Definition:Strictly Negative/Real Number

Definition

The strictly negative real numbers are the set defined as:

$\R_{<0} := \set {x \in \R: x < 0}$

That is, all the real numbers that are strictly less than zero.

Also denoted as

The $\mathsf{Pr} \infty \mathsf{fWiki}$-specific notation for the set of strictly negative real numbers $\R_{<0}$ 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 $\set {x \in \R: x \le 0}$.

Some sources, particularly those whose treatment is topological, use $\bar \R_+$.

Also known as

Some sources merely refer to a strictly negative real number as negative, as their treatments do not accept $0$ as being either negative or positive.

Also see

• Definition:Negative Real Number

• Definition:Strictly Positive Real Number
• Definition:Positive Real Number

• Results about strictly negative 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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): negative 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): negative number