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Decimal Representation

The decimal numeral system (also called base ten or occasionally denary) has ten as its base. It is the numerical base most widely used by modern civilizations.

Decimal notation often refers to a base-10 positional notation. 

Decimals also refer to decimal fractions, either separately or in contrast to vulgar fractions. In this context, a decimal is a tenth part, and decimals become a series of nested tenths. There was a notation in use like 'tenth-metre', meaning the tenth decimal of the metre, currently an Angstrom. The contrast here is between decimals and vulgar fractions, and decimal divisions and other divisions of measures, like the inch. It is possible to follow a decimal expansion with a vulgar fraction; this is done with the recent divisions of the troy ounce, which has three places of decimals, followed by a trinary place.

A decimal representation of a non-negative real number r is an expression of the form

where a0 is a nonnegative integer, and a1, a2, … are integers satisfying 0 ≤ ai ≤ 9, called the digits of the decimal representation. The sequence of digits specified may be finite, in which case any further digits ai are assumed to be 0. Some authors forbid decimal representations with an infinite sequence of digits 9.[1] This restriction still allows a decimal representation for each non-negative real number, but additionally makes such a representation unique. The number defined by a decimal representation is often written more briefly as

 

That is to say, a0 is the integer part of r, not necessarily between 0 and 9, and a1, a2, a3, … are the digits forming the fractional part of r.

Both notations above are, by definition, the following limit of a sequence:

.

Number Line

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.[1] 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.

It is divided into two symmetric halves by the origin, i.e. the number zero.

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. Customarily, positive numbers lie on the right side of zero, and negative numbers 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, as well as the integers, whole numbers, and the natural numbers (the counting numbers).

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.