# 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 quotient.

There is believed to be a mistake here, possibly a typo.In particular: the quotient as linked is not actually the quotient of integers, because for $n < -1$ we have that $x^{n + 1}$ is not generally an integer.You can help ProofWiki by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mistake}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

### 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): Entry:**index (indices)**