Fractions
1 Warning
This page is, from the point of view of teaching and learning, something of an experiment. Its main justification is that some experiments will be useful, when developing ways of learning and teaching that work better for blind students. See also the postscript to this page.
2 About
This page is for those who are preparing to study calculus. It assumes that you already know how to calculate with fractions whose numerator and denominator are integers. Here the numerator and denominator might be any sort of algebraic object, such as numbers, unknowns, variables and even functions.
You might find it helpful to read it twice. The first time, assume all the quantities are integers. So you have an abstract view of something familar. The second time, assume all the quantities are variables. This will help you understand better both the integer and variable cases.
3 Definition
Each ordered pair \((a, b)\) with \(b \neq 0\) represents a fraction. On this page we will always write the fraction associated with \((a, b)\) as
\[ \frac{a}{b} \]
- Words: fraction a over b
- Latex:
\frac{a}{b}
4 Numerator
In the fraction \(\frac{a}{b}\) the quantity \(a\) is called the numerator. The numerator can be any quantity.
5 Denominator
In the fraction \(\frac{a}{b}\) the quantity \(b\) is called the denominator. The denominator is not allowed to be zero. In symbols
\[ b \neq 0 \]
- Words: b is not equal to zero
- Latex:
b \neq 0
6 Equality (easy)
Suppose that \(\frac{a}{b}\) is a fraction. Suppose also that \(m \neq 0\). Then it is a rule that
\[ \frac{m \times a}{m \times b} = \frac{a}{b} \]
- Words: [I don’t know what to put here.]
- Latex:
\frac{m \times a}{m \times b}
7 Addition (easy)
Addition is easy when the two fractions have the same denominator. Suppose that \(\frac{a}{b}\) and \(\frac{c}{b}\) are both fractions. Here the common denominator is \(b\). In this case
\[ \frac{a}{b} + \frac{c}{b} = \frac{a+c}{b} \]
- Words: [I don’t know what to put here.]
- Latex:
\frac{a+c}{b}
8 Addition (hard)
Suppose we have two fraction \(\frac{a}{b}\) and \(\frac{c}{d}\). We do not assume that \(b\) and \(d\) are equal. In this situation we can use \(d\times b\) as a common denominator, and then use the easy case of addition.
\[ \frac{a}{b} = \frac{d\times a}{d \times b} \] and \[ \frac{c}{d} = \frac{b\times c}{b \times d} \] and finally \[ \frac{a}{b} + \frac{c}{d} = \frac{d \times a + b \times c}{d \times b} \]
9 Equality (hard)
When are two fractions equal? In other words write \[ \frac{a}{b} = \frac{c}{d} \] as a condition on \(a, b, c, d\). First note that as a general rule \(x - y\) is a shorthand for \(x + (-1)\times y\).
Therefore the difference \[ \frac{a}{b} - \frac{c}{d} = \frac{d \times a - b \times c}{d \times b} \] has as numerator \[ (d \times a - b \times c) \] and so \[ da - bc = 0 \] is precisely the condition we are looking for.
10 Postscript
I’m not sure about this page. What I have in mind is that the student is:
- Familiar with fractions where numerator and denominator are explicit integers. Such fractions are of course also known as rational numbers.
- Familiar with doing calculations with rational numbers.
- Preparing to study calculus.
I am intending this page to provide:
- A review of the rules for fractions.
- The extension of the rules to more complicated fractions.
- More awareness that calculation is a consequence of basic properties.
- More appreciation of the depth of mathematical ideas.
- Better recall and understanding of the rules for fractions.
I am concerned that:
- My pedagogic judgement is not well informed by contact with students.
- The student might not share my liking for abstraction.
On the other hand:
- Some blind students might find a more abstract approach less of a cognitive burden.
- This approach done well might help students with concentration and reasoning ability.