Fractions are Hard!
2.9 So … What is a fraction?
It feels like we are getting closer to being able to say what we mean by a fraction.
We have the Key Property of Fractions: \(\dfrac{xa}{xb}=\dfrac{a}{b}\), at least for positive whole numbers. But let’s allow \(x\) and \(a\) and \(b\) to be most any type of number.
A (positive) fraction is any answer to a division problem that is equivalent, via the Key Fraction Property, to a division problem \(\dfrac{a}{b}\) with \(a\) and \(b\) positive whole numbers.
And we like to believe that fractions are numbers in their own right following certain rules of arithmetic.
A (positive) fraction is an expression equivalent, via the Key Fraction Property, to one of the form \(\dfrac{a}{b}\) with \(a\) and \(b\) positive whole numbers accompanied with the following four Beliefs about them.
BELIEF 1: \(\dfrac{a}{a} = 1\)
BELIEF 2: \(\dfrac{a}{1}=a\)
BELIEF 3: \(x \times \dfrac{a}{b} = \dfrac{xa}{b}\) (Basic Multiplication Belief)
BELIEF 4: \(\dfrac{a}{b}=\dfrac{xa}{xb}\) (Key Fraction Belief)
And we are aware of some logical consequences of these beliefs.
CONSEQUENCE 5: \(\dfrac{a}{b} \times b = a\).
CONSEQUENCE 6: \(\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}\).
CONSEQUENCE 7: \(\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{bd}\).
(Also, we saw that Belief 1 follows as a logical consequence of Beliefs 2 and 4 and so it does not really need to be listed separately.)
Thus we seem to be willing to say that \(\dfrac{2}{3}\) is a fraction (it is already of good form)
and so is \(\dfrac{1.2}{3}\) (by the Key Fraction Property this is equivalent to \(\dfrac{1.2 \times 10}{3 \times 10}=\dfrac{12}{30}\)),
as is \(\dfrac{57\sqrt{2}}{61\sqrt{2}}\) (which is equivalent to \(\dfrac{57}{61}\) ).
Whole numbers themselves are fractions. For example, \(2\) is the result of the division problem \(\dfrac{2}{1}\).
So fractions are some new set of numbers we’ve created to expand the arithmetic system of whole numbers. We still haven’t really articulated what one is, but we have articulated very well the properties we believe this new system of numbers should follow – and we feel this new number system still has relevance to the real world: these numbers are connected with our intuition of parts of a whole.
THE TRUTH ABOUT FRACTIONS (OPTIONAL)
Now it is time for the truth about fractions. Unfortunately it is a university-level story. This section is optional reading.
We like to believe that there are numbers that deserve to be called “fractions.” And we like to believe that these are numbers that lie at positions between the whole numbers on the number line.
We like to believe that these numbers can be added, subtracted, divided, and multiplied, that is, they are numbers in their own right and have their own arithmetic rules.
But these things called fractions are hard! They are slippery and seem to slip around, through and between the models we create to describe them. And no one model seems to pin them down utterly and completely.
Nonetheless, all models for them start with the idea that fractions fill a philosophical gap in our understanding of numbers, that fractions “complete” our picture of multiplication just as negative numbers complete our understanding of addition. To explain:
In the world of positive counting numbers we can the solve equations
\(x+5=7\) and \(42 + w = 100\).
But the positive counting numbers fail to solve equations such as:
\(x+8=3\) and \(42+q=20\).
With the invention of negative numbers, however, all equations of the form \(x+a=b\) now have solutions.
So by expanding our number system from just counting numbers (positive whole numbers) to the set of all integers, we’ve filled an apparent deficiency: we can now solve all addition equations.
Wonderful! But we’re not out of the woods.
In the world of integers, some multiplicative equations have solutions:
\(5x=20\) and \(6q=-18\),
for example, do but many do not, such as:
\(7x=10\) and \(3w=2\).
So we expand our number system again so as to include all solutions to equations of this multiplicative type: \(bx=a\) with \(a\) and \(b\) (non-zero) whole numbers. We call the expanded system of the set of rational numbers – the whole numbers with fractions.
