Ordered Pairs of Real Numbers

While real numbers correspond to points on a line, ordered pairs of real numbers correspond to points on a plane. This correspondence gives us a way to draw pictures of calculus problems and to translate physical problems into the language of calculus. It is the starting point of the subject called analytic geometry.

An ordered pair of real numbers, (a, b), is given by the first number a and the second number b. For example, (1, 3), (3, 1), and (1, 1) are three different ordered pairs. Following tradition, we use the same symbol for the open interval (a, b) and the ordered pair (a, b). However the open interval and ordered pair are completely different things. It will always be quite obvious from the context whether (a, b) stands for the open interval or the ordered pair.

We now explain how ordered pairs of real numbers correspond to points in a plane. A system of rectangular coordinates in a plane is given by a horizontal and a vertical copy of the real line crossing at zero. The horizontal line is called the horizontal axis, or x-axis, while the vertical line is called the vertical axis, or y-axis. The point where the two axes meet is called the origin and corresponds to the ordered pair (0, 0). Now consider any point P in the plane. A vertical line through P will cross the x-axis at a real number x₀, and a horizontal line through P will cross the y-axis at a real number y₀. The ordered pair (x₀, y₀) obtained in this way corresponds to the point P. (See Figure 1.1.3.) We sometimes call P the point (x₀, y₀) and sometimes write P(x₀, y₀). x₀ is called the x-coordinate of P and y₀ the y-coordinate of P.

Figure 1. 1. 3

Conversely, given an ordered pair (x₀, y₀) of real numbers there is a corresponding point P(x₀, y₀) in the plane. P(x₀, y₀) is the point of intersection of the vertical line crossing the x-axis at x₀ and the horizontal line crossing the y-axis at y₀. We have described a one-to-one correspondence between all points in the plane and all ordered pairs of real numbers.

From now on, we shall simplify things by identifying points in the plane with ordered pairs of real numbers, as shown in Figure 1.1.4.

Figure 1. 1. 4