What are Some Properties of Zero?

Mary McMahon
Mary McMahon

Zero is a fascinating little number, and it has some very distinctive properties. Ever since zero was invented, mathematicians have struggled to define it and to use it in their work, with the properties of zero being arrived at through the use of mathematical proofs which are intended to illustrate those properties at work. Even with proofs to support the rationale behind some of the properties of zero, this number can be quite slippery.

Ever since zero was invented, mathematicians have struggled to define it and to use it in their work.
Ever since zero was invented, mathematicians have struggled to define it and to use it in their work.

People haven't always used zero. A crude form of zero as a placeholder appears to have been used by Babylonian mathematicians, but Indian mathematicians are usually credited with coming up with the idea of zero as a number, rather than just a placeholder. Almost immediately, people struggled to define the number and learn how it worked, and explorations into the properties of zero got quite complex.

Numbers can be classified as positive or negative, depending on whether they are greater or less than zero, but zero itself is neither. Zero is also even, something which comes as a surprise to some people when they learn about the properties of zero, as they often assume that it is either odd or outside of the even/odd dichotomy. In fact, extensive math could be used to show you how zero is classified as even, but the simplest way to show how zero is even is to think about what happens when you have a multiple digit number which ends in an even number. 1002 ends in a 2, an even number, so it is considered even. Likewise with 368, 426, and so forth. Numbers which end in zero are also treated as even, illustrating that zero is itself even.

The Addition Property of Zero states that adding 0 to a number does not change that number. 37+0 equals 37, for example. In the Multiplication Property of Zero, mathematicians state that multiplying a number by zero always ends in zero: if you multiply six oranges zero times, you end up with no oranges. Some other properties of zero have to with addition and subtraction. Subtracting a positive number from zero ends in a negative number, and subtracting a negative number from zero ends in a positive.

Zero has another property which is familiar to anyone who has tried to divide a number by zero with a graphing calculator. Division by zero is simply not allowed in mathematics, and if you attempt it, a calculator usually returns the message “undefined,” “not allowed,” or simply “error.” The Indians actually tried very hard to prove that you could divide by zero, but they were unsuccessful. However, you can divide zero by other numbers (although not by zero), although the result is always 0. 0/6, for example, equals 0.

Mary McMahon
Mary McMahon

Ever since she began contributing to the site several years ago, Mary has embraced the exciting challenge of being a wiseGEEK researcher and writer. Mary has a liberal arts degree from Goddard College and spends her free time reading, cooking, and exploring the great outdoors.

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Discussion Comments


I agree with anon and would like to see it presented mathematically. The only difference is: I see 6 x 0 as six of something being multiplied by nothing 6 x 0 = 6 and 0 x 6 as 0 of something being multiplied by 6. Since there is nothing to begin with then there is not anything to multiply therefore 0 x 6 = 0. Somehow, I think this benefits a capitalist economy.


So if we have 0 times 6 oranges (a group of 6 oranges), but we are saying there are 0 of those groups, it means no oranges, we have nothing.

I do not understand why is this such a hard concept. If we just listen to it, it makes sense, 0 times 6 oranges, means we have no times the amounts of oranges that we could have, meaning we have no oranges.


Six oranges multiplied by zero means that you ate the oranges. I just don't understand how people can't see that 6X0=0. It's like 6X2 is 12. Meaning: there is one 6 and there is another 6. There are 2 sixes, so to speak. All together, there are 12 items. Similarly, 6X0 is 0. Meaning there are no sixes at all


Mathematically, it doesn't matter how you arrange the terms in multiplication. 2 x 3 x 5 = 30 can be rewritten as 3 x 2 x 5 = 30 or 5 x 2 x 3 = 30 or 5 x 3 x 2 = 30. The answer is always 30 because the terms are always 2, 3, and 5.

Basically what you're saying with 6 x 0 is that there are no (0) groups of 6. 0 x 6 basically means the same thing, just rearranged slightly (6 groups of nothing compared to no groups of 6).

Basically, multiplying by 0 or dividing a number into 0 is saying that there was nothing in the first place.


If I have 6 oranges and multiply them by zero, I still have 6 oranges...they do not cease to exist or disappear...they neither increase or decrease, so 0 X 6 = 6 where as 6 x 0 = 0...I do not understand the property of multiplying something by zero (nothing) leaves nothing, since this is a short method of addition. 0 X 6=6 just as 0+6=6 but 6 x 0 is the same as 0+0+0+0+0+0=0. Can you show this mathematically as opposed to just stating in words?

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