Algebra Tutorials! Saturday 16th of February

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# Real Numbers and the Real Line

Real numbers can be represented by a coordinate system called the real line or x-axis. The real number corresponding to a point on the real line is the coordinate of the point. As the figure below shows, it is customary to identify those points whose coordinates are integers.

The point on the real line corresponding to zero is the origin and is denoted by 0. The positive direction (to the right) is denoted by an arrowhead and is the direction of increasing values of x. Numbers to the right of the origin are positive. Numbers to the left of the origin are negative. The term nonnegative describes a number that is positive or zero. The term nonpositive describes a number that is negative or zero. Each point on the real line corresponds to one and only one real number, and each real number corresponds to one and only one point on the real line. This type of relationship is called a one-to-one-correspondence.

Each of the four points in the figure below corresponds to a rational numberâ€”one that can be expressed as the ratio of two integers. (Note that and .) Rational numbers can be represented either by terminating decimals such as or by repeating decimals such as

Real numbers that are not rational are irrational. Irrational numbers cannot be represented as terminating or repeating decimals. In computations, irrational numbers are represented by decimal approximations. Here are three familiar examples.

See figure below.