Coordinate Lines

Coordinate Lines

In 1637 Rene' Descarts published a philosophical work called Discourse on the Method of Rightly Conducting the Reason . In the back of that book was an appendix that the British philosopher John Stuart Mill described as, ``The greatest single step ever made in the progress of the exact sciences." In that appendix Rene' Descartes linked together algebra and geometry, thereby creating a new subject called analytic geometry ; it gave a way of describing algebraic formula by geometric curves and, conversely , geometric curves by algebraic formulas .

Figure 1.1.4

     In analytic geometry , the key step is to establish a correspondence between real numbers and points on a line . This is done by arbitrarily designative one of the two directions along the line as the positive direction and the other as the negative direction  . The positive direction is usually marked with an arrowhead as in Figure 1.1.4 ; for horizontal lines the positive direction is generally taken to the right. A unit of measurements is then chosen and an arbitrary point , called the origin , is selected anywhere along the line. The line, the origin, the positive direction , and the unit of measurement define what is called a coordinate line or sometimes a real line. With each real number we can associate a point on the line as follows :

$\bullet$ Associate the origin with the number $0$
$\bullet$ Associate with each positive number $r$ the point that is a distance of $r$ units in the positive direction from the origin.
$\bullet$ Associate with each negative number $-r$ the point that is a distance of $r$ units in the negative direction from the origin

     The real number corresponding to a point on the line is called the coordinate of the point .


Example 1 In Figure 1.1.5 we have marked the locations of the points with coordinates $-4,-3,-1.75,-\cfrac{1}{2},\sqrt{2} ,\pi$ and $4$. The location of $\pi$ and $\sqrt{2}$ which are approximate , were obtained from their decimal approximations, $\pi\approx 3.14$ and $\sqrt{2}\approx 1.41$

Figure 1.1.5

     It is evident from the way in which real numbers and points on a coordinate line are related that each real number corresponds to a single point and each point corresponds to a single real number . To describe this fact we say that the real numbers and the points on a coordinate line are in one-to-one correspondence