Zero is even. It is twice the integer zero, and there is no integer k for which 0 = 2k + 1.
Zero satisfies the even definition
An integer is even if it equals twice an integer. Zero qualifies immediately:
0 = 2 × 0.
No special rule is needed. Zero meets the same condition as 8, which equals 2 × 4, and −8, which equals 2 × (−4). The division algorithm treatment in SUNY Schenectady’s discrete structures text classifies integers by their remainder after division by two.
To test whether zero could be odd, try the other definition: 0 = 2k + 1. Solving gives k = −1/2. That is not an integer, so zero does not satisfy the odd definition.
The pattern crosses zero without a gap
Read these consecutive integers from left to right:
| Integer | Classification |
|---|---|
| −3 | Odd |
| −2 | Even |
| −1 | Odd |
| 0 | Even |
| 1 | Odd |
| 2 | Even |
| 3 | Odd |
The alternating pattern continues through zero. If zero were excluded, the labels would develop an unnecessary hole exactly where the arithmetic stays perfectly ordinary.
A grouping picture works too. Zero objects can be divided into two equal groups containing zero objects each. There are no unmatched objects. Having nothing to pair is different from having one object left over.
Dividing zero is allowed
0 ÷ 2 = 0 is valid. The divisor is two, and the quotient is an integer. That is another way to recognize zero as even.
2 ÷ 0 is a different expression and is undefined in ordinary real-number arithmetic. Reversing the order of a division changes the question. The rule against dividing by zero does not prevent zero from being divided by a nonzero number.
Other facts agree with the answer
Adding zero leaves parity unchanged: 17 + 0 = 17 and 18 + 0 = 18. Adding any even integer has the same effect on parity. Multiplying an integer by zero produces zero, an even result.
Zero is neither positive nor negative, but sign and parity are separate classifications. A number can be neutral in sign while still being even.
In mathematical notation, 0.0 and −0 represent the same integer zero. The calculator uses integer notation without a decimal point, so enter 0. For related edge cases, see negative odd numbers and decimals versus integer values.