# 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 numbersis the set of counting numbers,

together with the operations of

addition(+) andmultiplication.

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 and ,

- The order of multiplication doesn’t matter. For any two numbers and ,
.

### Facts 3 & 4: Identity

- For any number

- For any number

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

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 and are nice objects as natural numbers but 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 , there exists a number called its

negativewhich satisfies.

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

### Definition: Integers

The

integersis the set of natural numbers and their negatives:,

together with the operations of addition,

subtraction(-) and multiplication.

### Definition: Subtraction

For numbers and define

.

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 and

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

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-zeronumber there exists a number called itsreciprocalwhich satisfies.

We can get all the fractions now by defining division:

### Definition: Division

For

non-zeronumbers and define.

### Theorem: Division by Minus, etc.

For any number and non-zero :

*Proof: *Consequence of the fact that . This is the case because

,

and if we would have contradicting the Minus-by-Plus Theorem

### Definition: Fractions

The set of

fractionsis set of multiples of reciprocals of non-zero integers:

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

### Fact 7: Multiplying Out/Factoring

For numbers

.

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

### Theorem: Multiplication by Zero

For any number ,

.

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

Therefore we can divide by zero if we can define . Consider though

.

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:

.

By Commutativity this equals:

.

Now using Reciprocals

.

Therefore, using Commutativity

,

and so by the Reciprocal Property we *must *have

.

Therefore

.

# Division

By the Definition of Division:

.

Note that

,

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

,

and so

.

# 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:

.

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

.

# Appendix

### Proof of Zero-Times-Anything is Zero

By Identity,

.

By Multiplying Out,

.

Therefore

.

By Negative Numbers there exists a number . Add this to both:

.

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October 28, 2016 at 1:32 pm

Solving ‘Easy’ Equations: Part I | J.P. McCarthy: Math Page[…] we had to use the definition of division, multiplying out and commutativity to properly divide a […]

October 4, 2017 at 2:44 pm

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