On **Z, **we now have unlimited subtraction. Can we divide in **Z? **In a limited sense, yes. Division is essentially an inverse problem. That is to say, would stand for a number which when multiplied by n gives m. As long as we are in **Z**, the question does not always have an answer. For example, we know that 5 multiplied by 2 gives 10 as the product, hence 10 divided by 5 gives 2 as the *quotient. * What if we asked which multiplied by 2 gives us 5? We know that there is no integer which when multiplied by 2 gives 5. Such questions frequently arise when we have to distribute m things equally among n persons. If m is not an integral multiple of n, then **Z **becomes inadequate for such distribution.

If 10 cakes are to be shared equally among 5 children, then each child gets 2 cakes. But how do we proceed if the same 10 cakes are to be shared equally among 4 children? One possibility is to give 2 cakes to each of the 4 children and then divide each of the remaining 2 cakes into two equal halves so that there are 4 halves. Each child can now be given half a cake.

What we are doing is introducing a new kind of numbers called “fractions”, meaning a “part” of the whole number. Each half of the cake is represented by . Suppose we had 3 cakes to be shared equally by 5 children. What we do is to divide each cake into 5 equal parts. So we have in all 15 similar pieces of cake. Now we can distribute these 15 pieces among 5 children, each getting a share of 3 pieces. Each child’s share is represented by . Suppose we divided each of the cakes into 10 equal parts instead, then we would have thirty equal pieces of cake. When shared equally by 5 children, each child gets a share of 6 pieces. Each child’s share can be represented by . The share of each child in both ways of division ought to be the same, as in both cases each child has an equal share and nothing of the original three cakes remains. This is what would amount to saying that and represent the same number. Geometrically, they will look identical.

Unless this “equivalence” of fractions is allowed, it would be hard to add fractions. For example, how does one add to , though we have no difficulty in adding to giving us ? We argue that since represents the same number as and , the same as , we have

.

This means that if somebody has three parts out of six parts, and next two parts out of six parts, then he has in all five parts out of six. We can perform subtraction similarly. Multiplication and division can be defined as we did in our elementary school. This presupposes that a fraction represents measurement of some physical quantity like cake, stick, pole, etc. But we should have a definition which should capture the essence of the process of “dividing a certain physical object like a cake into 5 equal parts”. To formalize our construction of fractions, we proceed as follows:

Consider the Cartesian product

and define the relation in $latex \textbf{Z} \times (\textbf{Z}-\{ 0 \})$ by if . It is easy to see that is an equivalence relation in .:

Now, we can decompose the set into disjoint equivalence classes. Let us denote the equivalence class containing by .

Suppose is another element of the same equivalence class. Then, we ought to denote it also by

. But, since we have . So, we can say, if . We can now define addition, subtraction, multiplication and division among the new numbers denoted by in the following way:

(i) for all and .

(ii) for all and .

(iii) for and

.

With these rules of addition, we have

for all and . This is to say that for every element behaves like the “zero element” which when added to any of the numbers of the form gives us the same number . Such an element is sometimes called *additive identity.* We have also

.

This tells us that for every element , we have an element of the same type, which when added to it gives the zero element. This is what we usually understand as the “negative” of the number which is sometimes called the *additive inverse *of . Again, the rules of multiplication tell us that

for all and .

This means that acts the way the *multiplicative identity *like 1 behaves in (which when multiplied by any number gives us back the same number). For simplicity of notation, we denote by 0 (note that ) and by 1 so that we may identify with . Since , when and , we conclude that every non-zero element of the form has a multiplicative inverse.

We call the new numbers * rational numbers *which is in fact, an extension of

**Z.**We denote the set of rational numbers by

In , we can add, subtract and multiply at will. We can also verify easily (**Exercise**):

(i) for all

(ii) for all

(iii) for all

(iv) for every , there is a unique such that .

We call this the negative of , denote it by , and write .

(v) for all

(vi) for all .

(vii) for all

(viii) for all

(ix) For every , , there is a unique such that . We denote or .

Given and , we define divided by , written as by . Note that for ,

**Remark.**

Note that if and , then there does not exist a such that . Thus, in this case, cannot be defined. Also, when and , any choice of will satisfy . Hence, again cannot be defined. So, division by zero is not defined.

**Remark.**

Any set with two operation satisfying all the nine properties listed above is called a * field. *This means that is a field. We will be doing a detailed discussion of fields later in these blogs.

The set is called the set of rational numbers. No doubt is a field, where usual arithmetic operations can be performed, but it has an additional feature that is the * order relation.* That is to say, given two rational numbers we can decide, if they are not equal, which is the larger of the two. To describe the modus operandi of this, we proceed as follows:

Let , the subset of positive rational numbers, then , is a subset of having the properties:

i) for either , or , or . In other words, .

ii) implies that , .

This is sometimes paraphrased by saying that is closed under addition and multiplication. We define an order relation in with the help of by saying that

for and

if , or if .

This is meaningful because either , or , or .

i) Given and , we have implies ; this is obvious because we have , and hence, .

ii) and

iii)

iv) ,

v) Given and , there is an such that .

For a proof of this, first note that if , then there is nothing to prove (we might take ). If, on the other hand, , set and , . This is true because .

Now, . Let , where (since ). Finally, we set so that we have

.

This last property is called the * Archimedean property *of . (This was used by Archimedes in his works, but he later credited Eudoxus for this.).

vi) Given , , there is a such that . For example, take .

vii) for all .

It can be verified that and for all .

More later,

Nalin Pithwa