Real number

In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to "imaginary number".

Related Topics:
Mathematics - Number - One-to-one correspondence - Points - Line - Number line - Retronym - Imaginary number

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Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero.

Related Topics:
Rational - Irrational - Algebraic - Transcendental - Positive - Negative - Zero

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Real numbers measure continuous quantities. They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; these are often (mis-)represented in the same form as 324.823211247… The three dots indicate that there would still be more digits to come, no matter how many more might be added at the end.

Related Topics:
Continuous - Decimal fraction - Three dots

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Measurements in the physical sciences are almost always conceived as approximations to real numbers. Writing them as decimal fractions (which are rational numbers that could be written as ratios, with an explicit denominator) is not only more compact, but to some extent conveys the sense of an underlying real number.

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The real numbers are the central object of study in real analysis.

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A real number is said to be computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, most real numbers are not computable. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable.

Related Topics:
Computable - Algorithm - Countably - Constructivists - Definable number

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Computers can only approximate most real numbers with rational numbers; these approximations are known as floating point numbers or fixed-point numbers; see real data type. Computer algebra systems are able to treat some real numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their decimal approximation.

Related Topics:
Computer - Floating point - Fixed-point - Real data type - Computer algebra

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Mathematicians use the symbol R (or alternatively, Bbb{R} , the letter "R" in blackboard bold) to represent the set of all real numbers. The notation Rn refers to an n-dimensional space of real numbers; for example, a value from R3 consists of three real numbers and specifies a location in 3-dimensional space.

Related Topics:
R - Blackboard bold - Set - Notation - Dimension

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In mathematics, real is used as an adjective, meaning that the underlying field is the field of real numbers. For example real matrix, real polynomial and real Lie algebra.

Related Topics:
Matrix - Polynomial - Lie algebra

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