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All rational numbers are real, but the converse is not true. Irrational numbers ( R ∖ Q {\displaystyle \mathbb {R} \setminus \mathbb {Q} } ): Real numbers that are not rational. Imaginary numbers : Numbers that equal the product of a real number and the imaginary unit i {\displaystyle i} , where i 2 = − 1 {\displaystyle i^{2}=-1} .
The term "real number" is a retronym coined in response to "imaginary number". Together with the p-adic numbers , the reals are a limit set of the rational numbers . Real numbers may be represented in a variety of ways: as binary numbers , as continued fractions , and in other interesting ways, such as rational zeta series .
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 following table classifies the various simple data types, associated distributions, permissible operations, etc. Regardless of the logical possible values, all of these data types are generally coded using real numbers, because the theory of random variables often explicitly assumes that they hold real numbers.
The whole numbers were synonymous with the integers up until the early 1950s. [23] [24] [25] In the late 1950s, as part of the New Math movement, [26] American elementary school teachers began teaching that whole numbers referred to the natural numbers, excluding negative numbers, while integer included the negative numbers.
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