With 2, the natural numbers divide into equal halves. One of which we call odd and the other even. And we use this property a lot in math.
If you do it with 3, then one group is going to be a third and the other two thirds (ignore that both sets are infinite, you may assume a continuous finite subset of the natural numbers for this argument).
And this imbalance only gets worse with bigger primes.
So yes, 2 is special. It is the first and smallest prime and it is the number that primarily underlies concepts such as balance, symmetry, duplication and equality.
But why would you divide the numbers to two sets? It is reasonable for when considering 2, but if you really want to generalize, for 3 you’d need to divide the numbers to three sets. One that divide by 3, one that has remainder of 1 and one that has remainder of 2. This way you have 3 symmetric sets of numbers and you can give them special names and find their special properties and assign importance to them.
This can also be done for 5 with 5 symmetric sets, 7, 11, and any other prime number.
Not sure about how relevant this in reality, but when it comes to alternating series, this might be relevant.
For example the Fourier series expansion of cosine and other trig function?
True, but normally, you’d introduce trig functions before complex numbers.
Anyhow: I appreciate the meme and the complete over the top discussion about it :D
Yes, but not really.
With 2, the natural numbers divide into equal halves. One of which we call odd and the other even. And we use this property a lot in math.
If you do it with 3, then one group is going to be a third and the other two thirds (ignore that both sets are infinite, you may assume a continuous finite subset of the natural numbers for this argument).
And this imbalance only gets worse with bigger primes.
So yes, 2 is special. It is the first and smallest prime and it is the number that primarily underlies concepts such as balance, symmetry, duplication and equality.
But why would you divide the numbers to two sets? It is reasonable for when considering 2, but if you really want to generalize, for 3 you’d need to divide the numbers to three sets. One that divide by 3, one that has remainder of 1 and one that has remainder of 2. This way you have 3 symmetric sets of numbers and you can give them special names and find their special properties and assign importance to them. This can also be done for 5 with 5 symmetric sets, 7, 11, and any other prime number.
Not sure about how relevant this in reality, but when it comes to alternating series, this might be relevant. For example the Fourier series expansion of cosine and other trig function?
But then it is more natural to use the complex version of the Fourier series, which has a neat symmetric notation
True, but normally, you’d introduce trig functions before complex numbers. Anyhow: I appreciate the meme and the complete over the top discussion about it :D
Complex numbers ftw
Then you have one set that contains multiples of 3 and two sets that do not, so it is not symmetric.
You’d have one set that are multiples of 3, one set that are multiples of 3 plus 1, and one stat that are multiples of 3 minus 1 (or plus 2)
How do you people even math.
You might as well use a composite number if you want to create useless sets of numbers.
I don’t know if it’s intentional or not, but you’re describing cyclical groups
Not intentionally, but yes group rise in many places unexpectedly. That’s why they’re so neat