[Marika (OP)]

Michael wants to know:

Is the number zero (0) an even number, an odd number, or neither?

What do you think?

[alancalverd]

An even number is an integer which is "evenly divisible" by two. This means that if the integer is divided by 2, it yields no remainder. Zero is an even number because zero divided by two equals zero, which despite not being a natural number, is an integer.

[chiralSPO]

Precisely what alancalverd said.

[Bored chemist]

Both by convention, and to avoid issues with things like "the sum of any two even numbers is also an even number", zero is even.

[Janus]

Also, by the rules of multiplication:
Odd x Odd = Odd
Odd x Even = Even
Even x Even = Even

Since  N x 0 = 0  whether N is odd or even, then 0 must be even.

[evan_au]

A binary computer-oriented explanation:
- If you represent any number as a binary number (if you want to include negative numbers, use the 2's complement scheme)
- You can decide if it is odd or even just by looking at the least-significant bit: If it is zero, the number is even
- The binary representation of 0 is all zeroes.
- In particular, the least-significant bit of 0 is zero, so it is even
See: https://en.wikipedia.org/wiki/Two%27s_complement

You can apply this to the decimal representation of numbers: Any decimal number ending in 2, 4, 6, 8 or 0 is even
- So 0 is even.

[guest4091]

0xn=0. It's the same thing as doing nothing, thus redundant.
It is the equivalent of the empty set {}, i.e. no elements. Also referred to as a place holder.
'0' does not divide anything nor does '1'. n/1=n.
It's also the source of nonsensical arguments, such as 0/1, and the related infinity.
The set of positive integers is allowed with or without '0'.
Math was interesting.

[evan_au]

0/1, and the related infinity

I think you mean "1/0, and the related infinity".

A lot of mathematicians survived over the centuries without accepting that zero was a real number, even if they accepted "0" as a placeholder in Indian/Arabic numerals.

[jeffreyH]

0/1, and the related infinity

I think you mean "1/0, and the related infinity".

A lot of mathematicians survived over the centuries without accepting that zero was a real number, even if they accepted "0" as a placeholder in Indian/Arabic numerals.

I believe what was meant was 0/1 and the related 1/0 infinity. The idea of zero being divided by anything else is a strange one when contemplated.

[Bored chemist]

The set of positive integers is allowed with or without '0'.
Math was interesting.

What do you get if you subtract six from six?
Zero turns out to be quite useful if you want to do anything but add up.

[guest4091]

The set of positive integers is allowed with or without '0'.
Math was interesting.

What do you get if you subtract six from six?
Zero turns out to be quite useful if you want to do anything but add up.

It is definitely useful, but that doesn't qualify it as an integer. Integers can be used in the formation of larger integers. What can you build using zeroes?
Zero is the container, which is different from its contents.
A cup can contain various forms of liquids, yet it is none of those things.

[chiralSPO]

0 is most definitely an integer. It is not a positive integer (nor is it negative), but it is nonetheless an integer (and an even integer at that!)

[Bored chemist]

It is definitely useful, but that doesn't qualify it as an integer. Integers can be used in the formation of larger integers. What can you build using zeroes?
Zero is the container, which is different from its contents.
A cup can contain various forms of liquids, yet it is none of those things.

You seem to be muddling two things;
the use of zero in our system for writing numbers where it makes the difference between 2 and 20 and 200
and
The number of sisters I have- which is a number and is zero.

[guest4091]

It is definitely useful, but that doesn't qualify it as an integer. Integers can be used in the formation of larger integers. What can you build using zeroes?
Zero is the container, which is different from its contents.
A cup can contain various forms of liquids, yet it is none of those things.

You seem to be muddling two things;
the use of zero in our system for writing numbers where it makes the difference between 2 and 20 and 200
and
The number of sisters I have- which is a number and is zero.

In the notation system with base 10, the symbol '0' informs the viewer there are no units in 20,
and no units or tens in 200.
The meaning is the same, the container '0' has no elements to consider.
This application would be true in any base. As a unique symbol, not included in the integers, it eliminates the problem of division by 0.

[evan_au]

The idea of zero being divided by anything else is a strange one when contemplated.

What if you had 3 people who agree to cooperate on some work, and agree to split the profits equally.

They are paid $1000, but the materials cost $1000, leaving $0 profit.
In this context, it makes perfect sense to divide $0 three ways, and $0/3=$0.

[Bored chemist]

It is definitely useful, but that doesn't qualify it as an integer. Integers can be used in the formation of larger integers. What can you build using zeroes?
Zero is the container, which is different from its contents.
A cup can contain various forms of liquids, yet it is none of those things.

You seem to be muddling two things;
the use of zero in our system for writing numbers where it makes the difference between 2 and 20 and 200
and
The number of sisters I have- which is a number and is zero.

In the notation system with base 10, the symbol '0' informs the viewer there are no units in 20,
and no units or tens in 200.
The meaning is the same, the container '0' has no elements to consider.
This application would be true in any base. As a unique symbol, not included in the integers, it eliminates the problem of division by 0.

The thing is that  in the number 456 there are 4 lots of a hundred... and so on.
The 4,5, and 6 are still integers.
Why would zero be different?
It's just a number  for counting stuff- like the number of times I have visited the moon.

Why do you think it's different from the other integers?

[alancalverd]

An integer is a counting number. It is a response to "how many?" I have 0 friends. 0 is an integer.

[alinasri]

its an even number

[RjMaan]

I think it is neither even nor odd. It is no thing.

[chiralSPO]

I think it is neither even nor odd. It is no thing.

That is demonstrably incorrect. It doesn't matter that zero represents no thing. It is still an even number.

A number is defined as an integer if it can be derived by adding or subtracting 1 from another integer, starting with the axiom that 1 is an integer. (From this it can be proven that the sum or difference of any two integers is another integer.)

Therefore, 1 – 1 = 0 establishes 0 as an integer.

An even integer is defined as any integer that when divided by 2 (1+1) produces another integer.

Therefore 0/2 = 0 establishes 0 as an even integer.

That is the proof.

It is also convenient (and supports the proof) that 0 has all the properties associated with other even integers.
The product of an even integer and any other integer is even. 0 x 3 = 0 √
The sum of two even integers is even. 0 + 28 = 28 √
The sum of an even integer and an odd integer is odd. 0 + 7 = 7 √
(something that arises from this last one is that even and odd integers alternate when counting by units of 1: 1 is odd, 2 is even, 3 is odd, 4 is even etc... 0 being even fits nicely into this scheme as 1 and –1 are both odd.)