Square Root Symbol

Beating the square root out of a radical sign with the number one

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.

What exactly is the square root of 36? » Mathematics for Teaching

Mathematics is already a difficult subject and most times we make it more difficult to students to make sense of with the confusing rules we tell them. For example, we insist that the square root of 36 is 6 and -6 yet we also insist that √36 is equal to 6 and not -6. A colleague said that by convention, when you ask ‘what is the  square root of 36?’ the answer is 6 and -6  but when you just write the symbol ‘√36 =?’ the answer must be 6. So, I said,  do you mean that if you just want the positive root, you do not read the symbol √36 otherwise it will have 2 values? Here’s an excerpt from an article I read recently about the matter. It’s title is “What root do you want to take?” by Derek Ball.

Anyway, solving quadratics is where things like √36 really come into their own. If you want to solve x (√4))/2. Now new questions arise. Am I allowed to write √(-4) and if I am what is its value? Is it 2i or -2i and why? Perhaps you want to say the answer is obviously 2i. In that case how about a quadratic equation whose solution involves √(3 – 4i). Is this – 2 + i or is it 2 – i? What I am saying is that we use symbols to help us solve problems. If we use in front of a square root sign this reminds us that in order to solve the quadratic equation completely we need to remember to take two different values for the square root. Knowing that √36 means 6 and not -6 does not matter at all, unless we are asked silly questions in pub quizzes or GCSE exams like ‘What is √36?’ and we are supposed (for some unknown reason) to know that we have to answer 6 and not -6.

Still not convinced? Well, am I allowed to write √-27 and is its value allowed to be -3? Its value could be other things too, of course. Does that complicate things? Does that make us think that we can only judge what a symbol means from the context? So, far be it from me to defend textbooks, but perhaps they have some justification for using the square root sign inconsistently. As for fractional powers, they seem to raise exactly the same issues as root signs.The moral of this tale is surely that we move away from asking questions towhich we want a correct answer, so thatwe can say ‘Right’ or ‘Wrong’, and instead solve problems that interest us, talk about mathematics and connect ideas together. And I hope we sometimes get confused, because confusion is often a spur to sorting our ideas out.

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