# Surds: what are they good for?

Here is a question I was asked:

Why is rearranging equations containing square roots on the curriculum for GCSE? What might it be useful for in later life?

This is a two-part question, one part of which is dynamite. When I put the question to Twitter, Paul Taylor @aPaulTaylor was the first to take the bait:

Is usefulness in later life a necessary condition for inclusion on the GCSE curriculum?

Let’s set that aside for now. Whether usefulness is necessary or not, asking what a topic might be useful for in later life is a perfectly valid question for a fourteen year old who is being asked to study that topic.

Surds is one of those confusing areas that I vaguely remember but have to look at a definition to recall properly. The BBC GCSE Bitesize website has “a square root which cannot be reduced to a whole number” and says “you need to be able to simplify expressions involving surds”. Rearranging surds, then, is the business of noticing that the square root of 12 multiplied by the square root of 3 can be combined to give the square root of 36, which is 6.

Surds, then, are a part of general algebraic fluency. I expected, therefore, that one answer would be that this is the kind of manipulation that helps generally with higher mathematics; though I wonder when such neat numbers arise in reality. I also expected to hear that surds were useful in very efficient computation. I remember once speaking to someone who was programming computers to go on board aeroplanes. These had very limited computing power and needed to work in real time; the programming involved all sorts of mental arithmetic tricks to minimise the complexity of calculations.

For the latter, I am not sure how relevant this is to modern engineering or programming. For the former, it might be that we are including this for every student at GCSE simply as part of the algebraic fluency that we hope of from incoming mathematics students at university. When I put the question to Twitter, two responses reflected my cynicism on this point. When are surds useful in later life?

Other, less cynical responses, were available. Early responses:

• Ian Robinson ‏@IanRobinson said: “it allows you to work with precise fractional values without rounding errors in calculations. Useful for engineering etc.”
Later, Colin Beveridge @icecolbeveridge suggested something similar: “in computing, roots are expensive — if you can consolidate them, you save computing time.”
This rings true for me but it was a mathematically-inclined structural engineer who asked the original question. Is this really used in engineering?
• Christian Perfect @christianp said “anything involving making rectangles” thinking particularly of “carpentry and landscaping“.

I put these suggestions – rounding errors and rectangles – to Twitter.

I think it’s unlikely anyone doing practical work would need the accuracy. Feels more pure Maths than Applied. But is it used? For engineers, landscape, carpentry etc expansion to a few decimal places so you can measure to reasonable accuracy is fine.

Carol Randall ‏@Caro_lann said: “engineering isn’t just measurement! There’s lots of heavy maths involved in getting a B.Eng (and beyond).”

John Read ‏@johndavidread asked: “where in Maths do equations with square roots come up that you’d want to simplify without calculating numerical value?”

To this, Daniel Colquitt ‏@danielcolquitt wrote what on Twitter must be considered an essay, a four tweet message (1, 2, 3, 4):

Very simple examples: Computing the eigenfrequencies of beams, or reciprocal lattice vectors & hence in various Fourier transforms. In this case, exact form is required, decimal expansion will not do. For the beam example, a numerical value can be computed for a given set of parameters, but if you want to know that frequency for *any* set of parameters, you need to know how to hand surds.

On algebraic fluency, Christine Corbett ‏@corbett_inc suggested “the umbrella of ‘simplifying equations'”.

To this, John Read ‏@johndavidread asked: “but then why not teach it as ‘simplifying equations’? No kid had heard of a surd in the 1980’s”.

Daniel Colquitt ‏@danielcolquitt replied: “For GCSE & roots of reals >0, I would tend to agree with you. Complex roots are somewhat different”.

But we’ve swayed back rather close to the dynamite, haven’t we? I’ll stop there.

My sense is that I haven’t had a satisfactory answer really. This sort of rearrangement is good for building up the background knowledge of the undergraduate mathematics student or perhaps engineering student, but no one seems to be claiming they are an engineer who uses this outside of the classroom. No one seems to have claimed this topic develops mathematical thinking in an interesting way, or that engineers who don’t think they are using it really are relying on it in the black box of software, or that the topic somehow contributes to an appreciation of the beauty of mathematics in the teenagers who are learning it. (This may be due to my experiences and the experiences of those who have replied, or the way I have misinterpreted their words.) It may be that there’s a bunch of stuff on the GCSE syllabus just for those who go onto A-level or degree-level mathematics, and perhaps that’s fine, but it would be nice to have a more satisfying answer to give. So, dear reader, are you satisfied with these answers? Do you have a better answer?

