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November 6th, 2017, 06:48 PM   #1
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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.
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December 31st, 2017, 08:33 AM   #2
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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.
Whole numbers is a term that exists in my country, but is no widely recognized. Integers is the proper term for $\mathbb{Z}$.
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.
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December 31st, 2017, 10:08 AM   #3
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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 non-negative integers and positive integers, respectively. This gets around any ambiguity that the symbol $\mathbb{N}$ has.
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December 31st, 2017, 10:30 AM   #4
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That said, I think it's pretty inconvenient there is no standard notation for $\{1,...,n\}$. I really feel this is lacking somehow.
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December 31st, 2017, 11:00 AM   #5
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That said, I think it's pretty inconvenient there is no standard notation for $\{1,...,n\}$. I really feel this is lacking somehow.
To avoid ambiguity, I use $\mathbb N^+ \equiv \{1,\ 2,\ ...\}.$

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 11:04 AM.
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January 1st, 2018, 06:43 PM   #6
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Well, I guess it's case closed, then.
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January 2nd, 2018, 01:32 AM   #7
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That said, I think it's pretty inconvenient there is no standard notation for $\{1,...,n\}$. I really feel this is lacking somehow.
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.
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January 2nd, 2018, 08:20 AM   #8
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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.
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 ..."
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January 2nd, 2018, 01:11 PM   #9
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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 .
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January 2nd, 2018, 03:22 PM   #10
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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 .
Not really, since Peano's first axiom is as easily stated with 1 as first element. And some books actually DO this.

You're right about the set theory thing though...

Last edited by Micrm@ss; January 2nd, 2018 at 03:25 PM.
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