Definition:Strictly Positive Real Number/Definition 1
Definition
The strictly positive real numbers are the set defined as:
- $\R_{>0} := \set {x \in \R: x > 0}$
That is, all the real numbers that are strictly greater than zero.
Also denoted as
The $\mathsf{Pr} \infty \mathsf{fWiki}$-specific notation $\R_{> 0}$ is actually non-standard.
The conventional symbol to denote this concept is $\R_+^*$.
Note that $\R^+$ is also seen sometimes, but this is usually interpreted as the set $\set {x \in \R: x \ge 0}$.
Also known as
Throughout Euclid's The Elements, the term magnitude is universally used for this concept.
It must of course be borne in mind that at that stage in the development of mathematics, neither of the concepts real number nor positive were fully understood except intuitively.
Some sources refer to this just as positive, as their treatments do not accept $0$ as being either positive or negative.
Also see
- Results about strictly positive real numbers can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.1$: Set Notation
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): positive number
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): positive number