I guess I should clarify a little bit.
For a given set of numbers that are increasing in value, there are infinite patterns that work since you can say "the pattern is that these numbers follow the x-intercepts of an equation ______" Since a pattern can be terminal in nature, the equation does not have to be infinitly long in nature.
ex: 1 3 5 7
1) f(x)=(x-1)(x-3)(x-5)(x-7)(x-9)
2) f(x)=(x-1)(x-3)(x-5)(x-7)(x-9)(x-10)
3) f(x)=(x-1)(x-3)(x-5)(x-7)(x-9)(x-10)(x-11)
4) f(x)=(x-1)(x-3)(x-5)(x-7)(x-9)(x-10)(x-11)(x-12)
5) f(x)=(x-1)(x-3)(x-5)(x-7)(x-10)
6) f(x)=(x-1)(x-3)(x-5)(x-7)(x-10)(x-11)
7) f(x)=(x-1)(x-3)(x-5)(x-7)(x-10)(x-11)(x-12)
8) f(x)=(x-1)(x-3)(x-5)(x-7)(x-1100)
9) f(x)=(x-1)(x-3)(x-5)(x-7)(x-1100)(x-10000)
in each of these examples they all follow the same set of numbers and obviously there are infinite numbers that can come after the 7, and infinte equations that will get any given number after 7.
What about sets of numbers that are not necessarily increasing?
since this is a pattern: 1 2 4 3 2 4 8 6 4 8 16 12.... any finite set of numbers can follow a pattern such as this. (hint: think of it as sets of 4 numbers)
say you are given: 1 4 2 85 12
following the type of pattern given above, there are infinite solutions to the next number in the pattern. here are some examples of the next numbers in a pattern:
( 1 4 2 85 12) 2 5 3 85 13 3 6 4 86 14
( 1 4 2 85 12) 56 2 5 3 85 13 57 3 6 4 86 14 58
(obviously there are infinite other options that just adding 1, but this is the simplest example for this type of pattern)
Now that we have established that any set of numbers can result in any number, I also believe that the next entity in the pattern does not have to even be a number. It can be APPLE if you so wanted.
for example this is a pattern:
1 2 C 4 5 F 7 8 I
given that same pattern above, this is another possible pattern that involves something other than a number:
(1 4 2 85 12) APPLE 2 5 3 85 13 APPLF 3 6 4 86 14 APPLG
so, in short, there are an infinte number of possible patterns and answers for any given set of entities wheather they be numbers pictures, words, etc etc etc
For a given set of numbers that are increasing in value, there are infinite patterns that work since you can say "the pattern is that these numbers follow the x-intercepts of an equation ______" Since a pattern can be terminal in nature, the equation does not have to be infinitly long in nature.
ex: 1 3 5 7
1) f(x)=(x-1)(x-3)(x-5)(x-7)(x-9)
2) f(x)=(x-1)(x-3)(x-5)(x-7)(x-9)(x-10)
3) f(x)=(x-1)(x-3)(x-5)(x-7)(x-9)(x-10)(x-11)
4) f(x)=(x-1)(x-3)(x-5)(x-7)(x-9)(x-10)(x-11)(x-12)
5) f(x)=(x-1)(x-3)(x-5)(x-7)(x-10)
6) f(x)=(x-1)(x-3)(x-5)(x-7)(x-10)(x-11)
7) f(x)=(x-1)(x-3)(x-5)(x-7)(x-10)(x-11)(x-12)
8) f(x)=(x-1)(x-3)(x-5)(x-7)(x-1100)
9) f(x)=(x-1)(x-3)(x-5)(x-7)(x-1100)(x-10000)
in each of these examples they all follow the same set of numbers and obviously there are infinite numbers that can come after the 7, and infinte equations that will get any given number after 7.
What about sets of numbers that are not necessarily increasing?
since this is a pattern: 1 2 4 3 2 4 8 6 4 8 16 12.... any finite set of numbers can follow a pattern such as this. (hint: think of it as sets of 4 numbers)
say you are given: 1 4 2 85 12
following the type of pattern given above, there are infinite solutions to the next number in the pattern. here are some examples of the next numbers in a pattern:
( 1 4 2 85 12) 2 5 3 85 13 3 6 4 86 14
( 1 4 2 85 12) 56 2 5 3 85 13 57 3 6 4 86 14 58
(obviously there are infinite other options that just adding 1, but this is the simplest example for this type of pattern)
Now that we have established that any set of numbers can result in any number, I also believe that the next entity in the pattern does not have to even be a number. It can be APPLE if you so wanted.
for example this is a pattern:
1 2 C 4 5 F 7 8 I
given that same pattern above, this is another possible pattern that involves something other than a number:
(1 4 2 85 12) APPLE 2 5 3 85 13 APPLF 3 6 4 86 14 APPLG
so, in short, there are an infinte number of possible patterns and answers for any given set of entities wheather they be numbers pictures, words, etc etc etc
Well aleph-nul and aleph-one (and so on) are the number of elements in infinite sets. And they are different numbers, or different infinities.
So when you say "there are an infinite number of patterns for any given set of entites, be they numbers, pictures, words, etc. etc,", I was just musing that those infinities are different, depending on the nature of the entities.
But I still disagree with your logic, in some ways:
going back to the original pattern
1, 1 1, 2 1, 1 2 1 1, ...
and taking those elements to be the roots of a polynomial,
I still maintain that there is only ONE polynomial that will generate that pattern, although it has an infinite number of roots.
Not quite the same as claiming an infinite number of polynomials.
But I think we are disagreeing more on a point of liguistics rather than mathematics.
So when you say "there are an infinite number of patterns for any given set of entites, be they numbers, pictures, words, etc. etc,", I was just musing that those infinities are different, depending on the nature of the entities.
But I still disagree with your logic, in some ways:
going back to the original pattern
1, 1 1, 2 1, 1 2 1 1, ...
and taking those elements to be the roots of a polynomial,
I still maintain that there is only ONE polynomial that will generate that pattern, although it has an infinite number of roots.
Not quite the same as claiming an infinite number of polynomials.
But I think we are disagreeing more on a point of liguistics rather than mathematics.
Chris8sirhC said:
Maybe the point of the exercise was to think of numeric symbols in a non-mathematical way.
There's another exercise where "the victim" reads a series of words. The words are the names of various colors. But the color of each printed word is different than the word's meaning. It isn't easy (for me) to do.
It's easy to be distracted or even blinded by expectation.
Just a (different) thought…
Looney
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