An image asking the question: Is Zero really even or odd?

Why bother with the question: “Is Zero really even or odd?” Well, I tracked down two historical events that make this question valid and relevant for any person.

Way back in 1977, due to thick smog in Paris, people with even- or odd-numbered license plates were allowed only on alternate days on public roads. There was confusion as to whether zero was even or odd, and even the authorities were not sure about how to handle the conundrum (source).

Similar to this situation, back in 2012, the United States was experiencing a hurricane nicknamed Superstorm Sandy. Amidst floods and power cuts, things got so tight that the New York City mayor of the time, Mr. Michael Bloomberg, had to ration fuel resources.

He announced that drivers who had license plates ending in an odd number would be able to buy fuel only on odd-numbered days. Furthermore, to avoid confusion as to whether zero is even or odd, he added that drivers with number plates ending in an even number or zero would be able to buy fuel only on even-numbered days (source).

So, naturally, mathematicians had to settle it once and for all — Is zero really even or odd? In this article, I will be answering this question via a step-by-step analytical approach. But before we begin the analysis, we need to look at what it means for a number to be even or odd.

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Parity in Mathematics — Even and Odd Numbers

Parity is a fancy-sounding term for one of the first things we learn in mathematics. It refers to the concept of splitting integers as either even or odd numbers. To do this, we are faced with the following 2 questions:

1. When is an integer classified as an even number?

2. When is an integer classified as an odd number?

The answer to the first question is as follows:

“An even number is an integer of the form n = 2k, where k is an integer.”

– OEIS A005843

In short, if an integer is a multiple of 2, it is an even number. Now, we turn to the second question. Any number that is not even is an odd number. Well, that was simple, wasn’t it? Right, now that we have established the relevant definitions, we may proceed with the analysis.

Is Zero a Multiple of 2?

Any integer is considered a multiple of 2 if it is the result of 2 (the integer) multiplied with any other integer. For example, we can illustrate that 10 and 50 are multiples of 2, and hence, even numbers:

10 = 2*5

50 = 2*25

The next question that springs to our mind is if zero is a multiple of 2. In other words, is zero the result of 2 multiplied by any other integer?

The answer is YES! When 2 is multiplied by zero, the resulting integer is zero.

2*0 = 0

Based on this, we could claim that zero is even. But in order to be thorough with our analysis, let us check for certain properties that are true for even numbers in general.

Divisibility by 2

It is a feature of all even numbers that they are divisible by 2. ‘Divisible’ by 2 means that if an integer is divided by 2, the remainder of the operation is zero. For example, 4 and 16 are both divisible by 2:

4/2 = 2

16/2 = 8

We get quotients of 2 and 8 respectively, but the remainder in both cases is zero.

When odd integers are divided by 2, we get non-integer quotients. Consider the following example:

25/2 = 12.5

In this case, 12.5 is not an integer. Now, let us see what happens when we divide zero by 2:

0/2 = 0

Here, we get a quotient of zero and a remainder of zero as well (both being integers). As such, there is no violation of the pattern observed so far. We’ll take a look at the additive properties of even numbers next.

Sum of Two Even Numbers

Whenever two even numbers are added, the resulting integer is also even. Consider the following examples:

256 + 10 = 266

120 + 756 = 876

We know that zero added to any number results in the same number. Consequently, any even number when added with zero results in the same even number:

2 + 0 = 2

78865986 + 0 = 78865986

From this analysis, we can conclude that zero passes the test of the sum of two even numbers.

Sum of an Even Number and an Odd Number

Whenever an even number is added to an odd number (or vice-versa), the result is an odd number. Consider the following examples:

25 + 12 = 37

365 + 20 = 385

Any odd number when added with zero will result in the same odd number:

25 + 0 = 25

365 + 0 = 365

Therefore, zero passes the test of the sum of an even number and odd number as well. As a final test, let us consider subtracting 2 from even numbers.

Subtracting 2 From Even Numbers

Whenever 2 is subtracted from an even number, the result is also an even number. Let us start with 8, and proceed downward:

8–2 = 6

6–2 = 4

4–2 = 2

In the above cases, we see that 6, 4, and 2 are all even numbers. Let’s keep proceeding downward:

2–2 = 0

0–2 = -2

-2–2 = -4

We see that zero smoothly fits with the property of subtracting 2 from even numbers without any contradictions.

The Parity of Zero

When you observe the integer line, any even number has an odd number on either side of it. In accordance with this alternating pattern, zero occurs between the two odd numbers: -1 and 1.

Is Zero Really Even Or Odd? An image of a number line with zero at its centre. The integers 1, 2, 3… occur to the right, and the integers -1, -2, -3, … occur to the left. Any even number is typically seen between two odd numbers. Similarly, zero occurs between -1 and 1, which are two odd numbers.
Illustration created by the author

Via our analysis, we have logically shown that zero fits the definition and properties of an even number without any contradictions. Considering also the advantages such a definition offers, we can conclude that zero is an even number.

Although the parity of zero helps situations like resource rationing, there are still real-life situations where zero is not treated as an even number. For example, in the game of roulette, zero is neither odd nor even. Similarly, there are other notions of bets in gambling that would be affected if the parity of zero is considered to be even.

So, I conclude by saying that while zero is an even number, there are certain specific situations in real life where it is not beneficial or preferred to treat zero as even. Hopefully, there will be universal alignment when it comes to the parity of zero in the future.

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Further reading that might interest you: What Really Happens When You Divide By Zero? and Does Division By Zero Really Lead To Infinity?

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