# Definition:Power (Algebra)/Integer

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## Definition

Let $x \in \R$ be a real number.

Let $n \in \Z$ be an integer.

The expression $x^n$ is called **$x$ to the power of $n$**.

$x^n$ is defined recursively as:

- $x^n = \begin {cases} 1 & : n = 0 \\ & \\ x \times x^{n - 1} & : n > 0 \\ & \\ \dfrac {x^{n + 1} } x & : n < 0 \end {cases}$

where $\dfrac {x^{n + 1} } x$ denotes division.

### Even Power

Let $x \in \R$ be a real number.

Let $n \in \Z$ be an even integer.

Then $x^n$ is called an **even power of $x$**.

### Odd Power

Let $x \in \R$ be a real number.

Let $n \in \Z$ be an odd integer.

Then $x^n$ is called an **odd power of $x$**

## Also known as

The expression $x^n$ is vocalised in a number of other ways:

**the $n$th power of $x$****$x$ to the $n$th power****$x$ to the $n$th****$x$ to the $n$**.

### Knuth Uparrow Notation

In certain contexts in number theory, the symbol $\uparrow$ is used to denote the (usually) integer power operation:

- $x \uparrow y := x^y$

This notation is usually referred to as **Knuth (uparrow) notation**.

## Examples

### Negative Power: $-3^{-3}$

- $-3^{-3} = -\dfrac 1 {27}$

## Also see

- Definition:Power of Zero for the definition of $x^n$ where $x = 0$.

- Definition:Power of Group Element, where the operation is defined in a general group and shown to be consistent with the definition given here.

## Historical Note

The concept of an integer power to a negative exponent was introduced by John Wallis in the $17$th century.

## Sources

- 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.1$. Arithmetic: Example $1$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.9$: Roots - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: $(4)$ - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.2.5$: An aside: proof by contradiction - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**index (indices)**