The Real Number Line

Real numbers can be represented by points on a line called the real number line.

-2 -1 - 1 _ 2 Scale 1 unit O 0 Negative real numbers Zero Positive real numbers 1 _ 2 1 √2 2
Figure 1. Real number line

Pick a point on the line somewhere in the center, and label it O. This point, called the origin, corresponds to the real number 0.

The point 1 unit to the right of O corresponds to the number 1. The distance between 0 and 1 determines the scale of the number line.

The negative real numbers are the coordinates of points to the left of the origin O.

The real number zero is the coordinate of the origin O.

The positive real numbers are the coordinates of points to the right of the origin O.

Graph Inequalities

An important property of the real number line follows from the fact that, given two numbers (points) a and b, either a is to the left of b, or a is at the same location as b, or a is to the right of b.

a b a < b a b a = b b a b < a
Figure 2.

If a is to the left of b, we say that “a is less than b” and write a < b.

If a is to the right of b, we say that “a is greater than b” and write a > b.

If a is at the same location as b, then a = b.

If a is either less than or equal to b, we write a ≤ b. Similarly, a ≥ b means that a is either greater than or equal to b.

Collectively, the symbols >, <, ≤, and ≥ are called inequality symbols.

-4 -2 0 2 4 (b) x ≥ 1 -4 -2 0 2 4 (a) x < 1
Figure 3.

(a) Notice that we use a right parenthesis to indicate that the number 1 is not part of the graph.

(b) Notice that we use a left bracket to indicate that the number 1 is part of the graph.

Find Distance on the Real Number Line

-4 -2 0 2 4 3 units 4 units
Figure 4.

The absolute value of a real number a, denoted by the symbol |a| , is defined by the rules

|a| = a if a > 0 and |a| = -a if a < 0

If P and Q are two points on a real number line with coordinates a and b, respectively, the distance between P and Q, denoted by d(P, Q), is

d(P, Q) = |b - a|

Since |b - a| = |a - b|, it follows that d(P, Q) = d(Q, P).