Mathematics is already a difficult subject so let us not 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.
Teachers and books make the most extraordinary statements about quadratic equations. All quadratic equations have two solutions, they say. How about x2 – 8x + 16 = 0? Does that have two solutions? “Yes”, they say, “it has the repeated solution 4. If you put x = 4 the equation is true.” Fine. “And if you put x=4 …” Yes, I heard you the first time.
Of course, I can mock, but it is quite useful for some purposes to think of this equation having a ‘repeated solution’, just as it is quite useful for other purposes to think of it as having just one solution. Confused? So you should be. To add to the confusion I shall ask you this question: ‘What is the square root of 9?’ Depending on context you might answer ‘3 or -3’. If I ask you: ‘What is the square root of zero’, what are you to answer: ‘Zero’ or ‘Zero and zero’ or ‘Zero and minus zero’?
Anyway, solving quadratics is where things like ?36 really come into their own. If you want to solve x2 – x – 1 = 0, you can use ‘the formula’ and obtain the solution x = (1 + ?5)/2. What I need to remember when interpreting this solution is that ?5 is positive. Or do I? Why do I? And if every collection of symbols is supposed to represent (at most) one number, what about‘+?5’?
And sooner or later you may want to solve equations like z2 – 4z + 5 = 0 and you may perhaps use the quadratic formula and obtain z = (4 +(?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 3?-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.
Amen to that. For more confusing rules that we give to our students read my post on Mistakes vs. Misconceptions. You may also want to know the more about Algebra Errors.
, and e are not only difficult to work with, they are also difficult to understand conceptually.
where p and q are integers and q 
0. Irrational numbers are all the numbers that cannot be expressed in the form of 
is an irrational number, how do we show them that it fits the definition without resorting to indirect proof or proof of impossibility? What I am saying here is it is not pedagogically sound to start with definitions because definitions are already abstraction of the concept. I would say the same for all other mathematical concepts.