And just to continue the story …
It was a shock to mankind that some equations of the form \(x^{n}=a\) can be solved in the world of rational numbers, such as \(x^{2}=\frac{49}{25}\), and \(w^{3}=8\), but not all such equations: \(x^{2}=2\), for example. (It is not at all obvious that \(\sqrt{2}\) fails to be a fraction – hence the shock to mankind!)
So the system of rationals was also expanded to the set of all algebraic numbers so as to incorporate solutions to equations of these type too. By working with the decimal expansions of numbers, this system was expanded further to the set of all infinite decimals, the real numbers.
But the story doesn’t end there, as some equations still fail to have solutions in the set of reals, the most famous being \(x^{2}=-1\).
With the invention of the complex numbers, we now have a system of numbers for which each and every polynomial equation \(ax+b=c\), \(ax^{2}+bx+c=d\), \(ax^{3}+bx^{2}+cx+d=e\), and so on, is certain to have solutions. (This is an astounding feature of the complex numbers: they represent the end of this expanding story. Phew!)
The result that no more types of numbers, beyond the complex numbers, are needed to solve all polynomial equations in algebra is called the Fundamental Theorem of Algebra.
Back to fractions:
The truth about fractions is that these quantities are actually an abstract concept designed to solve an abstract problem: make sure all multiplication problems have solutions. Thus mathematicians take the following approach to fractions:
Given any equation of the form \(bx=a\) with \(a\) and \(b\) whole numbers, we’ll say there exists a unique number which solves the equation. We denote this number \(\dfrac{a}{b}\).
Notice right away we have a belief: \(\dfrac{a}{b}\) is the solution to the equation \(bx=a\) and so we have:
(1) \(b \times \dfrac{a}{b}=a\) . |
One problem:
The equation \(bx=a\) is equivalent to the equation \(2bx=2a\). So this means the fraction \(\dfrac{2a}{2b}\) must be deemed the same as the fraction \(\dfrac{a}{b}\). Similarly the fraction \(\dfrac{3a}{3b}\) must equal \(\dfrac{a}{b}\) as it is the solution to \(3bx=3a\), which is just the equation \(bx=a\) in disguise again. And so on. That is, we need to assume that our Key Fraction Property holds:
(2) \(\dfrac{a\lambda}{b\lambda}=\dfrac{a}{b}\) . |
Obviously the equation \(1\cdot x=a\) has solution \(a\) so it follows that:
(3) \(\dfrac{a}{1}=a\). |
And the equation \(ax=a\) has solution \(1\):
(4) \(\dfrac{a}{a}=1\) . |
Going further…
I claim that \(\dfrac{e}{b}+\dfrac{f}{b}\) is a solution to the equation \(bx=\left(e+f\right)\).
Let’s check:
\(b\left(\dfrac{e}{b}+\dfrac{f}{b}\right) = b \times \dfrac{e}{b}+ b\times \dfrac{f}{b}\) by the usual distributive property
\(= e+f\) by (1) above.
This means that \(\dfrac{e}{b}+\dfrac{f}{b}\) is \(\dfrac{e+f}{b}\). We have just proven:
(5) \(\dfrac{e}{b}+\dfrac{f}{b} = \dfrac{e+f}{b}\). |
So we know now how to add fractions with a common denominator. The Key Fraction Property (2) then suggests a method for adding fractions with different denominators.
How do we divide fractions? What is \(\dfrac{a}{b} \div \dfrac{c}{d}\)? Let’s assume the answer exists for the moment and call it \(x\). Then:
\(\dfrac{\dfrac{a}{b}}{\dfrac{c}{d}}=x\).
This means we want
\(\dfrac{a}{b}=x\dfrac{c}{d}\).
Multiplying through by \(b\) and by \(d\) with the aid of (1) means \(x\) is the solution to the equation \(ad=xbc\). The solution to this is, by definition, \(\dfrac{ad}{bc}\).
