In teaching Middle School students (often around year 8 or 9), the topic of surds comes up here (I have to teach this topic) - and is often met with derision on commencement of the topic and during the topic. When asked, the students' derision is usually related to the fact this is a 'dry' subject with a perception of it being not practical and not of any use (usually with the 'when are we ever going to use this in our life?' being asked).

I have found that the assessment (test) results are not all that great compared to other topics - with questions about these either unanswered or not completed.

There are a bounty of websites that have the basic tutorials as to what surds are - for example, the Math is Fun page on surds provides a colorful, yet still dry representation of the definition

When we can't simplify a number to remove a square root (or cube root etc) then it is a surd.

and that

The surds have a decimal which goes on forever without repeating, and are Irrational Numbers.

With examples, such as:

$\sqrt{4} = 2$ is not a surd
$\sqrt{2} = 1.4142135...$ is a surd

Is there a practical means to teach surds?

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    $\begingroup$ As a professional mathematician, I can say I have never known of the word "surd" until just now. The reference in the wikipedia article claims (books.google.com/…), claims that $\sqrt{\sqrt{3}}$ is not a surd, but $\sqrt[4]{3}$ is. If this is truly the case, I think the concept is beyond worthless. $\endgroup$ – Steven Gubkin Nov 8 '15 at 2:32
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    $\begingroup$ Surds are very commonly taught under that name where I am from. $\endgroup$ – user5799 Nov 8 '15 at 2:34
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    $\begingroup$ @StevenGubkin Page 25 of that book states, "Every surd is an irrational number but every irrational number is not a surd." It may be that the problem is not with surds but with the terrible exposition of a writer who doesn't even understand the difference between "Every X is not Y" and "Not every X is Y". $\endgroup$ – David Richerby Nov 8 '15 at 12:03
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    $\begingroup$ @StevenGubkin Also, even if the claims in that dubious book are correct, the fact that being a surd is a property of the way something is written down doesn't necessarily make it worthless. For example, $\sum_{i=0}^\infty (x^n/x!)$ is a Maclaurin series but $\mathrm{e}^x$ is not a Maclaurin series, even though $\mathrm{e}^x = \sum_{i=0}^\infty (x^n/x!)$. $\endgroup$ – David Richerby Nov 8 '15 at 12:15
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    $\begingroup$ @StevenGubkin Hope this question on the origin and definition of surd helps, though not sure math.stackexchange.com/questions/84075/… $\endgroup$ – Amir Asghari Nov 8 '15 at 12:32

Chapter 2 of Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Edwards, H. M.) is dedicated to surds and their application to Euler's treatment of the problem of whether there are integers $x,y,z$ such that $x^3+y^3=z^3.$ I would look into whether this book, or the chapter in question, can be read or captured in an accessible form in the class.

If the problem is student engagement, a lot of people are willing to go along with what you're doing if it's relevant to a story, accepting the motivations of historical characters as valid. Especially if they see the struggle from their point of view - the simplicity of the statement, accessibility to reasoning, and the centuries of perspective shifts it took to prove the thing. The chapter also deals with proof and mistakes in reasoning, which are often overlooked aspects of the nature of mathematics at that level of education.

It deals with square roots of negative numbers. It doesn't assume the background of an education in complex numbers, but rather introduces the way they were manipulated in terms of "this is what they did, and they did it by defining the operations this way, and it worked."


In my experience (remedial and college algebra), these topics are always resisted and show poor performance -- very analogously to the topic of fractions (irreducible divisions). One thing I do is show the proof that $\sqrt{2}$ is irrational and tell the story of Hippasus; the lesson, as Lowell put it, "Thoughts that great hearts once broke for, we/ Breathe cheaply in the common air". That's my one stab at making the subject visceral and memorable.

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    $\begingroup$ The dust we trample heedlessly / Throbbed once in saints and heroes rare. $\endgroup$ – Joseph O'Rourke Nov 8 '15 at 1:18

IF you have time (which you may not), I recommend doing something with continued fractions and surds. There is a nice theorem (which you won't have to prove or even state) that quadratic surds and repeating continued fractions are nicely related.

In your context, of course, you wouldn't say that or mention Galois. But creating an experimental worksheet where they use their calculators to create some continued fractions (this is actually nowhere near as hard as it sounds) out of e.g. $\sqrt{5}$, $\sqrt{10}$, $\sqrt{17}$, $\sqrt{26}$ will lead to nice patterns. So at least they see something pretty and maybe that helps with motivation.

(Going the other direction, from a repeated continued fraction to a quadratic surd, would be a lot harder because to have that make sense you need things like $$1+1/(1+1/(1+1/(1+\cdots)))=1+1/(itself)$$ which invokes all kinds of nasty infinitude.)

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    $\begingroup$ Experimental based mathematics work very nicely with the class I have. $\endgroup$ – user5799 Nov 10 '15 at 20:17
  • $\begingroup$ By the way, the dot notation in that document you link to scares me, as no doubt the overline notation I learned would scare those Down Under... $\endgroup$ – kcrisman Nov 10 '15 at 20:17
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    $\begingroup$ @Ghost - awesome, then you should do that. I also would suggest that after they do it "by hand", you could have them use SageMath or something like that to do bigger experiments. $\endgroup$ – kcrisman Nov 10 '15 at 20:18
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    $\begingroup$ There are a lot of things in Maths that are scary - even for us teachers (like year 9 class on a Friday afternoon) $\endgroup$ – user5799 Nov 10 '15 at 20:18

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