Me, a genius at five

Can you compute square roots by heart ? How fast, and with what precision ? Would you be surprised if your colleague could take five seconds to tell you that the square root of 123,456 is 351.35, give or take a hundredth ? Probably not much. But what if your colleague’s five-year-old kid did ?

Marcoscan ( http://www.marcoscan.com ) in a recent post was quoting from the hilarious biography “Surely you’re joking, Mr. Feynman!”, where the matter of computing squares by heart is discussed. I thus recalled I used to be good at it as a kid (well, I always have since then). I left a comment to his post there, mentioning the fact, but his reply forced me to recall harder.

He basically asked: So, ok. You could do square roots by heart pretty easily as a five years old child. But exactly, how did you learn to do that ?

I remember (or I remember my mother told me) I was attracted to numbers and to counting as soon as I learned about their existence, when I was three – and I have no clue why. But I believe it wasn’t until I saw my older brothers struggling on their homework to compute square roots that my curiosity really caught fire.

Square roots! What a marvelous invention for a child of five. I knew how to add, and I also knew how to do multiplications on paper – something else I had learned directly from observing my brothers. But square roots fascinated me as soon as I understood what it was about.

Square roots to me were like a piece of magic. Tell me the result, and I will tell you the question. 25 ? That’s 5 times itself. Easy.

Most kids my age would have stopped there. I think I did not because I somehow found in the numbers an escape from the anxiousness caused by my parents’ imploding marriage. And a smart kid with a lot of free time can work out a series of facts by himself if he has enough stimuli.

First, one starts realizing that if five times five is 25, and if 50 times 50 is 2500, then one easily guesses what is 80 times 80 if 8 times 8 is 64. And from there, another small steps leads to mastering infinitely large numbers! 500 times 500 is surely 250000… Mastering zeroes is a child trick, indeed, but what a giant step in the understanding of integers!

And then it was soon clear to me that the first thing to do in order to compute a square root was to determine whether the number of digits was even or odd. Even = look at the first two digits for a start. Odd = look at the first three digits. So, if I had to compute the root of 2345, I would start by 23, which is little less than 5, and I would guess that the result was a little less than 50. If instead the question was sqrt(23456), I would look at 234 and, remembering 225 as the perfect square of 15, would guess that the result would be 150 plus something.

Of course, in the meantime I was indeed learning perfect squares by heart. The trick was to get to know at least one every four or five up to 100. That will do. For instance, if you know that 20 times 20 is 400, and you know that 30 times 30 is 900, but you do not remember any in between, you will have a hard time computing the square root of 666. But if you remember 20, 25, and 30, it’s already much easier.

Refinements came later, when I started to be able to really do quadratic interpolations. But it was a natural development of a continuing curiosity in the matter. And I became an attraction to my family: an anecdote everybody remembers is my uncle Ciccio, an engineer of some success, picking me up and asking me: “Tommaso, what is the square root of 9103 ?”, when I stare at the ceiling for five seconds, then say “it’s 95.4”. To which, my uncle puts me back on the floor, rushes to his brand new calculator bought in Japan, and people hears a profanity in the background as he finds out I am right.

I would no doubt have become a problem child if my parents had pushed me into developing further my ability, studying more and more, and giving up the normal activity of a small child. I must give credit to my mother for that: she played it down softly, and by the time I was a teenager, I was (almost) normal again.