1. ## [SOLVED] Zero

I can never remember if zero is an integer. Is it?
What is the definition of an integer?

2. ## Definition of an Integer

An integer is a whole number that can be either greater than 0, called positive, or less than 0, called negative. Zero is neither positive nor negative nor an integer.

FYI:
Two integers that are the same distance from zero in opposite directions are called opposites.

Every integer on the number line has an absolute value, which is its distance from zero.

Math Guru

3. Mathworld and many other sources claim that 0 is an integer (denoted by blackboard bold "Z").

4. "Zero is not an integer."

This is false!
Zero IS an integer, that is 0 is in Z, the set of integer numbers
0 is in N, the set of natural numbers

bye
the prof

5. {...-3,-2,-1,0,1,2,3,...} is the set of Integers.
{0,1,2,3,...} is the set of Nonnegative Integers.
{1,2,3,...} is the set of Natural Numbers.

But the definition of the set of Naturals seems to vary from book to book and place to place. So, {0,1,2,3,...} is probably also a common definition for the set of Naturals.

6. ## Peano's axioms

Originally Posted by beepnoodle
{...-3,-2,-1,0,1,2,3,...} is the set of Integers.
{0,1,2,3,...} is the set of Nonnegative Integers.
{1,2,3,...} is the set of Natural Numbers.

But the definition of the set of Naturals seems to vary from book to book and place to place. So, {0,1,2,3,...} is probably also a common definition for the set of Naturals.
Not only "probable". It IS the definition. Look here:
http://mathworld.wolfram.com/PeanosAxioms.html

7. ## sci.math

Originally Posted by beepnoodle
Let's hear from other people http://tinyurl.com/a3km9

"God created integer numbers, all the rest is made by men"
Leopold Kronecker (1823-1891)

IMVHO God did'n create anything but the Universe. Actually Natural (and Integer) numbers are created by men.
I didn't quote Peano only because he was Italian from Torino like me: natural numbers DO contain zero, at least in the mind of most matematician I know, like Leonardo Pisano, AKA Fibonacci.
In his well known book "Liber Abaci" [1202] he introduces arabic digits. He says:
"These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated."
I remark the geniality of introducing digits in descending order! So it is natural what comes 'first'
I think that if there is no doubt that 0 is a digit why 0 should not be a natural number?

bye

8. ## Zero is an integer

Zero is an integer. My authority is that I'm a graduate student of mathematics, heavily read in number theory, and I've taught 5 different algebra classes. I taught them about the natural numbers [1..inf),the integers which range (-inf...inf), and the whole numbers [0..inf). You can remember it, I taught them, as well as did the teachers of the large lecture halls of 500+ students, by the fact that "whole" has an "o" in the middle that looks like a zero.

9. Thought I'd chuck my two-pence in.

My understanding is that zero is not in the Set of Natural Numbers. When we define addition on the SET of natural numbers we take 2 numbers from the set, A, B and make the new element A + B. Under the natural numbers a problem arises should we wish A + B = A, there are no two natural numbers that fit this description. The way round this is to invent an element B called the additive identity and labelled 0. The new set {0, N} include the natural numbers and this new element called 0 such that A + 0 = A. The next obvious question is can we find an element within the set such that A + B = 0 (the identity element). The set fails to provide such an element B so we invent one called the additive inverse (-A) so A + (-A) = 0 This new set is called the integers. Without the additive identity element the integers don't really make sense.

That's the way I've always understood things.

10. This is the sort of discussion that tends to generate more heat than light. There are a number of reasons in general why we define things in mathematics: such as convenience, consistency with other notions, consistency with historical terminology, consistency with other workers. One might argue that it is more convenient to include zero in the natural numbers because then they can count every finite set and we would have the convenience of being able to state a theorem that "the natural numbers are precisely the finite cardinals". One might argue that the Peano axioms are easier to set up starting from zero. One might argue that the majority of texts in the past excluded zero, or that the majority of workers in the field today include zero (I don't know whether either of these is true!). One might argue that if your teacher does, then so should you (at least if you want to get good marks). FWIW, I have always preferred to include zero myself. But please, if you want to argue, make clear the grounds you're resting on...

One piece of terminology that is standard, I believe, is

positive integers: 1,2,3,...
negative integers: -1,-2,-3,...
non-negative integers: 0,1,2,3,...

11. The problem about whether 0 is an integer or not has a quite stupid source. When people first used numbers they all started by 1. 0 was equal to nothing and that's the way they wrote it. If it was 0 they wrote nothing. As a result the 0 was not mentioned in any math book in Europe until the 14th century when somebgody had the brilliant idea to give a sign to the value of nothing. That's where the problem comes from. The definition what finally an Integer number is, was introduced earlier than 0.

Well I know that the 0 was known to the indians and arabians already long before someone in Europe thought about it, but at that time, Europe was far too arrogant to admit that other people could be more intelligent than them.

As for my opinion, the numbers are defined the following way
$\mathbb{N} = 0, 1, 2, 3, 4, ...$ Naturals with 0
$\mathbb{N}* = 1, 2, 3, 4, ...$ Naturals without 0
$\mathbb{Z} = 0, 1, 2, 3, 4, ...$ Integers with 0
$\mathbb{Z}* = 1, 2, 3, 4, ...$ Integers without 0
...

That way you admit that 0 is an integer and if there is a reason why it should work better without 0, you just change your set and you have what you would like. It just depends on what you actually need.

12. Originally Posted by hoeltgman
Well I know that the 0 was known to the indians and arabians already long before someone in Europe thought about it, but at that time, Europe was far too arrogant to admit that other people could be more intelligent than them.
That's rather far from the truth. Civilisation in Europe in the 14th century was recovering from a long period of low economic and cultural activity, following on the collapse on the Roman Empire in the West after pressure from Germanic and other tribes and the loss of many of the richest and most highly cultured provinces in the East after Islamic conquest. The Black Death, which killed almost half the population, didn't exactly help either. Much of the Graeco-Roman cultural heritage had either destroyed or rendered inaccessible until the limited contacts between Christian and Muslim states began to make possible the recovery of that heritage. It wasn't so much intellectual arrogance as it was poverty and distance. Europeans no doubt can be exactly as arrogant as any other group of people but in the 14th century the few who could afford the leisure to worry about such things were quite well aware that their civilisation had suffered from centuries of decline or stagnation and were making efforts to get it back -- the college at which I had the privilege to teach was founded in 1350 precisely to help recover from the Black Death. Before the invention of printing it wasn't easy to recover the lost learning.

13. Not only is zero an integer, it's also a natural number. I refer you to Peano's Axioms:

Let S be a set.

1. There is a distinguished element in S, called zero.
2. The sucessor s(n) if any number n is also a number.
3. If s(n) = s(m), then n=m.
4. Zero is not the successor of any number.
5. If zero is in S, and n in S implies s(n) in S, then S contains every number.

14. Zero is an integer, yes it is.

What is an integer? You can informally think of it as a whole number. I like to think of a natural number as a symbol which represents a cardinality of a finite set. Even more formally are the Peano Axioms from set theory.

Why zero? Well, in Abstract Algebra there is an important structure called a "group" in a group there must exist an identity element-it has a property such as A+0=0+A=0. Thus, math has a zero for the sake of creating a group over the integers.

How can I remember zero is an integer? Easy zero DOES NOT RHYME with integer.

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