Pattern description: powers of a negative number

[samir]

Given the following pattern.

$$(-1)^{0}=1$$
$$(-1)^{1}=-1$$
$$(-1)^{2}=1$$
$$(-1)^{3}=-1$$
$$(-1)^{4}=1$$
$$(-1)^{5}=-1$$
$$(-1)^{6}=1$$
$$(-1)^{7}=-1$$
$$\ldots$$

In words, we might say that the power of a negative number is:

• Positive for even exponents.
• Negative for odd exponents.

How can we concisely describe this pattern using mathematical notation and symbols?

I almost asked the question "what area of mathematics studies patterns?" That would be silly! :p Almost all of them study patterns to some extent. Mathematics is all about patterns and relations. What I meant to ask is, what is the name of the area of mathematics that seeks to find a formal, symbolic way of describing recurring patterns such as these?

Is "series" and "sequences" what I'm looking for? Is this the actual name of the area? Is this a sub-category of calculus?

 

[I like Serena]

Given the following pattern.

$$(-1)^{0}=1$$
$$(-1)^{1}=-1$$
$$(-1)^{2}=1$$
$$(-1)^{3}=-1$$
$$(-1)^{4}=1$$
$$(-1)^{5}=-1$$
$$(-1)^{6}=1$$
$$(-1)^{7}=-1$$
$$\ldots$$

In words, we might say that the power of a negative number is:

• Positive for even exponents.
• Negative for odd exponents.

How can we concisely describe this pattern using mathematical notation and symbols?

I almost asked the question "what area of mathematics studies patterns?" That would be silly! :p Almost all of them study patterns to some extent. Mathematics is all about patterns and relations. What I meant to ask is, what is the name of the area of mathematics that seeks to find a formal, symbolic way of describing recurring patterns such as these?

Is "series" and "sequences" what I'm looking for? Is this the actual name of the area? Is this a sub-category of calculus?

Hi samir! ;)

We might write it as:

$(-1)^{2k} = 1$ and $(-1)^{2k+1} = -1$ where $k$ is an integer.

Or:
$(-1)^n = \begin{cases}1&\text{if $n$ even} \\ -1 & \text{if $n$ odd}\end{cases}$

Or as recurrence relation:

$a_{n+1} = -a_n, a_0 = 1$

If we're only talking about patterns with whole numbers, the typical area would be Number Theory.
If we're more generally talking about recurrence relations, like $a_{n+1} = -a_n$, we might end up in Discrete Mathematics.
If we're talking about real numbers, typically combined with limits, the typical area is indeed Calculus.
We might call it a sub-category of Sequences and Series, or Limits, but these types of sub categories are not really formalized.
Even the distinction between the main areas can be a bit blurry, as we can see here.

 

[samir]

Hi! :)

This is what I was getting at. I like both description, but I would probably prefer the second description. I think you know me by now! I like symbols! :D

$(-1)^n = \begin{cases}1&\text{if $n$ even} \\ -1 & \text{if $n$ odd}\end{cases}$

This looks a lot like a piece-wise function?... is it? Piece-wise relation perhaps?

 

[I like Serena]

Hi! :)

This is what I was getting at. I like both description, but I would probably prefer the second description. I think you know me by now! I like symbols! :D

This looks a lot like a piece-wise function?... is it? Piece-wise relation perhaps?

Yep. It's a piece-wise function. (Nod)
And as Deveno already mentioned, a function is a relation, a special one.