Posted by: Gary Ernest Davis on: August 16, 2011

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On the last Twitter #mathchat there seemed to me to be a fair amount of confusion about what constitutes a fraction.

Commonly, people were treating any expression of the form as a fraction, no matter what were .

Confusion about fractions is something I’ve experienced in many places, in many contexts. The confusion seems to stem from the ‘bar’ notation; anything that has two numbers separated by a horizontal or sloping bar is a fraction it would seem.

The problem lies in the divisibility properties of integers.

Technically, the integers form a commutative ring and not a field; there is no integer .

So, if divide 2 liters of water between 3 people, each person will not get an integer number of liters of water. We could do with numbers other than integers to describe how many liters each person gets.

We could adopt the ancient Greek practice of just saying 2 liters for every 3 people. However Muḥammad ibn Mūsā al-Khwārizmī introduced the Hindu idea of fractions as numbers to represent ratios around 830 AD, and ever since fractions have been part of arithmetic.

The basic idea of fractions is that if we cannot divide 2 by 3, for example, then we invent a new “number” as if we could. So we invent the symbol which stands for the result of dividing 2 things into 3 equal parts.

When we invent expressions for all pairs of integers we get the field of fractions of the ring of integers, usually denoted by .

Strictly speaking, does not consist of all expressions , because we recognize, for example, from our liquid division problem, that we should treat as the same “number” as .

So, strictly, consists of expressions .

Alternatively, we can take to consist of all expressions and we re-define “equality” between these expressions to mean .

With this understanding every fraction can be written in the form thanks to Euclid’s algorithm for the greatest common divisor , since .

The fractions are are constructed from integers , yet the integers now appear, in disguise as it were, as special cases of fractions: fractions of the form or more generally .

So, now integers appear as special cases of fractions.

We use the same name for the integer, as we do for the fraction . These are essentially different objects, but we give them the same name for obvious reasons, and then we confound them – we act as if they are the same thing.

This situation, in which previous entities appear in new guises, is common in mathematics: it happens again, for example, in the construction of complex numbers from the real numbers. The real numbers appear as special cases of complex numbers.

The real numbers are tricky. We can think of them as decimal strings where the are not all 9’s from some point on.

A big problem here is that addition, subtraction, multiplication and division are not easy to define due to the need for infinite carrying. It can be done, but is not so simple.

The problem is avoided in school mathematics by only dealing with decimals that are **finite**: that is, those for which from some point on. Unfortunately, these so-called decimal fractions do not even cover the ordinary fractions such as .

But the decimals do form a field: division, except by 0, is always possible, and it makes sense to divide 0.4 by 0.19 for example: . This is **not** a fraction: 0.4 and 0.19 are not integers. The expression indicates a division in the field of decimal numbers.

Similarly an expression like is not a fraction: it is a division, in the field of decimal numbers.

But now we observe that the fractions, and so the integers, are hiding in disguise in the decimal numbers: , so we can ask of the fraction , does this mean the same as dividing the decimal 2 by the decimal 3? And the answer is “yes”.

So even though the bar notation for fractions was just a notation, it ends up representing actual division when we move to the larger field of fractions. A notation has become an operation.

This might not be, as Paul Solomon (@lostinrecursion) says, how students think. But the reason I wrote the post was to ask the question of teachers of mathematics: how do you think about fractions?

In terms of ways of thinking about fractions (as opposed to the abstract definition of the field of fractions of the integers), I recommend:

http://math.berkeley.edu/~wu/fractions1998.pdf

As I always preach, how I treat something is different to each audience. For Daughter (who’s 2yo next month) fractions are “2 of the 3 blocks are red.” To you, Gary, it would be more like what you’ve written. (It’s way sexy seeing commutative ring written – I was into non-commutative rings for my Masters, but I don’t mind a little commutativity on occasion.)

To a random student, it would be whatever strikes me to start with. Then I would watch how it lands on them. I’d expand or contract what I was explaining based on their facial and body language.

The idea is to get people to discover and declare math in their own way. As a teacher, I don’t get to tell them how to do that. Merely help them figure it out on their own.

When I teach the fractions unit in my content course for future K-8 teachers I teach the definition of rational numbers and emphasize the conceptual understanding of part to whole as well as the notation being the operation of division.

1 | Dave Radcliffe

August 16, 2011 at 11:19 am

A “fraction” is just an instance of fraction notation. It is an expression that consists of two numbers, called the numerator and the denominator, separated by a horizontal line or a forward slash. The numerical value of this expression is the quotient of the numerator by the denominator.

In my opinion it is not strictly correct to say that 4 is a fraction, although we can write a fraction whose value is 4, namely 4/1.

Gary Ernest Davis

August 16, 2011 at 1:09 pm

Dave, a/b is only the quotient of a by b when it is possible to divide. So, as you know, we first have to construct some place, larger than the integers, in which division, except by 0, is always possible. Whether we write things in this larger space as pairs (a,b) or as a/b, or as a#b, or a%b, or a!!b, … is beside the point. It’s a formal device that captures what we want. Then, lo and behold, a/b is now the quotient of a by b.