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The long real line pastes together ℵ 1 * + ℵ 1 copies of the real line plus a single point (here ℵ 1 * denotes the reversed ordering of ℵ 1) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ 1 in the long real line but not in the real ...
The set which contains the values produced is called the codomain, but the set of actual values attained by the operation is its codomain of definition, active codomain, image or range. [12] For example, in the real numbers, the squaring operation only produces non-negative numbers; the codomain is the set of real numbers, but the range is the ...
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
Older books, when they use the word "range", tend to use it to mean what is now called the codomain. [1] More modern books, if they use the word "range" at all, generally use it to mean what is now called the image. [2] To avoid any confusion, a number of modern books don't use the word "range" at all. [3]
Algebraic numbers on the complex plane colored by degree (red=1, green=2, blue=3, yellow=4). A real number is called a real algebraic number if there is a polynomial (), with only integer coefficients, so that is a root of , that is, () =.
R – real numbers. ran – range of a function. rank – rank of a matrix. (Also written as rk.) Re – real part of a complex number. [2] (Also written.) resp – respectively. RHS – right-hand side of an equation. rk – rank. (Also written as rank.) RMS, rms – root mean square. rng – non-unital ring. rot – rotor of a vector field.
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Range (statistics), the difference between the highest and the lowest values in a set; Interval (mathematics), also called range, a set of real numbers that includes all numbers between any two numbers in the set; Column space, also called the range of a matrix, is the set of all possible linear combinations of the column vectors of the matrix