# Definition:Strictly Positive/Integer

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## Contents

## Definition

The **strictly positive integers** are the set defined as:

- $\Z_{> 0} := \set {x \in \Z: x > 0}$

That is, all the integers that are strictly greater than zero:

- $\Z_{> 0} := \set {1, 2, 3, \ldots}$

## Also known as

Some sources to not treat $0$ as a positive integer, and so refer to:

- $\Z_{> 0} := \set {1, 2, 3, \ldots}$

as **the positive integers**.

Consequently the term **non-negative integers** tends to be used in such sources for:

- $\Z_{\ge 0} := \set {0, 1, 2, 3, \ldots}$

Sources which are not concerned with the axiomatic foundation of mathematics frequently identify the **positive integers** with the **natural numbers**, which is usually completely appropriate.

Writers whose aim is specialised may refer to the **positive integers** as just **numbers**, on the grounds that these are the only type of number they are going to be discussing.

## Also see

## Sources

- 1964: J. Hunter:
*Number Theory*... (next): Chapter $\text {I}$: Number Systems and Algebraic Structures: $1$. Introduction - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 4$: Number systems $\text{I}$: The rational integers - 1974: Murray R. Spiegel:
*Theory and Problems of Advanced Calculus*(SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Real Numbers: $1$ - 1979: G.H. Hardy and E.M. Wright:
*An Introduction to the Theory of Numbers*(5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.1$ Divisibility of integers - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): Chapter $1$: Complex Numbers: The Real Number System: $1$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes