A Need for New Number

A problem always arises


Same operation as adding natural numbers; and if we call the elements in our model ‘fractional numbers’, we must be equally clear that these may or may not have the same properties as natural numbers. We use the same words as for natural numbers because we are hoping to generalize the corresponding ideas, in the manner described in Chapter 4, page 61.
In particular, we would like a notation for fractional numbers which (i) is based on same numerals as are used for natural numbers; (ii) allows us to use the same method for adding as those which we have learnt for natural numbers; either as they stand, or extending them by learning a few extra procedures, as we did when we developed our short-multiplication schema into long multiplication.

Fraction
The model we are going to develop will be the same, except for the nature of the units, for all the physical qualities for which we are using it. So when we refer to ‘cutting up’, we are using it in the generalized sense of any way of breaking up into parts. (we want to keep ‘division’ for the mathematical operation: compare ‘uniting’ sets and ‘adding’ numbers.)
This represents any standard object,


And this represents it cup up into five.



Clearly this way of cutting up is of no use for measurement, however, since we do not know how big* the bits are; and whether or not we get a match with our given object will depend on which bit we choose.
If we cut up our standard object into bits which match each other, according to whatever quality we are trying to measure, this gets over the second problem, of which one we choose. How big the bits are will then depend on how many of them there are. A cutting-up of this kind we will call sharing; bits which are equivalent (i.e. which match in the way described) we will call parts; and we will describe that size of the parts by saying into how many of these parts we have shared our standard object. So this represents a standard object** shared into fifth parts.



This, eighth parts.


This, third parts.


This represents the result of sharing into eighth parts, and then combining*** three of these parts.


We call this fractional parts of an object, or for short ‘three eighths’ of it. A fractional parts is thus a part which is obtained by a double action of sharing and combining. Abstracting what is common to all these double actions, we get in realm 3 a mathematical double operation which is called a fraction.

The mathematical notation for this double operation is (read this as ‘three over eight’). Since the numeral below the line tells us the name of the parts represented, - whether they are fifth…………..
* In the generalized sense; i.e. how heavy, how long, how great an e.m.f.,etc
** Thought it has been emphasized that this is a generalized object on which generalized action of sharing and combining are to be done, it is nevertheless a help to one’s thinking to imagine it as something more concrete; such as a cake which we are literally cutting up into fair shares – i.e. equal volumes of cake.
*** In whatever way is appropriate to the physical quality concerned. If weight, by putting in same scale pan. If volume of cake, the same person eats it.

Part, eighth part, thirrd part, etc,- this is called the denominator of the fraction. The numeral above the line tells us how many such parts are combined, and is called the numerator.
The nation might appear to suggest, both from our habit of reading downwards and from its often being written as for convenience in printing or typing, that the combining is done first, whereas in the foregoing description we first share into 8 part and then combine 3 of these. Hewever, we shell see that this actions are commutative- we get the same result whichever we do first. So the notation may be taken as representing simultancously booth of the two possibel orders of the mathematical double operation.
Start with a standart object.



Share into 8 parts.


Coimbine 3 of these eighy parts: result. Three eighth-parts of an object.



Now the other way about. Start with a standard object.


Combine 3 of these standard object.




Share this into 8 parts: result, one eight part of three objects.




Except for their arragement(which does not affect the quantity), the shaded part is the same as before. So the fraction represents(÷, 8 x 3), as embodied in the first set of diagrams above, and (×, 3 : 8), as embodied in the second set of diagrams. This is one reason for reading as ‘three over eight’, rathere than ‘three eighths’, which implies only the first of these alternative orders.


Equivalent fractions
We shall now reverse the process. By using embodiments of these doble operations which we call fractions and equivalence relation between fractions.

Fraction Embediment











And so on; the pattern is clear.
Though the fraction themselves are different, they correspond to same amount of whatever physical quality we are concorned with. If we applied the corresponding actions of sharing and combining to a standard object, the resulting part-objects would match. Whit units attached, the fraction represent equal measures. (Equal amounts of cake, in the concret example). In this respect they are therefore equifalent; and we may collect them together into the equivalent class { .
In the same way we can find other sets of equivalent fractions. For example:








Set of equivalent fractions: { .
Another example, this time without diagrams: {
Not only is the pattern of each equiovalence class clear, but a general method of forming them is beginning to energe.

Start with any fraction,
Double both upper and lower numbers
Treble both upper and lower numbers,
Etc.

Equivalence class

And, in general, if a, b, kare natural numbers, then the fraction (is equivalent to) .

This works both ways; since also , we can get another fraction equivalent to any given fraction by either multiplying or dividing the numerator by the same natural number. The former we can always do; the latter, which is the well-known ‘cancelling’ rule, sometimes.
Example: .
We also have: .
So these also belong to the last named equivalence class, which we can now write:
{ .

