Introduction

In Ireland at least, we first encounter fractions at age 6-8. At this age, because of our maturity, while we might be capable of some conceptional understanding, by and large we are doing things by rote and, for example, multiplying fractions is just something that we do without ever questioning why fractions multiply together like that. This piece is aimed at second and third level students who want to understand why the ‘calculus’ of fractions is like it is.

Mathematicians can in a rigorous way, write down what a fraction is… this piece is pitched somewhere in between these constructions — perhaps seen in an undergraduate mathematics degree — and the presentation of fractions presented in primary school. It is closer in spirit to a rigorous approach but makes no claims at absolute rigour (indeed it will make no attempt at rigour in places). The facts are real number axioms.

Defining Fractions

We will define fractions in terms of integers and multiplication.

To get the integers we first define the natural numbers.

Definition 1: Natural Numbers

The set of natural numbers is the set of counting numbers

\mathbb{N}=0,1,2,3,4,\dots,

together with the operations of addition (+) and multiplication \times.

We tend not to have any ongoing problems with the set of natural numbers and how we add them and multiply them. We note four facts at this point.

The facts we present will be true for all real numbers and so we don’t refer to natural numbers, etc. but just numbers.

Facts 1 & 2: Commutativity

  • The order of addition doesn’t matter. For any two numbers n and m,

n+m=m+n

  • The order of multiplication doesn’t matter. For any two numbers n and m,

n\times m=m\times n.

Facts 3 & 4: Identity

  • For any number n

n+0=n

  • For any number n

n\times 1=n

We can add and multiply to our heart’s content but at some point we will meet the notion of subtraction. Subtraction doesn’t cause us too many issues until we are asked to consider quantities such as

1-2

This quantity is no longer a natural number but we can enlarge our collection of numbers to include negative numbers. A principle, which I am going to call the Niceness Principle, says:

Nice objects combined in nice ways should remain nice.

Both 1 and 2 are nice objects as natural numbers but 1-2 is no longer nice. Either we declare that subtraction is not nice… or we extend in this context our definition of what it means for a number to be nice. We do this by introducing negative numbers.

Fact 5: Negative Numbers

For each number n, there exists a number -n called its negative which satisfies

n+(-n)=0.

This allows us to define a new set of numbers that of the integers.

Definition: Integers

The integers is the set of natural numbers and their negatives:

\mathbb{Z}=\{\dots,-2,-1,0,1,2,\dots\},

together with the operations of addition, subtraction (-) and multiplication.

Definition: Subtraction

For numbers a and b define

a-b:=a+(-b).

We might state a theorem about the multiplication of integers. The proof is given here.

Theorem: Minus-By-Plus and Minus-By-Minus

For numbers a and b

  • (-a)\times b=-(a\times b)=a\times (-b)
  • (-a)\times (-b)=a\times b

We can add, subtract and multiply to our heart’s content but at some point we will meet the notion of division. Division doesn’t cause us too many issues until we are asked to consider quantities such as

3\div 2

This quantity is no longer an integer so using the Niceness Principle either we declare that division isn’t nice or we extend our definition of what a nice number is (There is one situation where division isn’t nice… we will discuss that later when we properly define division.) We do this by defining reciprocal numbers.

Fact 6: Reciprocal Numbers

For each non-zero number a there exists a number \displaystyle \frac{1}{a} called its reciprocal which satisfies 

\displaystyle a\times \frac{1}{a}=1.

We can get all the fractions now by defining division:

Definition: Division

For non-zero numbers a and b define

\displaystyle a\div b:=a\times \frac{1}{b}:=\frac{a}{b}.

Theorem: Division by Minus, etc.

For any number a and non-zero b:

  • \displaystyle \frac{-a}{b}=-\frac{a}{b}=\frac{a}{-b}
  • \frac{-a}{-b}=\frac{a}{b}

Proof: Consequence of the fact that \displaystyle \frac{1}{-b}=-\frac{1}{b}. This is the case because

\displaystyle (-b)\times \frac{1}{(-b)}=1,

and if \displaystyle \frac{1}{-b}=\frac{1}{b} we would have (-)(+)=1=(+) contradicting the Minus-by-Plus Theorem \bullet

Definition: Fractions

The set of fractions is set of multiples of reciprocals of non-zero integers:

\displaystyle \mathbb{Q}=\left\{\frac{a}{b}\,:\,a,b\in\mathbb{Z},\,b\neq 0\right\}

To explain why we don’t allow division by zero we need a fact:

Fact 7: Multiplying Out/Factoring

For numbers x,\,y\text{ and }z

x\times (y+z)=x\times y+x\times z.

