How an old man’s ravings awakened the mathematician in me
The number problem that won’t go away
Education conferences often serve up quirky exhibitors, speakers and attendees. At an event this week, I had expected to meet with 300 school leaders, all billed serious-minded educators. As it turned out, I met 299 of them. 299 and one crazed old man. A crazed old man who would set me on the course to mathematical obsession.
Initially I thought little of it. As he sauntered past our exhibition booth, there was little to distinguish the quaintly dressed elderly gentleman from the rest of the crowd. But then he spoke. Rather, he bellowed. Having caught sight of our mathematically themed Whizz Education stand, the man proclaimed — with every ounce of strength his lungs could muster — that:
‘Mathematicians are too precise. You know what you should put on your stand? That 49/20 may as well be the square root of 6.’
I scarcely had a moment to register these words before he was on his merry way. A brief moment of intrigue on my part swiftly subsided as I engaged a separate throng of school leaders.
As the day’s proceedings wore down, that brief moment had eluded my thoughts altogether. And then, from somewhere, the man reappeared once more. This time:
‘And you know what else? 17/12 is as good as you need for the square root of 2.’
After letting out the most maniacal of manic laughs, he was gone. This was his parting shot; a mischievous way of bidding me farewell.
If his goal was to arouse my curiosity, it worked. I hastily scribbled two statements, ready to ponder on the journey home:
49/20 ≈ √6
and
17/12 ≈ √2
The three-hour car ride back was intended to be a winding down period. With my colleague Mark at the wheel, I had envisioned an event debrief between the two of us, interspersed with naps between the one of me. That dream died as my mind processed the remnants of a maths problem handed down by this seemingly crazed individual. Was there hidden depth to the old man’s ramblings?
My body was crying for rest, but my mind was buzzing. I set to work.
I knew right away that 49/20 could not possibly equal the square root of 6, and that 17/12 is no way the square root of 2. In each case, the square roots yield irrational numbers (those that can not be expressed as a fraction). This basic insight already gave some meaning to the man’s claims. He was chastising us mathematicians for our precision. In its place, he was inviting the alternative: approximation. So the question became:
In what sense is 49/20 a good approximation of √6 (and 17/12 of √2)?
The mathematician in me — largely dormant since 2011 — was slowly awakening. My first instinct was to bend the rules a little. Let’s suppose these values are in fact equal, even though we’ve confirmed otherwise. Let’s flirt with absurdity just to see where it gets us.
If 49/20 = √6, then squaring both sides gives:
2401/400 = 6
Interesting. 2400/400 is bang on 6. This simple exploration gives some credence to the old man’s approximation. It shows that, subject to a remainder of 1, 49/20 is a pretty good estimate of √6.
Is that what he was getting at? Was the old man’s point to just ignore the remainder 1? Since he had obliged with a second example, I could test-drive this idea further — let’s see what happens in the case of 17/12 and √2:
If 17/12 = √2, then squaring both sides:
289/144 = 2
Again, this statement is true up to that remainder of 1.
Having fun yet? I certainly was. The more I delved into the problem, the more I felt I was getting into the old guy’s head.
Sadly, the man had departed before I even had a chance to get his name. I would have no way of following up. Then again, I was making this problem my own. By now, I was asking sharper questions — questions such as:
How interesting is this property of numbers?
Can we find other such examples? How many?
It was here that I formalised the problem. Moving beyond the specifics of the two example, I asked:
For which positive integers a,b,c does a²/b² differ from c by a remainder of 1?
Some basic algebraic manipulation reduces this statement to the equation:
(a-1) * (a+1) = b² * c
This formulation lands the problem squarely in the domain of number theory. The most interesting question of all is just how interesting this relationship is: how often it occurs, and under what circumstances. Fermat mused over a vaguely similar problem almost 400 years ago; the rest is the stuff of mathematical legend. Our problem seems much narrower in scope; the approximation feels too crude to be of any real significance. Still, it was worth digging.
After a while (two hours, give or take) I retreated from the problem to focus on my two other commitments: the event debrief and the nap. Neither took place because, it transpired, I was not the only one fixated by the problem. My colleague Mark professed a long-term enjoyment of mathematics, curtailed only by the excessive focus of school exams. And now he was my collaborator. The interaction was telling; Mark was able to suggest creative and honest-to-god useful strategies, which combined nicely with my specific knowledge of prime decompositions to deliver a series of mini breakthroughs.
I will spare the reader all the details, except to say that one such mini breakthrough allowed us to generate an infinite supply of such examples. Great though this sounds, it need not mean we found them all. In fact, that procedure did not even recover the example of 17/12. The old man’s logic was not yet in our grasp.
I did not sleep well that night. This was a draining trip; I should have been out like a light. But this problem had given me a second wind. By the next morning, I was obsessively drawn to the same problem, even in the midst of heavy work commitments. I warned my wife I was a recovering mathematician; now she could see first-hand what that meant. Is this what they mean by addiction? I certainly felt helpless as I devoted the entirety of my morning commute to probing the problem some more.
My efforts bore fruit; I eventually figured out how to derive all examples of this peculiar property. More specifically, I showed that:
Given a positive integer a, we can find all pairs of positive integers b and c such that (a-1)*(a+1)=b²*c.
In other words, I found a way to generate all such examples, including both presented to me by the raving old man. Ahhh, delight. Relief. Joy.
But a true mathematician never stops questioning. Maths would be interminably dull otherwise. I have at least two questions yet to be resolved:
· How does the accuracy of this approximation depend on the values of a,b,c?
· For which values c does such an approximation exist (in other words, when can you approximate square roots according to the remainder 1 trick)?
For the latter, I realised quickly that c can not be a square number. Beyond that, I’m just a bit stuck. So I did what any twenty-first mathematician worth their salt does: I Googled it. But Google is no panacea (no, really); I have not yet found a search term that delivers relevant results. Methods for approximating square roots abound; this particular method may be too obscure, or too ordinary, to have made its way online. I just don’t know.
So I’m posting here to see if anyone can help. I hope this article does for you what the old man did for me — awaken the mathematician in you. Perhaps the most intriguing question of all is what went through his mind as he declared his fleeting proclamations. We may never know; I’ll sure be looking out for him at future events.
If nothing else, the experience reminds me that my inner mathematician is always lurking. I feel a sense of vulnerability towards knowing that it can come out at any moment, directing my emotions and distracting me from sleep, work and marriage. And yet that is also where the joy of mathematics resides — in suspending our worldly sense of what is important and yielding to the lure of a problem, driven only by curiosity and the innate desire to discover new truths.
