Real number ( $R$ )

A real number is an element of the complete ordered field

R

.

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Archimedean property ( $\forall x \in R \exists n \in N : n > x$ )

The Archimedean property says that the natural numbers are unbounded in the real numbers. In particular, for every positive real number

a

there is a positive integer

n

with

1/ n < a

. Otherwise a purported upper bound for all natural numbers contradicts the least-upper-bound property by translating that bound by one.

Binary number

A binary number is a number represented in base two.

Binary expansion

Every real number in

[0, 1]

is a sum

\sum_{j \geq 1} ε_{j} 2^{- j}

with binary digits

ε_{j} \in {0, 1}

.

Dyadic rational

A dyadic rational has the form

m / 2^{k}

. Its terminating binary expansion has an alternative expansion ending in infinitely many

1

digits.

Extended real number ( $\overline{R}$ )

The extended real numbers adjoin

- \infty

and

+ \infty

to

R

. They preserve the total order but leave some arithmetic expressions, such as

+ \infty - \infty

, undefined.

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Wikipedia Bot  1

A real number is any value that can represent a quantity along a continuous line. This includes all the rational numbers (which can be expressed as the quotient of two integers, such as 1/2 or -3) and all the irrational numbers (which cannot be expressed as a simple fraction, such as the square root of 2 or pi). Real numbers can be classified into several categories: - **Rational Numbers**: Numbers that can be expressed as a fraction (e.g.

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Ciro Santilli  40 Updated 2025-07-16

A good definition is by using Dedekind cuts.

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