• #### Peter Rowlett

Peter Rowlett teaches mathematics at university. His views do not represent those of his employer. His column at The Aperiodical is Travels in a Mathematical World.

### 12 Responses to “Surds: what are they good for?”

1. Anonymous

I went to school in the 70s and 80s…don’t think I heard the word surd until my teenage kids brought it home. It sounds like part of the toolbox of tricks, like “chunking”, that they were given to make manipulating equations and doing arithmetic etc easier than it ever was for us.

As to whether it’s useful, perhaps if your calculator batteries fail when you’re triangulating up on the Moors?

2. Chris

I did my GCEs in the 60s and learnt about surds – so I don’t think this is a recent phenomenon.
Much of the discussion in the article seems to be about whether using surds is better than using a numerical approximation such as 1.414 or 1.732.
If you understand that the length of the diagonal of a unit square is the square root of 2, then you know more than if you just know that it is about 1.414. Expressing the answer as a surd gives you some idea where the number came from.
Surely understanding is more important than utility?

3. Meg

Math is the language we use to articulate our physical world the same way we use words to articulate our thoughts. And just like learning how to conjugate verbs properly so we communicate well, we need to learn how to manipulate variables to express the environment around us. It is hard to really understand this just by learning how to make change for a \$20 dollar bill. Students need to see and use the numbers, operators, variables, properties – mathematical language and grammar to understand how to manipulate the language to properly describe our environment. Learning that some numbers lose their whole number attributes when the square root operation is performed is like learning that replacing a word with its antonym changes the meaning of the sentence. So learn surds so you can understand the mathematical language enough to do the equivalent of asking where the bathroom is in a foreign country.

4. gary thomson

Tempted to say “Absolutely nuthin’, say it again, whoa…” but…We did surds at school in the 70s and I was a bit surprised to discover my partner’s 14 year old daughter asking me to help with them a few months back (Standard Grade syllabus in Scotland). This was because other than doing a bit of maths teaching for a short time in the early nineties, I’d never used them at all in my professional life, but that was software engineering so maybe not likely to meet them in that context. However I think that learning to work with them is useful for showing that it is still possible to reason with
entities that are “irrational”.

5. Anonymous

I went to school in the 60’s and we didn’t use the term surds although we learnt the concepts for GSE. I’ve only come across the word now helping my grandson with his Maths homework. But the study of surds is just part of understanding basic mathematical concepts needed to grasp the subject. As to their practical use, after having spent a lifetime in engineering, I only ever needed a result in terms of the numerical evaluation of a surd not the surd expression itself. But i needed to understand the basic maths to get to the result. Generally, I don’t think you can apply a principle that if you don;t need a specific piece of knowledge later in life its not worth teaching it in school. In Maths you learn to develop skills of logical analysis which you can apply later in life in all sorts of situations not limited to your lessons to school.

6. Anonymous

To the author: This is a question I am currently working on for GCSE teaching purposes – on that basis I am approaching it from the fundamental use of roots in real world examples – as an engineering graduate I am sure of a solution based on the absolute simplicity of the relation of one root to another encapsulated in an honestly meaningful example. I WILL POST WHEN RESOLVED GLJ

7. David

The question being what are… used for? is one I ask everyday of a multitude of things. The answer then being used to try and assign some level of importance or priority to needing it. There are some good inferences to why we might need to know and be able to use surds but surds for surds sake I’m not sure :). There must be a real world application to surds so I too will attempt to find an example to post. Back soon I hope.

8. Maths Hater

I am a 14 year old and I asked my maths teacher, what would you actually use surds for in life (other than tests) and she had no idea, even the teacher that hates my soul and would do anything to embarass me did not know. I have no idea how to do surds so this really annoyed me. Down with surds!

P.S I got moved down a set later that year

9. John Edmunds

For 99% of us surds, which I have just googled, are of no use whatsoever. I have never needed to deal with a square that cannot be reduced to an integer in my whole adult life. My failure to do so has not got in the way of my career thus far. Nor, I doubt, with most successful people in the world. What it has done is reduce a 16 year old girl to tears as she faces her maths GCSE. Well, maybe that’s the point of surds. And maths teacher. I recall my maths teacher in the 70s being unable to explain what most mathematical concepts were for. “To pass the exam” was her reply. Maths teachers: give what you teach a point that is relevant to most of the people you are teaching. Those up for the Fields Medal are a minority.