A page from a 19th Century math textbook by Peter Duveen
PETERS NEW YORK, Tuesday, June 14, 2011--Willard Gibbs (1839-1903) was a towering figure in late 19th Century physics. He was responsible for promoting a form of vector calculus that is close to the one taught in schools today. Even more importantly, he introduced revolutionary concepts and made new discoveries in the fields of thermodynamics and its sister science, statistical mechanics. But none of these accomplishments prevented him from falling victim to a rather glaring deficiency.
According to an account in a biography by Muriel Rukeyser, a young man who later became a prominent banker was present during a consultation between Gibbs and a real estate agent. Gibbs, it is said, was keen to figure out the diagonal breadth of a property from the given rectangular boundaries. Beginning with the Pythagorean theorem, he got as far as adding the squares of the two sides of the property, but his calculations had to stop there. We are told, quite shockingly, that he "confessed that he had forgotten the rule for finding the square root!"
The rule? "Method" might be a better word. A forgivable oversight, considering the person who used the term "rule" was merely a banker, and there is no indication from the account that he was a savvy mathematician. But Gibbs? Unable to compute a square root?
In my occasional role as high school teacher, I recently introduced to math students over a two-day period, an algorithm I developed for finding the square root of a number. These days students rely heavily on electronic devices to perform calculations, but most have at least the rudimentary ability to add, subtract, multiply and divide without this technology.
I have always viewed square roots as mysterious. The number for which the square root is to be determined is trapped under a symbol, and resists being coaxed from its cage into a more commensurable form. Take the square root of two, for example. We are often told that we do not need to simplify it further. It must remain caged, lest it escape and awaken the power of one's critical thinking. Do not play with it. Do not attempt to simplify or understand the number. It was thus that square roots have maintained in my psyche a rather mystical and untouchable status.
Recently I had the opportunity to challenge this peculiarity of square roots. I had been introduced to an algorithm that allows one to calculate the logarithm of a number--the power that the number ten must be raised to, to yield that number. It was a primitive algorithm requiring some reworking to make it fully transparent, but useful mainly for demonstration purposes. It got me to thinking that a method might also be found to free the square root of a number from its little typographic prison.
After much trial and error, accompanied by the use of an inordinate quantity of paper and ink, I stumbled upon a method that I have not seen used before. It is simple and transparent, and when applied thrice, generally yields results to within a few thousandths of the value generated by electronic devices. Surely this is revisiting old territory that dates as far back as the ancient Egypt and Babylon, Assyria. But these days our educational system does not provide such knowledge, I am afraid to say, and not merely because of the advent of electronic devices. Such a method of calculation seemed not to be part of the syllabi at least since I was first introduced to the subject in the 1950s. And while methods for extracting square roots are commonly provided in mathematics texts dating to the middle of the 19th Century, complete versions of which may be found on the internet or in local used book stores, I find these presentations somewhat abstruse. In other words, I have failed to master them.
After having developed my own approach, I mentioned to a fellow teacher my intention to share this method with my students. The teacher warned me that students were only interested in using calculators, and that I would be wasting my time. I naturally ignored what was probably a well-founded supposition, preferring instead to test the waters myself.
I began the Algebra-Trigonometry class that I was to host for a couple of days by asking students to pick any number between two and one hundred. Making no promises, I told them that, over the class period, I would attempt to find the square root of the number they gave me. I was reminded by my circumstances of the stage magician The Amazing Randi, now better known for his penchant for "debunking" what he deems false theories or tricks. But back in the 1960s, he was acclaimed for dazzling feats such as being hoisted by a crane into mid air while fully bound in a straight jacket, handcuffs, etc., and managing an escape during a live televised appearance. Here I boasted that I could release any number from two to one hundred, from its mathematically secure niche under the radical sign. My performance would be scrutinized. Could I pull it off?
As the class proceeded along the usual lines, I would go back from time to time to the problem, and eventually found a number that was quite close to the values established by the electronic calculators available to us. I repeated the performance, asking once again for the students to provide me with a number the square root of which we were to seek. But from time to time, as numbers crowded the blackboard, I relied on a a student with a calculator to perform some of the challenging divisions and multiplications the method employs.
