :)

You have seen it before. Yes, this is a reincarnation of the infamous 48/2(9+3). 

If you have not seen it, take a few seconds to work out the answer. In fact, even if you have seen it, try to solve the equation again in your head. 

Now that you have got it, let's check the answer. The "correct" answers are 288 and 9 respectively. Now, I will explain the quotation marks in a second.

When the original 48/2(9+3) question was released into the World Wide Web, it cooked up a storm as people hotly debated whether the answer should be 288 or 2. The "correct" answer is derived based on the strict interpretation of the "BODMAS" rule, which stands for Brackets, Order (Exponent), Division and Multiplication, Addition and Subtraction. Based on this rule, the operation should be
48 / 2 (9+3)
= 48 / 2 (12)
= 24 (12)
= 288
On the other hand, the proponent of the answer of 2 works it out this way:
48 / 2 (9+3)
= 48 / 2 (12)
= 48 / 24
= 2
Evidences have been thrown about in support of each argument. WolframAlpha and Google's default calculators both give the "correct" answer of 288. [1][2] However, different scientific calculators give either versions of the answer depending on brands and models. 

So which answer do I think is correct? I think the first one is technically correct, but the second one is not wrong either. The biggest mistake is in fact the person who wrote such an ambiguous expression in the first place. 

Before we go any further, let me introduce you to this video by vihart (starting from 2:32)

"I would like some juice or water with ice - do you mean you want either juice with no ice or water with ice, or do you mean you want either juice with ice, or water with ice?"

Essentially this argument is pointless. It detracts from the true spirit of mathematics which is to derive and discern fascinating pattern and relationship in nature based on a set of axioms. All this argument does is to delve into syntax which evolved arbitrarily in the evolution of mathematical notation  - it has NOTHING to do with whether the maths is right or wrong.

The conflict comes from the fact that "BODMAS" is taught in primary schools when we still use the sign "X" to mean multiplication, and the sign ÷ to mean division, and if you wrote out this equation 6 ÷ 2 X (1 + 2), then no one would have gotten it "wrong" based on the simple BODMAS rule. 

However, as we progress in our mathematical education, X is replaced by simply having two entities written next to each other, and ÷ is replaced by writing out the expression as a fraction. Because of this, when first presented with this writing of 6 ÷ 2(1+2) [also note the very intentionally misleading spacing in the original photo], the intuition in anyone who have become familiar with the advanced mathematics notation would automatically translate this into:
especially due to our natural instinct of grouping the multiplication together when a bracket is involved. So this is where the mistake came from. 

Just throwing a last question here to illustrate my point: what is 1/2x? Is it half of x, or is it the inverse of 2x? The answer is, it is neither, it's just a poorly constructed mathematical expression, and the debate on semantics is just a waste of time.