MHB Pattern description: powers of a negative number

  • Thread starter Thread starter samir
  • Start date Start date
  • Tags Tags
    Negative
samir
Messages
27
Reaction score
0
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?
 
Physics news on Phys.org
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.
 
Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is that in roulette there will invariably be variance. In other words, if A comes up 5 times in a row, B will be due to come up soon. However I have been proven wrong many times, and I have seen some...
Thread 'Detail of Diagonalization Lemma'
The following is more or less taken from page 6 of C. Smorynski's "Self-Reference and Modal Logic". (Springer, 1985) (I couldn't get raised brackets to indicate codification (Gödel numbering), so I use a box. The overline is assigning a name. The detail I would like clarification on is in the second step in the last line, where we have an m-overlined, and we substitute the expression for m. Are we saying that the name of a coded term is the same as the coded term? Thanks in advance.

Similar threads

Replies
3
Views
5K
2
Replies
78
Views
6K
Replies
6
Views
1K
Replies
11
Views
3K
Replies
4
Views
4K
Replies
1
Views
2K
Replies
2
Views
1K
Back
Top