Linked Questions

1
vote
3answers
589 views

Why do some accept zero as a natural number but others don't? [duplicate]

I have had many teachers who have told me that zero is a natural number but then there is those teachers who say its not. why is that ?
1
vote
2answers
898 views

0 as an element of the natural numbers [duplicate]

For what reasons would or wouldn't one want 0 to be the start of the natural numbers as opposed to 1? Why would one want it to be 1, or why wouldn't one?
0
votes
3answers
131 views

Set question - $ ℤ^+ = ℕ$ [duplicate]

I am not sure whether the following statement is true: $ ℤ^+ = ℕ$ if not, why? Thank you in advance! I appreciate your help!
1
vote
0answers
334 views

Do the natural numbers include zero? And what should we call them, with or without zero? [duplicate]

Possible Duplicate: Is 0 a natural number? There seems to be no consensus, although perhaps one is gathering over the centuries to say yes to the first question and identify $\mathbb{N}$ with $\...
1
vote
1answer
137 views

Ambiguous definition of the set of Natural Number [duplicate]

According to the book "An introduction to the analysis of algorithms (written by Michael Soltys)", the author says in chapter 1 as follows. Let $\mathbb N = \{0, 1, 2,...\}$ be the set of natural ...
3
votes
0answers
77 views

Isn't zero natural enough to be included in the set of natural numbers? [duplicate]

I always define $\mathbb{N}$ to include $0$ but some authors don't. Since the elements of $\mathbb{N}$ are used for counting, shouldn't $0\in\mathbb{N}$? $0$ is the number of cows in a classroom for ...
0
votes
0answers
37 views

Natural number starts with 1 or zero? [duplicate]

I know positive integers are 1,2,3,4,....;and the sequence of whole numbers are -3,-2,-1,0,1,2,3,...;But the Natural numbers start with one 0r zero ?
35
votes
12answers
6k views

Is this a valid proof that there are infinitely many natural numbers?

I remember reading a simple proof that natural numbers are infinite which goes like the following: Let $ℕ$ be the set of natural numbers. Assume that $ℕ$ is finite. Now consider an arbitrary number $...
31
votes
9answers
4k views

Why is $x^0 = 1$ except when $x = 0$?

Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.
13
votes
4answers
1k views

Simple combinatorics question - caught off guard!

Prove that ${{2n}\choose{n}}$ is even for $n \in \mathbb{N}$. This one caught me off-guard when answering (or attempting to answer!) this for a student today. I tried this approach: $${{2n}\choose{...
6
votes
5answers
1k views

How many $1$s are in the first $1023$ binary numbers?

How many $1$s are in the first $1023$ binary numbers? I'm not to sure how to approach this question. An idea, formula, or solution is appreciated!
3
votes
6answers
1k views

Can a number be non-imaginary and non-real?

I passed across this chart on the web and got confused. The diagram implies that there are numbers that are neither real nor imaginary. Is that possible, or is it just an incorrect diagram? I ...
9
votes
4answers
460 views

Should $\mathbb{N}$ contain $0$? [closed]

This is a classical question, that has led to many a heated argument: Should the symbol $\mathbb{N}$ stand for $0,1,2,3,\dots$ or $1,2,3,\dots$? It is immediately obvious that the question is not ...
28
votes
1answer
589 views

Power towers: to infinity and all the way back

In the following, let $n$ be a positive integer, all other variables be real (furthermore, $a>1$), all functions be real-valued, and logarithms of negative arguments be undefined. Let $\log^n(x)...
8
votes
4answers
232 views

$k$ with an even sum of digits for all multiples of $k$?

Is there a number $k\in\mathbb{N}$ such that $k\cdot n$ has an even sum of digits for all $n\in\mathbb{N}$? I would be grateful for any ideas of how to attack this problem...

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