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November 6th, 2017, 05:48 PM  #1 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 1,998 Thanks: 132 Math Focus: Trigonometry  International Notations for Number Sets
Okay, so I know that the international notations for the set of natural numbers, whole numbers, rational numbers, real numbers, and complex numbers are N, Z, Q, R, and C, respectively. However, what are the international notations for the set of integers (natural numbers and zero), even numbers, odd, numbers, prime numbers, composite numbers, and cube numbers? Thanks in advance.

December 31st, 2017, 07:33 AM  #2  
Senior Member Joined: Oct 2009 Posts: 430 Thanks: 144  Quote:
The natural numbers is debated. Some think it should have 0 (for example in set theory), others think it shouldn't. You'll find $\mathbb{N}$ to refer to both sets in the literature. Personally, as a set theorist, I prefer $\mathbb{N}$ to include 0. As for even numbers and odd numbers, I would use $2\mathbb{N}$ and $2\mathbb{N}+1$. Or if you accept negative numbers to be even/odd (as is common), you have $2\mathbb{Z}$ or $2\mathbb{Z}+1$. As for primes, composites, cubes, squares, I haven't encounter a proper notation that is standard. Sometimes I've seen $\mathbb{P}$ for primes, but this is not standard at all.  
December 31st, 2017, 09:08 AM  #3 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 204 Thanks: 60 Math Focus: Algebraic Number Theory, Arithmetic Geometry 
I don't think it's international/standard notation, but I personally like to use $\mathbb{Z}_{\geq 0}$ and $\mathbb{Z}_{> 0}$ for the sets of nonnegative integers and positive integers, respectively. This gets around any ambiguity that the symbol $\mathbb{N}$ has.

December 31st, 2017, 09:30 AM  #4 
Senior Member Joined: Oct 2009 Posts: 430 Thanks: 144 
That said, I think it's pretty inconvenient there is no standard notation for $\{1,...,n\}$. I really feel this is lacking somehow.

December 31st, 2017, 10:00 AM  #5  
Senior Member Joined: May 2016 From: USA Posts: 1,053 Thanks: 431  Quote:
This does not seem to be my invention, but it may not be a well accepted convention. EDIT: I could happily live with $\mathbb Z_{\ge 0} \text { and } \mathbb Z_{>0}.$ Last edited by JeffM1; December 31st, 2017 at 10:04 AM.  
January 1st, 2018, 05:43 PM  #6 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 1,998 Thanks: 132 Math Focus: Trigonometry 
Well, I guess it's case closed, then.

January 2nd, 2018, 12:32 AM  #7 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,825 Thanks: 1051 Math Focus: Elementary mathematics and beyond  I agree. And as we may count zero objects I think $\mathbf{N}$ should include zero. So $\mathbf{N}_0$ and $\mathbf{N}_1$ seems natural.

January 2nd, 2018, 07:20 AM  #8 
Senior Member Joined: Oct 2009 Posts: 430 Thanks: 144  According to set theory, the proper notation for $\{0,...,n\}$ is $n+1$. But somehow this is not popular, I wonder why "Take $m\in n+1$ and ..."

January 2nd, 2018, 12:11 PM  #9 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,600 Thanks: 546 Math Focus: Yet to find out. 
I’ve seen a similar thread on here once before. I’m confused what the fuss is. Doesn’t Peanos first axiom say that 0 is a natural number? In sets 0 ={}, 1 = {{}} etc. ? Or did I miss something .

January 2nd, 2018, 02:22 PM  #10  
Senior Member Joined: Oct 2009 Posts: 430 Thanks: 144  Quote:
You're right about the set theory thing though... Last edited by Micrm@ss; January 2nd, 2018 at 02:25 PM.  

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