Using this fact we can prove the following Theorem (proved at the end):

Theorem: Multiplication by Zero

For any number x,

x\times 0=0.

Now suppose that we could divide a number x by zero. By definition dividing by zero is multiplying by \displaystyle \frac10:

\displaystyle x\div 0=x\times \frac{1}{0}.

Therefore we can divide by zero if we can define \displaystyle 1\div 0=\frac10. Consider though

\displaystyle\frac{0}{0}=0\times \frac{1}{0}.

What should this equal? The Reciprocal Property says it must equal one. The Multiplication by Zero Theorem says it must equal zero. Therefore if we allow division by zero Reciprocals and the Theorem contradict each other. We are not prepared to let either the Reciprocal Property nor the Theorem go so we must reject division by zero.

Multiplication of Fractions

Now the question is begged, how do we multiply fractions together:

\displaystyle\frac{a}{b}\times \frac{c}{d}=a\times \frac{1}{b}\times c\times\frac{1}{d}.

By Commutativity this equals:

\displaystyle a\times c\times \frac{1}{b}\times \frac{1}{d}.

Now using Reciprocals

\displaystyle d\times b\times \frac{1}{b}\times \frac{1}{d}=d\times 1\times \frac{1}{d}

\displaystyle =d\times \frac{1}{d}=1.

Therefore, using Commutativity

\displaystyle b\times d\times\left(\frac{1}{b}\times \frac{1}{d}\right)=1,

and so by the Reciprocal Property we must have

\displaystyle \frac{1}{b}\times \frac{1}{d}=\frac{1}{b\times d}.

Therefore

\displaystyle\frac{a}{b}\times \frac{c}{d}=a\times c\times\frac{1}{b\times d}=\frac{a\times c}{b\times d}.

Division

By the Definition of Division:

\displaystyle \frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\times \frac{1}{\frac{c}{d}}.

Note that

\displaystyle\frac{c}{d}\times \frac{d}{c}=\frac{c\times d}{d\times c},

is equal to one by Commutativity and the Reciprocal Property. Therefore

\displaystyle\displaystyle \frac{1}{\frac{c}{d}}=\frac{d}{c},

and so

\displaystyle \frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\times \frac{d}{c}=\frac{a\times d}{b\times c}.

Addition

First of all fractions that haven’t the same ‘bottoms’/denominators can’t quite be added together… in the same way that apples and oranges can’t quite be added together.

When fractions do have the same bottoms, Factorising and Commutativity can be used:

\displaystyle \frac{a}{c}+\frac{b}{c}=a\times \frac{1}{c}+b\times\frac{1}{c}

\displaystyle =\frac{1}{c}\times(a+b)=\frac{a+b}{c}.

If fractions don’t share a common ‘bottom’ we can use Reciprocals and Identity to make them ‘share a bottom’:

\displaystyle \frac{a}{b}+\frac{c}{d}=\frac{a}{b}\times \frac{d}{d}+\frac{b}{b}\times\frac{c}{d}

\displaystyle =\frac{a\times d}{b\times d}+\frac{b\times c}{b\times d}

\displaystyle =\frac{a\times d+b\times c}{b\times d}.

Appendix

Proof of Zero-Times-Anything is Zero

By Identity,

x\times(0+0)=x\times 0.

By Multiplying Out,

x\times (0+0)=x\times 0+x\times 0.

Therefore

x\times 0+x\times 0=x\times 0.

By Negative Numbers there exists a number -x\times 0. Add this to both:

x\times 0+x\times 0+(-x\times 0)=x\times 0+(-x\times 0)

\Rightarrow x\times 0+0=0

\Rightarrow x\times 0=0.