Initially we think of the square root symbol as meaning the “positive” square root of a positive number. But, what does it mean to be a “positive” square root of a negative or complex number?

Tanton has some interesting commentary on this. In particular the breakdown of the usual square root rules. Something along the lines of: -1 = i x i = root(-1) x root(-1) = root(-1 x -1) = 1.

]]>About a century ago, due to several crises in the foundation of mathematics brought on mainly due to the much earlier invention of the differential calculus, it was decided that all mathematics should be derived from logic, essentially making mathematics a branch of logic. Russell and Whitehead produced a huge body of work attempting this feat. Later, through the work of people like Godel, it was shown that such an attempt would fail. Since then mathematicians and logicians have largely inhabited different worlds. This is not the whole story, however.

Another thread in this story is the mere fact that problems in mathematics are often not posed in an ideal manner. Only when the solution is obtained do we realize what we really should have been asking. As well, methods that avoid things like roots of negatives or other unpleasantries, are far more cumbersome than necessary – using quick and dirty tricks will get you the desired answer, at a ‘smaller’ cost.

Mathematical problems do not fall out of the sky, to be solved exactly as given. They are formulated for a specific purpose, and often formulated badly at first. A good student doesn’t just solve the problem as given, but questions why and how the problem is asked, if necessary, even re-formulating the problem. Many teachers are too cynical to give students credit for thinking in such innovative ways.

It is very important to remember that once a proof of a formula is obtained, the mathematician’s work is not done. I won’t go into details here, but recently I saw a proof of a formula by mathematical induction. Obviously, the formula is then only valid for positive integers. Not so! By using the ‘finitude’ of the formula, and the ‘infinite presence’ of the positive integers in the reals, it was shown that the formula was valid for all reals – and might even be extended to complex numbers, once we know what the meaning of that would be. There is even a method popularized by Herbert S. Wilf called “Proof by Example.” How often have we told students that one example does not suffice to prove a theorem in general? Well it turns out that, sometimes, a well-chosen example, does suffice.

Instances of ‘rule-bending’ abound in mathematics: take, for example, the umbral calculus, reputedly so named for the ‘shady’ methods it employs. There are also more prosaic instances. Often we use diagrams to motivate a problem, where an unknown distance might represent a real number quantity. When we solve the problem algebraically, one of the solutions may turn out to be negative, surely a throw-away. Sometimes not. A careful re-interpretation of the problem sometimes gives meaning to such a solution.

It is still much harder to ask the right question in mathematics than to answer that question. That is why a question should not be given an independent existence, automatically leading us to the right answer. A question is only a starting point to encourage exploration. Often we find that, in the end, the question deserves no answer – it becomes irrelevant.

It helps to remember that universal education was first set up to properly prepare children (actually boys then) to enter the military. Recruits were found to be too unruly, not disciplined enough, and lacking basic literacy. Universal education was to be the remedy. Later corporations found that getting children to march in formation was excellent preparation for the factory and office. However, times have radically changed. Strict conformity and adherence to rules for their own sake will just not cut it anymore. In today’s entrepreneurial world, everyone must march to the beat of their own drum. It is no longer the role of the teacher to instill ‘right thinking’ in students. Students must be encouraged to think and act for themselves. This means, that even in mathematics, I hope that I have convinced you, there is not just one ‘right answer,’ It has actually been this way ever since the first herdsmen used pebbles to keep count of their sheep, but the previous, now irrelevant, demands of government and industry, had temporarily altered perceived reality, and certainly for far too long. I am not saying that the needs of government and industry no longer matter in education, only that those needs have changed with technology and society, and that many more of our students are destined to make paths that do not lead to employment but to entrepreneurial ventures instead.

So, yes, I would be happy with either answer. They are equally correct, much like those two roads that lay equal in that famous Robert Frost poem.

]]>Have you ever given any thought to how imaginary numbers came about? It was in a situation much like we have here. At some stage in a calculation, a square root of a negative was found. Now quitting the calculation and muttering that something has gone seriously wrong is one option. The other option is to ask,”what if we just continued?” And lo and behold, after some more manipulation, the root of the negative disappears, and a “real” result obtains. That is the kind of blue sky thinking, asking what if?, plowing through that must be encouraged in our students.

Yes, all mathematics needs to be rigorous,…eventually. Scarcely anything new will be found if we blindly adhere to rules.

As to the point that this was a test question, thus only a review of learned material is warranted. During a normal lecture, normally less than half a student’s attention is engaged, but during an exam, a student is fully (or as close to fully as you’re ever going to get) engaged. Talk about a teachable moment! Personally, I have always felt disappointed walking away from an exam where I didn’t learn something new. In fact, if a student doesn’t learn something new for every moment that you’ve got their attention, you’ve wasted their time as a teacher.

]]>Simple example: solving linear differential equations using the characteristic polynomial came about when the differential operator was treated as a simple variable, factored out, set to zero, and the roots found. This method was only justified much later.

Another example: the calculus as invented by Newton and Liebniz. It took a couple of centuries for full justification. ]]>

I emphatically disagree with this.

For people who use mathematics with real-world applications, it can be very dangerous to not check if intermediate values are complex (much less negative). If your bank offers to quadruple your interest rate for a one-time flat fee, what do you want it to do if that flat fee is more than your current savings?

(a) start growing your debt at an increased rate

(b) keep your account at 0 with a constant amount owed

(c) tell you you can’t take part because you can’t pay the fee upfront

Now imagine it’s a company programming flight control software or an engineer designing a bridge. BAD things happen when you don’t check for domain issues.

Furthermore, for actual math professors who do research, mathematical rigor is absolutely crucial. Maybe it is not necessary to check every step when first approaching a problem, but if you try to publish a result without an actual proof (or strong statistical evidence), you won’t be taken seriously.

]]>There is a good account of an infuriating back-and-forth with the state about an absolutely embarrassing question here: http://jd2718.org/2011/06/23/ny-state-backs-down-on-inverse-flub-no-geometry-gaffes-until-later-today/

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