Fractional numbers

The caracteristic property of any set ot equifalent fraction we call a fractional number. Whit a unit attached, each fraction in an equivalence class represent the same measure; and, without the unit, it represent the same number. This means that we can use any fraction from the set as a name for the number of that set; and, although this invites confosion if we do not know what is going on, if we do know, it has considerable advantages for purposes of calculation.
So if we are talking about fractions, which are double operations,

If we are talking about fractional numbers,

For each denotes the same equivalence class. The sign in the middle therefore indicates which of the two is meant.

Adding fractional numbers. We want this matematical operation to correspond to combining part-objects. This is straightforward if the numbers are represented by fractions having the same denominator, for we are then combining part-objects of the same kind; an essential already note on page 187. but we have to remember that adding does not mean quite the same for fractional numbers as for natural numbers. To remind ourselves of this we use for the new kind of addition, and + for the old kind.
Example:







If the denominators are not equal, this is where the interchange-ability principle within equivalences sets (page 176) comes to our help. Since all the fractions in a equivalence set stand for the same number, we can choose whichever ones suit us best for some other purpose, in this case a calculation.
Suppose that we want to add (say)
Replace by these equivalent fractions
Which stand for the same numbers
As before. For denominator, we choose
4 9=36.
Now we can add.
=
It should, of course, make no difference which fractions we us as replacements, provided that they stand for the original numbers and have the same denominators. Let us try the calculation by a different route.
First we will replace
The original fraction by =
Equivalent ones using =
The canceling rule.
Now we can find a =
Smaller common =
Denominator, namely =
2 3=6. =
This answer looks different, but of course represents the same fractional number as , since = = . So we have verified that the interchangeability principle works in this case. A general proof is not difficult, but requires the use algebra. Multiplying fractional numbers. As yet we have no meaning for ‘multiplying’ in the new context of fractional numbers. We could of course decide to do without a meaning – there are plenty of mathematical system which have only one operation. But we shall then not have generalized the natural number system completely, so we ought to try. We can either look for a meaning for ‘multiplication’ which is satisfactory in the realm of pure mathematics, and then see whether it provides a useful working model for realm 1; or we can use the requirement of a satisfactory working models to suggest a meaning, and then check whether it is mathematically acceptable. Both approaches have their merits. The latter, being less abstract, is the one we shall use here.
Start as usual with a standard object.



Then this object represents the fraction


In natural numbers, 3 4 when embodied in physical objects means : start with a 3 set











And combine 4 of these.












So in fractional numbers, might reasonably mean : start with two third-parts of an object,





And take four fifth-parts of this.






In natural numbers ‘calculate 3 4’ means ‘find the number of the resulting set’. In fractional numbers, ’calculate ’ might therefore reasonably mean ‘find what fractional part of the object the resulting part-object is’. The resulting part-object is shown by the cross-hatched area. The original object has now been shared into 15 parts (3 5), and the cross-hatched area combines 8 (2 4) of these.






This suggests that

Would be a reasonable way to multiply these fractions ; reasonable, in the sense that it gives a good working model for part-objects. It also satisfies requirements (i) and (ii) on page 186 very well.
These two methods, for addition and multiplication of fractional numbers, are of course those which have been agreed by mathematicians – we have been pretending we did not know in order to try to see how they were arrived at. Stated generally, if a, b, c, d are natural numbers, then the method for adding is:

And the method for multiplying is

Where and refer to operations on fractional numbers, and + and to those on natural numbers.
There is still much unsaid about fractional numbers. Techniques for manipulating them have not been systematized, and decimal notation – which can greatly simplify some of these manipulations – has not been introduced. Neither of these will be done here, since the present aim is comprehension rather than skill at computation. Also, we have not checked that the fractional numbers have the five properties of a number system which we found in Chapter 9 to be so important. This we must certainly do. Since the treatment is algebraic, is has been put into an appendix to this chapter. The reader who does not think easily in algebraic terms may take it on trust, since he already has the ideas, and only requires to be assured that they also hold good for fractional numbers. There is also a third matter of importance, which is whether and to what extent natural and fractional numbers can be intermixed. This last point will be discussed in Chapter 12, with the help of the ideas of isomorphism and mathematical generalization.

APPENDIX
Fractional numbers have the five properties of a number system
Let a, b, c, d, e, f, x, y ,stand for any natural numbers.
Than , etc. will represent fractional numbers.
ADDITION IS COMMUTATIVE
We can only add if the denominators are equal.

This property follows immediately from the corresponding property for natural numbers, and the same is true for all the other properties.
ADDITION IS ASSOCIATIVE

MULTIPLICATION IS COMMUTATIVE

MULTIPLICATION IS ASSOCIATION

MULTIPLICATION IS DISTRIBUTIVE OVER ADDITION