Beating the square root out of a radical sign with the number one
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 I was attacking, 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.”
Let's take as our challenge obtaining the square root of 22. Twenty two is an awkward number because it is a bit distant from the closest perfect square, 16, and will require more calculation to get it right than say, the number 17.
We first determine the closest integral value not exceeding 22 which is itself the square of an integer. We notice that 42 is 16, but 52 is 25, exceeding 22, so we choose 16. We rewrite our value as Ö 16/16 x 22. In other words, all we have effectively done is to multiply the number by a value equal to one, which, by our elementary mathematics, does not change the value of the number being multiplied.
Now we do a very clever and tricky thing. We rewrite the number as follows: Ö (16 x 22/16). I think most of us will admit that this is allowed. But we remember that the square root of the product of two numbers is equal to the product of the square root of the numbers. For example, Ö (4 x 25) = Ö 4 x Ö 25. Let's see if this is true.
4 x 25 = 100; Ö 100 = 10. Or we can individually find the square roots of 4 and 25, and multiply the result.
Ö 4 = 2; Ö 25 = 5; therefore: Ö 4 x Ö 25 = 2 x 5 =10.
Having illustrated the theorem, we will apply it to transform our expression into an equivalent one, the square root of sixteen times the square root of 22/16. The first of these we know to be 4, so we are left with the expression
Ö (16x 22/16 = Ö 16 x Ö 22/16 = 4 Ö 22/16.
Our next step is to turn our fractional expression into a decimal, using the division indicated. The decimal expression for 22/16 happens to be 1.375.
Our expression now has the form 4 Ö 1.375. We then apply the same method to this number under the square root sign that we did for "22." We find that (1.1)2 = 1.21, but that (1.2)2 = 1.44, which is larger than the number under the square root sign. So, as our plan is to approach from a lower values, thus guaranteeing a positive remainder upon which to successively perform our operations, we use the first value. We replace Ö 1.375 with Ö 1.21x1.375/1.21. We of course know the square root of 1.21, since we just calculated it, so we merely write it outside the “Ö ” sign, and are left with the following expression: 4x1.1 Ö (1.375/1.21). We carry out the division indicated on the inside of the radical sign, to yield a decimal number, which is 1.1363636...an infinitely repeating decimal. We rewrite our expression as
4 x 1.1 x Ö 1.13636.....
Now we find by trial and error, more or less, that the square of 1.06 = 1.1236, whereas the square of 1.07 is 1.1449, the first less than our value, the second exceeding our value of 1.136363636...... So we rewrite our expression once more, this time as 4x1.1x Ö (1.1236 x 1.1363636.../1.1236). We take the square root of the first number under the radical sign, which we already know to be 1.06.and rewrite our expression as
4 x 1.1 x 1.06 x Ö 1.13636.../1.1236.
Doing the indicated division yields about 1.0113604. If we square 1.005, we get 1.01025, while squaring 1.006 yields 1.012036. The first being less than the number for which we seek the square, and the second being greater, we choose the first. Thus, we rewrite our expression 4x1.1x1.06 Ö (1.01025 x 1.0113604/1.01025) =
4x1.1x1.06x1.005 Ö 1.0113604/1.01025.
We’re almost through. We’ll just do this one more time. Carrying out the division under the radical sign yields approximately 1.0013.... Now 1.00062 = 1.00120036, while 1.00072 = 1.00140049. The second exceeds the number we want to find the square root of, while the first is less than it, so we choose the first. Finally, we have
4 x 1.1 x 1.06 x 1.005 x 1.0006 Ö (1.0013/1.00120036).
Ok, now that we are getting the hang of this, we’ll do the last as an approximation. Finally, 1.0013/1.0012 = about 1.00009. The number which when squared approximates this is 1.00004, giving us 1.00008. It ought to be apparent that squaring 0.00005 yields approximately 1.00010, which exceeds what we are looking for.
So we just bring the former quantity out of the radical sign, and, neglecting the contribution from what is within the radical sign at this point, which becomes less and less for each calculation anyway (closer to 1, that is), we shall leave our approximation at
4x1.1 x 1.06 x 1.005 x 1.0006 x 1.00004 = 4.69031...
Now if we use our calculator to find the square root of 22, we get 4.69041... The error is about 1/50,000.
This same method can be used to find the cube root of a number, but the calculations become more burdensome. We could have made life a lot easier for the reader by choosing a number closer to a perfect square, such as 17, or 5, for example. Try it with other numbers, see how it works out. Enjoy!
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