On the second day of our class, filled with new resolve, I vowed to make no use of a calculator whatsoever, but to find the square root of the number by the power of slate and chalk alone. This being a side issue to their review for a major state exam, not every student was fully engaged in this auxiliary task, but neither did I encounter much opposition or disinterest. All appeared to be aware of the problem at hand, and evinced some curiosity. They were, of course, participants, as they had provided the number that became the subject of our calculations, so they knew my performance could not be from wrote, but rather would depend upon the principles applied. They also helped by performing calculations on their hand-held devices when it became necessary. Finally, with a click clack of the chalk, a number emerged that was within a few thousandths of the calculator-generated value.
I have to say that these students expressed a muted but sincere amazement as the bell rang and they went on to their next classes. Indeed, they had assigned their best mathematician to review the method I was using, and this young man gave his assent to the results produced as having been established in a bona fide manner.
Now why is it important for students, or any of us, for that matter, to have a simple way to find a square root? As far back as our learning the Pythagorean theorem, square roots have held an important place in our minds. To find the diagonal length of a rectangle, as Gibbs attempted to do, when the squares of the sides do not add up to a perfect square becomes an important problem. If students are not shown that it is solvable, they may end up, like me, assigning a sort of mystical aura to the numbers under the square root sign. When actual values are required, it is usually a matter of referring to a table or a calculator, but to actually calculate the values the way one does in, say, dividing one number by another, is an art that has fallen by the wayside. The journalist in me longs to reveal to the public, even if that public consists of a mere handful of students, something it may not have known about.
Are you are waiting with bated breath to find out what my method is for finding square roots? Let me share it with you. The official name of it is "Beating the square root out of a radical sign with the number one." Because some websites cannot reproduce a square root symbol, we shall spell everything out, but for a more detailed analysis with radical signs, please go to Beating a square root out of a radical sign with the number 1.
First, find the perfect square that is closest to, but that does not exceed, the number under the radical sign. For the number 5, for example, the number 4 would be the closest perfect square, 4 being the square of 2. Then simply multiply and divide the number under the radical sign by that closest perfect square, then take the perfect square out of the radical sign, where it becomes its square root. Thus, Square root 5 = square root 4 x 5/4 = 2 square root 5/4. If you continually repeat this same process to the number remaining under the radical sign, you will gradually evolve a better and better approximation of the square root of your original number, the approximation being equal to the numbers outside the radical sign and the operations that connect them.
Let's continue a bit. 2 x square root of 5/4 = 2 x square root 1.25. The perfect square closest to, but not exceeding, 1.25, is 1.21, which is 1.1 squared. 1.2 squared, or 1.44, exceeds the value of our number, so we don't want to use that. Therefore, 2 x square root 1.25 = 2 x square root 1.21 x 1.25/1.21 = 2 x 1.1 x square root 1.25/1.21. Do you want to try once more? 1.25/1.21 = approximately 1.033. 1.01 x 1.01 = 1.0201, whereas 1.02 x 1.02 = 1.0404. The second number is larger than the one under the radical sign, so we'll use the first, 1.0201. So we have 2 x 1.1 x square root 1.0201 x 1.033/1.0201. That of course is equal to 2 x 1.1 x 1.01 x square root 1.033/1.0201. Let's multiply the numbers outside the radical sign as indicated. Our approximation up to this point is 2.222, whereas the calculator value is 2.236... Ok, still not convinced? Let's try one more time. We'll make life simple by approximating, and divide 1.033 by 1.02, which yields 1.011 to a sufficient approximation for our purposes. 1.005 x 1.005 = 1.010025. That's close enough to 1.01, so our number taken outside the radical sign is 1.005. In other words, square root 1.011 = square root 1.01 x 1.011/1.01 = 1.005 x square root 1.011/1.01. Our final approximation is now 2 x 1.1 x 1.01 x 1.005 = 2.233110. You can see we are getting increasingly closer to the calculator value of 2.236... for each time we apply the method. We started with 2, then 2.2, then 2.222, and now 2.233... For our final approximation, the difference between it and the calculator value is about .003. The percentage difference is 3/2236 = about 2/1,000ths. Of course, one must judiciously choose the sort of shortcut approximations one uses, because it will affect the accuracy of the results.
Keep repeating this process, and the numbers outside the radical sign, and the operations that connect them, will more and more closely approximate the value of the square root of the number you started out with. Try it out yourself. Enjoy!