1 + 1 = ?

1 + 1 = ?
By 6 commentsCategory: MathematicsTags: , , , ,
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What is 1+1 ? Isn’t that the simplest no-brainer Math?

Not exactly… Actually, the answer to this is (as with most of life’s questions of consequence) – ‘it depends’!

1+1 is 2, only in certain ‘contexts’. There are many other ‘contexts’ (alternate worlds?), where 1+1 is not 2. It all depends on what the two 1’s represent and what ‘adding’ really means. Let us examine this…

The most original treatise on this that I have come across, is in the famous Malayalam novel ‘Balyakala Sakhi’ (‘childhood sweetheart’) by Basheer (Basheer – 1944).

In the story, when the teacher asks the protagonist first-grader the answer for ‘1+1’, the boy thinks thus : “when two rivers merge, what would you get”? “A ‘slightly bigger’ river. So naturally, 1+1 is ’a slightly bigger 1′.

The author probably meant it (and is also often quoted) as a classic example of how ‘dumb’ souls misunderstand scholarly concepts. But with due respect to the veteran writer, I would say the boy’s thinking was brilliant! That is precisely what 1+1 is, but in that particular ‘context’.

Now, consider two Vectors in Vector Algebra.  When you add them, you get another Vector, typically longer – yes, a ‘bigger’ Vector!  Does this not look like ’a slightly bigger 1’? 

What are some other such ‘contexts’?

Let us talk about the world of Probability. The ‘Probability value’ of an event is a number between 0 and 1 and it represents the chance that the event occurs. A  value of ‘1’ means that the event is ‘certain‘ to occur (and a value 0 means it is certain not to occur). ‘Adding’ the probability values of two events would give you the ‘chance’ that ‘at least one of the two’ events occurs. So now, what does ‘1+1’ mean in this context? It would represent the chance that ‘at least one of two certain events occurs’ – which is also obviously ‘certain’.

Thus 1 + 1 = 1

Let us make it a bit more earthly. Take 1 kilogram of pebbles in a cup and 1 kilogram of beach sand in another (picture). What would you get if you mix (add) them? Well, 2 kilograms.. So finally we have 2 for a 1+1 (I see you are relieved!).

But wait – let us make it 1 cup of pebbles and 1 cup of sand. Now if you add them – what do we have? Well, definitely not 2 cups of the mixture! Depending on the size of the pebbles, you can have as little as just 1 cup. So again, it depends!

Finally –  suppose there is one chicken sitting in a room. Now imagine one fox being ‘added’ to the room. What would be the most likely result of this ‘1+1’?
(Pardon the Schrodinger-ish grim scenario!)

Can you think of some other ‘contexts’ where there are other interesting answers to 1 + 1 ?

Bibliography

Basheer, Vaikom Muhammad. Balyakalasakhi., DC Books, Kottayam, India, 1944, https://dcbookstore.com/books/balyakalasakhi

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About the Author

Sankar is the founder and CEO of Silver Pi.
He had worked in Banking and Technology for about three decades before switching to Education. He is an Electrical Engineer by training, with a career in Technology and Banking across Japan, USA, Singapore and India.

Sankar's interests include Holistic Learning, Technology-enabled learning, Mathematics, Science and Scientific approach, Financial Literacy, Poetry, Cultural travel, Folklore & Folklife, Japanese language & culture etc.

6 comments

  1. Ramachandran - Reply

    1+1=2 is not always true in politics as well.The merging or two political parties may not always resul t in the exact addition of their vote shares,instead it may result in a loss .

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    • Sankar M - Reply

      True that!
      In fact, most of the things in life do not follow that 1+1=2 logic. But we forget that and expect and behave that way!

      From an academic point of view, our students are not taught that 1+1 is context-specific. All the cases other than a few are taught as exceptions. With too many such exceptions, they get confused and disillusioned..

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  2. Sreejith KT - Reply

    Interesting line of thought. Putting this into my familiar field of process engineering, putting 2 poeple together (1+1) to work on a task wouldn’t necessarily yield a 2 person output but slightly more than 1, unless they are put in context/framework of standard work with defined job responsibilities, tools and procedures. It can also happen that a suitable framework can yield 1+1>2 in terms of outputs per man-hour.

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    • Sankar M - Reply

      Thanks Sreejith!
      You are right – I missed to mention the whole set of cases where the sum of the parts is greater than the whole. In these cases, the addition typically represents an organic and ‘virtuous’ process.
      Again, the meaning of ‘1+1’ depends on how you define the two 1’s (the factors) and also the process of ‘addition’.

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  3. Sabarinathan Ganapathisubramani - Reply

    As a teacher in a management school what I find most worrying is the sense of uniqueness students of STEM to look for in answers to every question in the world. The idea that there can be one right answer to every problem in the world. 1+1=2 is a manifestation of that expectation. The irony is that in business life for sure and in much of one’s personal life there are more than one right answers, the correctness of an answer being so specific to the whole of the context in which the answer is being provided. To the average engineer or student of STEM who enters a business that is a highly unsettling proposition. While rigour in thinking is essential it is not the same as inability to provide multiple answers to a question. On the contrary the more the rigorous the thinking the more one realises that there can and will be more than one right answer.

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    • Sankar M - Reply

      Yes. I think you have  articulated it in a more broader and clearer way.
      All such ‘formulae’ are originally devised with a clear set of underlying assumptions and in ‘a context’. But over time, we lose track of that context and start looking at (and more importantly propagate) them as absolute truths. Thus we end up juggling many ‘exceptions’ and apparent contradictions within what we learn. All of this poke decisive holes in our confidence about our learning.

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