This suggests we should define \(\dfrac{a}{c}\div\dfrac{c}{d}=\dfrac{ad}{bc}\) (just as the result of applying the Key Fraction Property to \(\dfrac{a/b}{c/d}\) suggests) .
How should we define the product of two fractions, \(\dfrac{a}{b}\times \dfrac{c}{d}\) ? Let’s assume the answer exists for the moment and call it \(x\). Then:
\(\dfrac{a}{b}\cdot\dfrac{c}{d}=x\).
Multiply by \(b\) and by \(d\) to rewrite this, using (1) to see we have the equation \(ac=xbd\). The solution to this is, by definition, \(x=\dfrac{ac}{bd}\).
IN SUMMARY:
Analyzing solutions to equations of the form \(bx=a\) with \(a\) and \(b\) whole numbers (and we should declare \(b\) to be non-zero) suggests there are numbers, called fractions, of the form \(\dfrac{a}{b}\) , with the following properties:
Fractions come in equivalent forms: \(\dfrac{a\lambda}{b\lambda}=\dfrac{a}{b}\). Fractions add (and subtract) by: \(\dfrac{a}{b} \pm \dfrac{c}{d}=\dfrac{ad}{bd} \pm \dfrac{bc}{bd}=\dfrac{ad \pm bc}{bd}\).
Fractions multiply and divide by: \(\dfrac{a}{b}\times\dfrac{c}{d} = \dfrac{ac}{bd}\) \(\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{ad}{bc}\). |
In scary really formal language: A single fraction is a whole equivalence class of a set of ordered pairs of the form \(\left(a,b\right)\) , with \(a\) and \(b\) each integers, \(b\not= 0\), under the equivalence relation \(\left(a,b\right) \sim \left(ka,kb\right)\) for each non-zero integer \(k\). Their arithmetic operations are define by \(\left(a,b\right)\pm\left(c,d\right) = \left(ad \pm bc, bd\right)\), \(\left(a,b\right) \times \left(c,d\right) = \left(ac,bd\right)\) , and \(\left(a,b\right)^{-1} = \left(b,a\right)\) for \(a\not= 0\), \(b \not= 0\). (And one checks that these arithmetic definitions are independent of the chosen representatives for each equivalence class.)
Done!
This approach is not very intuitive, nor exciting. We have to define the arithmetic rules of fractions, in the end, via the cold mechanical rules, and not as consequences of a clever model. It is just the reality of what fractions are.
And this why fractions are so hard for students.
We cannot share this university-level mathematical story with beginning students: it is not at all pedagogically appropriate. So we must present models for these objects, and change models as our sophistication of work with fractions changes.
Each model is inadequate to some degree, and we and our students can sense that. But we don’t have the means to do anything about it – we never, in the end, share with students what a fraction actually is:
A fraction is number we like to believe exists that solves an equation of the form \(bx=a\). These are the equations that usually arise in division or sharing problems.
To make matters worse:
Since we like to believe that solutions to these equations are unique, this forces to say that many different looking fractions are actually equivalent:
\(\dfrac{a}{b}=\dfrac{2a}{2b}=\dfrac{3a}{3b}=\cdots=\dfrac{-a}{-b}=\dfrac{-2a}{-2b}=\cdots=\dfrac{0.1a}{0.1b}=\dfrac{\sqrt{2}a}{\sqrt{2}b}=\cdots\).
And for the savvy student, we see that there is a potential problem with our definitions of addition, subtraction, division, and multiplication that needs to be attended to. (It is the parenthetical remark in the scary formal definition.) Here is the issue …
Question:
a) In our definition of addition for fractions we wrote:
\(\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{bd}\).
If we worked with an equivalent version of the fraction \(\dfrac{a}{b}\) and an equivalent version of the fraction \(\dfrac{c}{d}\) , and applied this addition definition to those equivalent fractions, is the result sure to be a fraction equivalent to \(\dfrac{ad+bc}{bd}\) ?
b) Similarly, are our definitions for multiplication and division also “well defined”? |
Yeesh! Fractions are hard!
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