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In basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point. Often the integers are shown as specially-marked points evenly spaced on the line. Although this image only shows the integers from −9 to 9, the line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers.
In advanced mathematics, the expressions real number line, or real line are typically used to indicate the above-mentioned concept that every point on a straight line corresponds to a single real number, and vice versa.
The number line is usually represented as being horizontal. Positive numbers always lie on the right side of zero, and negative numbers always lie on the left side of zero. An arrowhead on either end of the drawing is meant to suggest that the line continues indefinitely in the positive and negative real numbers, denoted by . The real numbers consist of irrational numbers and rational numbers, the latter of which include the integers, which in turn include the natural numbers (also called the counting numbers or whole numbers).
If a particular number is farther to the right on the number line than is another number, then the first number is greater than the second (equivalently, the second is less than the first). The distance between them is the magnitude of their difference—that is, it measures the first number minus the second one, or equivalently the absolute value of the second number minus the first one.
Thus, for example, the length of a line segment between 0 and some other number represents the magnitude of the latter number.
Numbers can be added by "picking up" the length from 0 to one of the numbers, and putting it down again with the end that was 0 placed on top of the other number.
The section of the number line between two numbers is called an interval. If the section includes both numbers it is said to be a closed interval, while if it excludes both numbers it is called an open interval. If it includes one of the numbers but not the other one, it is called a half-open interval.
All the points extending forever in one direction from a particular point are together known as a ray. If the ray includes the particular point, it is a closed ray; otherwise it is an open ray.
A line drawn through the origin at right angles to the real number line can be used to represent the imaginary numbers. This line, called imaginary line, extends the number line to a complex number plane, with points representing complex numbers.
Alternatively, one real number line can be drawn horizontally to denote potential values of one real number, commonly called x, and another real number line can be drawn vertically to denote potential values of another real number, commonly called y. Together these lines form what is known as the Cartesian coordinate system, and any point in the Cartesian plane denotes the values of a pair of real numbers.
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