MHB Pattern description: powers of a negative number

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The discussion centers on the pattern of powers of negative one, where even exponents yield positive results and odd exponents yield negative results. Participants suggest mathematical notations to describe this pattern, including $(-1)^{2k} = 1$ for even integers and $(-1)^{2k+1} = -1$ for odd integers. They also explore the concept of piece-wise functions to represent this relationship, confirming that the notation resembles a piece-wise function. The conversation touches on the broader mathematical areas involved, such as Number Theory, Discrete Mathematics, and Calculus, while acknowledging the overlap between these fields. Overall, the focus is on formalizing the description of recurring patterns in mathematics.
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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?
 
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samir said:
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.
 
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

I like Serena said:
$(-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?
 
samir said:
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.
 

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