Quora Answers 18/5/2020: Is Zero a Number?

Today we are starting a new series where I share some of my more interesting Quora answers. Today we are talking about nothing. Well, the concept of zero anyway.

How will this work? I will go through Quora every once in a while and answer questions. And make one of these posts if I have enough interesting worth sharing here.

How often will I do this? I am not sure yet. I will try do this at least a few times a week, trying to answer more than one question at a time.

Today I answered this question:


Is it possible that Zero is just a space and not a number in mathematics?

My Answer

What do you mean by a “space”? Without knowing what you think this means, it is hard to say that it is or is not a space. But, it does not fit any definition of “space” that I am aware of.

Zero is a number. It is that number which you can add to any number and get that number. When other people here say zero has “additive identity”, that is what they are talking about.

Granted, it is a somewhat interesting number, at least compared to the natural numbers. The natural numbers are the whole numbers, but not zero (although some people will include zero as a natural number).

The natural numbers are easy to grasp. They refer to easily observable quantities of things.

For instance, you can grasp the concept of “one” by seeing a singular entity. Two is that unit there and that other one. Three is those two entities and that other one over there. And so forth.

Perceptually, zero is a lack of entities, it corresponds to the absence of something. It means that there are not any of the perceptual entities we are talking about.

But, it is still a number. In this very perceptual, low-level sense, it refers to a lack of the entities being counted.

Of course, numbers are not just used to count concrete objects. They are used to measure things. Zero is a number on the number line. Although, you can say a few interesting things about it.

It lays between the negative and positive numbers. When measuring things, it can indicate“no units” of this. And/or act as an intermediate value between “positive units of this” and “negative units”.

Time for some more abstract mathematics, although nothing overly complicated.

Zero times any number is zero. Any number plus or minus zero is that same number.

Division is a little more complicated. You cannot divide by zero. It is easy to see why if you think about “x divided by y” as “taking x and cutting it into y pieces”.

Imagine trying to cut a pie into zero pieces? What does that even mean?

What then, does it mean to take x and divide it into zero pieces? Um, nothing, this is a contradiction in terms, it makes no sense. This is why we say that dividing by zero is undefined, as it makes no sense to talk about trying to do this.

Another way to see this is to see m / n as trying to see how many times n goes into m.

So what then is 5 / zero? How many times does zero go into 5? Does this even make any sense? No. Not really. Zero does not go into any number any number of times. It makes no sense to ask “how many zeroes are in a number?”

The number zero is an incredibly useful concept in mathematics. There is a lot to say about nothing…

Below is a simplified version of the more technical answer. Feel free to skip this, since I think it is already clear why division by zero makes no sense.

We are going to use the * symbol for multiplication. It is what I am used to having being trained in computer science and I find it adds readability.

Suppose you have m * n = p. Let us suppose we have 5 * 3 = 15.

We can view multiplication as having a certain number and adding it together a certain number of times.

For instance, 5 * 3 is equivalent to 3 lots of 5 and adding all those together to get 15.

In general if you have m * n = p, then m * n is the same as:

Take m and add it together n times to get p.

So, suppose we have 5 * 3 = 15

But we want to figure out to get that 3. In other words, we want to figure out how many times to add 5 to itself to get 15.

So, division is the inverse operation of multiplication. We want to find out for m * n = p, what is the value of n?

In other words, n = p / m, in this case 15 / 5

for 5 * n = 15, we know it is 3.

What then is 15 / zero? Well, in this case, m = 0 and p = 15.

So, this is equivalent to:

zero * n = 15

But, that makes no sense there are no values for n for which zero * n = 15.

Look at this way, zero * n = p is like saying, how many times do you add zero together to get p, where p is not zero?

This makes no sense. Zero of anythings is always zero, not some other number.

It does not matter what your values of p or m, you have a similar problem, division by zero is equivalent to

zero * n = p, where p not equal to zero.

But that is a contradiction, as zero times anything is zero. Therefore, division by zero makes no sense.

What about zero * zero = zero? Well, how many times do you add zero to zero to get zero? That is meaningless. You cannot add zero to zero any number of times. So, we say that division by zero is always undefined.

If you liked this, you can find out more about mathematics here.

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