Real numbers, sequences and limits: part IV

Representation of Q on the Number Line.

We have seen earlier that every element of can be represented on a straight line with a certain point representing 0. Positive integers are represented by points to the right of this marked point at equal lengths, and negative integers are similarly represented by points to the left of this.

We can use the same straight line to represent the newly constructed rational numbers as follows.

We know that we can divide a line segment, using a straight edge and compass, into q equal parts. So, the line segment between 0 to 1 can be divided into q equal parts for every positive integer q.

The points of division will now represent the numbers \frac{1}{q}, \frac{2}{q}, \ldots , \frac{q-1}{q}. It is now clear that any rational number of the form \frac{p}{q} can be represented on the straight line. We have seen that between any two distinct rational numbers r and r^{'}, no matter how close they are, we can always find another rational number between them, example, \frac{r+r^{"}}{2}. Geometrically, this means that between any two distinct points on the line representing rational numbers, there is a point between them representing a rational number.

One can ask: does every point on the line correspond to a rational number? Note that we also have a point on the line representing length of the diagonal of a square of side 1. But, we shall show in the next blog that the length of the diagonal of unit square does not correspond to a rational number.

To prepare the ground for such discussions, let us start with a few definitions:

A subset A \subset Q is said to be bounded below if there exists a rational number \alpha such that

\alpha \leq \beta for every \beta \in A

and \alpha is called a lower bound of A. Similarly, A \subset Q is said to be bounded above if there is a \gamma \in Q such that \beta \leq \gamma, and \gamma is called an upper bound of A. A set which is bounded above and also bounded below is called a bounded set. Observe that if a set has a lower bound \alpha then it has many more like \alpha -1, \alpha -2, \ldots. Similarly, a set that is bounded above has many upper bounds. If there is a least element of all these upper bounds, then we call it the least upper bound of the set. But, it is not true that every set of rational numbers which is bounded above has a least upper bound. We can say similar things about sets which are bounded below. Some sets may have least upper bound. For example, the set of negative rational numbers is certainly bounded above and it is easy to see that 0 is the least upper bound.

More later,

Nalin Pithwa

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