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# Surds, Trapezoids and Global Math(s)

March 29, 2018

About half a year ago we celebrated the incredible Global Math Week. It was a week where over a million students stepped into the zone of joyful mathematics. The focus was Exploding Dots, a fascinating and fun way to explore numbers, arithmetic, algebra and many related concepts. It was global, truly global! But was it "math" or was it "maths"?

In the United States, mathematics is abbreviated as "math"; however, in many other parts of the English speaking world, it is "maths".  So should we say "Global Math Week" or "Global Maths Week"?

Does it actually matter?

Mathematics is a study of ideas and thoughts. It is the development of logical conclusions that arise from earlier assumptions and explorations. In its pure form, mathematics is universal. It can be used to describe many natural and physical phenomena. It helps create and engineer advanced, cutting-edge algorithms and systems. For some, it is recreational and can be seen as an art form; for others it is a core activity used to predict, plan, and optimize. As observed in the Global Math(s) Week, it can bring joy to many inquisitive minds of all ages, genders and nationalities. So perhaps it doesn't matter what you call it or how you spell it. Just do it!

Since mathematics can be built from basic axioms and assumptions, it is in many ways universal. But what about the language used to describe the concepts? Well, there are more than 400 languages in the world that are each spoken by more than a million people. So clearly, describing mathematics will sound different in each language. However, for our purposes, let's focus on the English language, spoken by about a fifth of the world population. Specifically, we wish to seek some peculiar differences between British mathematics terminology and American mathematics terminology. We've already pointed out "maths" vs. "math". On that matter, you may also want to watch this Numberphile video with Lynne Murphy

Within the English speaking world, what terms in mathematics are not global?
Do any terms clash?

Well in fact, there aren't many stark differences, only a handful. Sure, things are sometimes said a bit differently in different places. For example, our organization's name,  "One on Epsilon" is naturally interpreted in the UK and Australia as dividing 1 by the variable "Epsilon (ε)". However in North America, you would probably say "One divided by Epsilon".  A British mathematician may also tell you that Epsilon (ε) is a positive quantity very close to "naught". An American mathematician probably wouldn't say "naught" but just "zero". See this interesting Wikipedia entry for a discussion of names of zero.

You've also got other historical differences, such as the fact that in the UK, a "billion" used to stand for a "million million". But wait, isn't that a "trillion"?  Check out this Oxford Dictionaries video:

Yet, these differences are historical. Today, the world has agreed on the value of a billion as 10⁹ and the value of a trillion as 10¹². More on that from Numberphile.

Now, putting historical differences and slight word abbreviations aside, you may ask:

What are the current significant differences in mathematics
between UK English and North American English?

Well, that question has been asked and answered plenty of times. A cute comprehensive description is in this Math with Bad Drawings blog posted by Ben Orlin, where he focuses on commonly used classroom terminology. Based on our investigation, we would like to point out two major differences that we found:

(1) The phrase "surd" is not used in North American mathematics, however in UK and Australian terminology it is popular.

(2) "Trapezium" and "trapezoid" take on completely different meanings in the UK and North America.

"Surds" are a bit "Absurd"

When we, the authors Coco and Yoni, aren't working on One on Epsilon stuff, we are busy in our "day jobs" in the formal education sector. Coco is a secondary school teacher in Australia, and Yoni is a university lecturer and researcher. With that as background, see how this discussion between us evolves:

Coco: Hey Yoni, have you heard of "surds"?

Yoni: It is not a term that I use, but I know the term is used in secondary schools in the UK, Australia and a few other places outside of North America. Can you remind me what a "surd" is?

Coco: Sure. Here is one common definition:

Yoni: Ah, I see. For example, if n=2 (square root) and x=3, we have √3, and according to my calculator:

Coco: Indeed, this is an irrational number because the digits after the decimal go on forever without a repeating pattern.

Yoni: But you know Coco, this term isn't used in North America at all. I even ran an informal Facebook poll on it in a math education forum. The results  strengthened my belief that most North Americans have never heard of it.

Coco: Wow, that is surprising, in Australian schools (and the UK) we use this term.

Yoni: Well, in mathematics academic circles all over the globe, it isn't a commonly used term. Frankly, in my view, it is kind of an "absurd" term.

Coco: Funny that you say that. The word "surd" comes from the latin word "surdus" which stands for "deaf" or "mute". The same goes for the origin of the word "absurd". I guess irrational numbers, as real as they are, may at times seem absurd. But why do you think surds are absurd?

Yoni: Well, look at this,

Coco: Oh, so what?

Yoni: You see, the expression on the left and the expression on the right are equal and are irrational. However, the one on the left is a surd (n=4, x=2) and the one on the right (n=2, x=√2)  isn't a surd because x is not rational. Isn’t that absurd?

Coco: Well, while the values are the same, calling a mathematical expression a "surd" has to do with a particular form. Frankly, in teaching the term is often used loosely as follows:

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

Yoni: Oh, I see. This agrees with discussions on the maths is fun page. What else do you know about surds?

Coco: Well, let me ask you another question. You can take a square root of a number and get a rational number that isn't an integer, right?

Yoni: I guess so. Like √1.44. Yes, that would be equal to 1.2.

Coco: Right. Before, I asked you to take a square root of a number and now, I want you to take a square root of a positive integer. Now, tell me, can you get a result that is a rational number but not an integer?

Yoni: Ah... you don't want an integer, so I should exclude √4, √9, √16, √25, √36 .... and think of square roots of integers that aren't perfect squares: √2√3√5√6√7√8√10√11√12,....

Coco: Yes, exactly. Now here is a lovely claim: All those roots (of non-perfect squares) are going to be irrational numbers. It can be proved by contradiction, but we won't discuss the proof now - we'll leave that for a different day.

Yoni: Oh, that is kind of cool to know. I'd love to see the proof, but assuming it is correct, I guess there is a bit of sense in the phrase "surd". This is because:

A square root of an integer is either a whole number or a surd!

Trapezium vs. Trapezoid

Moving on from surds, consider this shape:

In North America, it is a "Trapezoid" because it is a quadrilateral with a pair of parallel sides. However, in the UK, this shape is called "Trapezium". So you see, this shape has a different name in the UK and the US. See this cool Mathantics video (you may start at the time 3:40):

However, be careful! In American English the word "trapezium" also has a meaning:

In North America, a quadrilateral with no parallel sides is sometimes called a "trapezium".

Confusing? Still, let's take it all in perspective. This is perhaps the most significant difference in mathematics between UK English and American English. Is it manageable? It probably is! If the differences within the English language don't go beyond what we presented above, then maybe Mathematics really is global? What do you think? Let us know.

Stay tuned for the 2018 Global Math Week, starting October 10, 2018. One on Epsilon and our free app, Epsilon Stream, will continue to support the Global Math(s) Project every step of the way.

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