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Real numbers

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The set of real numbers is a totally ordered and complete (though not algebraically closed) uncountable set of points, which can be thought of as constituting a [doubly] infinitely long line called the number line or real line, where the points corresponding to rational numbers form a dense and countable subset of the reals, and where the points corresponding to rational integers are equally spaced by one and include 0. Some real numbers include the integer –47, the rational number \scriptstyle \frac{854513}{138} \,, the (arithmetic) algebraic numbers \sqrt[3]{2} \, and the "simple" (although the number of [+, −, *, /, , ^, √] operations must be finite, it is unbounded)

\frac{\sqrt[101]{19} + 127 \sqrt[29]{257 + \sqrt[67]{13 + \sqrt{71}}}}{87 - 64 \sqrt[997]{97}},

[the real part of] the (nonarithmetic) algebraic integers which are the roots of the quintic monic polynomial x5 + x + 1 = 0, and the transcendental number \scriptstyle \pi \,. The computable real numbers constitute a countable subset of the real numbers, which implies that most real numbers are uncomputable.

The set of real numbers is usually denoted \scriptstyle \R \,.[1]

The set of [real] algebraic numbers \mathbb A \cap \R (real roots of \Z(x)) is a dense (though not complete) countable subset of the set of real numbers. The set of rational numbers \scriptstyle \Q \, (quotient field of the ring of integers) is a dense (though not complete) countable subset of the set of real numbers.

The set of complex numbers is the algebraic closure of the reals. A complex number z has a real part \scriptstyle \Re(z) \, and an imaginary part \scriptstyle \Im(z) \, (either or both may be 0). For example, if \scriptstyle z \, is one of the complex cubic roots of –12, then \scriptstyle \Re(z) \,\approx\, 1.14471424 \,.

Notes

  1. Miller, Steven J.; Takloo-Bighash, Ramin (2006). An Invitation to Modern Number Theory. Princeton and Oxford: Princeton University Press. p. xix. 

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