Posted by: Alexandre Borovik | March 31, 2009

What is the difference between addition and subtraction?

One more fascinating childhood story, from BS.

I respond to your request for childhood stories about learning mathematics. I don’t think my story is at all notable, but I take it you want to collect as much data as possible.

Two episodes

Ag e= 6 Year = 1948
Gender= male
Language = english, my mother-tongue
Final produc t =  doctorate in mathematics and a lifetime teaching maths in university setting

(A)  Each morning in class we spent about 90 minutes on arithmetic, and were usually issued with postcards containing three additions (of 3 two or three digit numbers) and three subtractions (of three digit numbers). I never got to the subtractions, being too slow: but finally worked out that they must be easier, as they involved only two numbers. So I started doing the three subtractions first; of course I got them all wrong because I just added the two numbers. I hadn’t realised it was a different sort of “sum”.

(B)  About three months after the above, I’d mastered the algorithms for addition and subtraction in simple cases.  We progressed to subtractions of the sort 72-55.  The teacher asked  “Now  if we try to take  5 from 2 we can’t, so what can we do? Does anyone know ?” I offered “Take the 2 from the 5”, and was not pleased to be told that wasn’t right. I was very indignant, even after being told the “correct” process; mine was shorter, easier and gave an answer of the required sort!

Comment: I had no idea that “Addition” and “Subtraction” were other than formal algorithmic processes to write down an answer that was “right”, really no idea at all that there was “applied arithmetic” so as to speak.

Further comment: I don’t think that I got over this difficulty until very late in my maths education. I am pretty sure that as a third year student at university I still found it difficult to disentangle conceptually numbers and numerals, addition/multiplication from  the usual algorithms to compute them. Clearly that’s got a lot to do with the way I learned arithmetic.

My brief response: these episodes are yet another confirmation of what increasingly appear to be a general principle: every error made by children because they thought that “it was natural to do it that way” is developed in the “grown up” mathematics into a serious theory. In your case, non-negative integers with a 2-valued binary operation

(x,y) –> [x+y, |x-y|]

(where [] denotes a multiset, that is, an unordered set with possibly repeated elements) is one of the basic examples of the so-called n-valued groups in the sense of Novikov and Buchstaber. The origins of their theory are in algebraic topology.

I do not know why this is happening; maybe mathematics is rich enough to contain an analogue of everything—or maybe there are some intrinsic reasons when every “simple and natural” alternative to “canonical” mathematics generates a rich mathematical theory.

And the final mystery for me is the selectivity of memory: why do mathematicians remember these particular episodes from their childhood?


  1. I remember being indignant at the age of around 8 when to the problem

    x - 2 = 3

    I answered

    x = 2 + 3 = 5

    and it was marked wrong. Presumably the teacher wanted to train us to do the same to both sides of the equation, i.e., add 2.

    I imagine I was thinking of subtraction as used in something like ’21 – 18′. Count on from 18 to 21. Answer 3. So 21 – 18 = 3, because 21 = 18 + 3.

    My teacher was forcing me down the route useful for subtraction such as ’21 – 3′. Count back 3 from 21. So 21 – 3 = 18, because 21 = 18 + 3.

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