“Two Kinds” of Zero: Same But Not The Same?

I recently got into a protracted discussion in which the other person insisted that the fact that the character 0 is used in place value notation is merely a place holder is evidence that zero is not a number, but rather a concept we use to indicate the lack of a number.

I have long been of the opinion that the character 0 actually represents two different things in mathematics: A “mere” place-holder and a numeric value. The numeric value is the number that is one more than 1 and one less than 1. The place holder is what distinguishes 1 from 10, for instance.

However, that conversation did lead me to change my thinking about the diminution of zero in 10 to “a place holder”, which comes from detaching what 10 actually means mathematically from the value it represents.

What does 10 actually mean? It means that, in the base in question (usually ten), there is one grouping of “b” items and zero additional items. This is actually how we teach place value in Kindergarten; I know, because this is what my five-year-old son brings home. 23 is two rubber-banded groups of ten popsicle sticks and three loose sticks. 30 is three rubber-banded groups of ten popsicle sticks and no loose sticks.

So, in base ten, 12 is this: ||||||||||||. There are other ways we can represent |||||||||||. In base four, it’s 30: Each group of sticks has four, and we need four groups of sticks. In base twelve, it’s 10: One group of twelve.

In base three, it’s 110. Because three is so much smaller than ten, we have to start grouping our groups. It’s one group of three groups of three sticks, one group of three sticks, and that’s it. In base two, it’s 1100: We have to have groups of our groups of groups, because each base group only has two sticks in it.

This is the way our place value system works: Each digit to the left of the first one represents a nested group: A group of sticks, or a group of a group of sticks, or a group of a group of a group of sticks, and so on.

In each place, we indicate how many groups of that size we need. In those cases where we don’t need any groups of that size at all, we put a 0. “As a place-holder”, we’re told.

But is that the best way to see it? The beauty, the key of our place-holder system is that we can break very large numbers down into manageable pieces. Rather than needing a unique symbol for every integer we might encounter, we have this mechanism for writing integers other than 1, 2, 3, 4, 5, 6, 7, 8, and 9 in terms of those integers (and 0). When we forget that, we get confused and say that 10 is ten.

10 is not ten: These are different mathematical objects. Ten is a numeric value; 10 is a conventional and useful way of representing that value. 10 is “one group of ten and no groups of one”; in reality, it’s “one group of the base unit and nothing left over”. Because the base unit is ten (by default), we learn to see 10 and think “ten”, which causes trouble for people who then learn about other bases.

From this perspective, 0-as-placeholder gains meaning beyond “a placeholder”: It means we have ten (not nine) basic numeric values which we can then use to create all non-negative integers. Note my phrasing in the previous sentence: “Non-negative”, not “positive”.

I’ll talk about negative integers in a moment, and non-integer real numbers even later, but let’s proceed with non-negative integers first. In base ten, 1035 describes how many of each of four sizes of groups we have: We have one group of group of group of ten, no groups of group of ten, three groups of ten, and five left over. It’s unlikely that we really have a stack of 1035 things that are properly grouped; the idea is that we have a pile of 1035 things that we could group in that way if we wanted.

We don’t have to mention the things that we have zero of, and we in fact don’t mention the groups of groups of groups of groups that we don’t have any of unless we have an even larger group. The part about not having to mention the things that we have zero of is a significant reason why we took so long to formalize a zero in the first place, compared to the other numbers. If I’m accounting the livestock I’m selling you, I don’t really need to mark that I don’t have any goats at all, as long as we agree that I’m only selling you what I’ve specifically listed.

And this is the mindset that leads to seeing 0 as a place-holder: If I mention how many groups of size basen I have, I also have to mention how many groups of size basen-1 I have, all the way down to base0 (where 0 is clearly a value, not a place-holder… right?). But, as a rule, I don’t mention how many groups of size basen+1 I have, because it doesn’t matter. Which is to say: 1 must mean one, 10 must mean ten, and 01 is not considered well-formed in standard mathematics.

The issue is, for me, that the “place-holder” mentality then leads to the view that 0 in this context doesn’t have any actual meaning. That’s not true. In 102, 0 has the same depth of meaning that 1 has, and it’s equally contextual. 1 “means” one hundred; 0 “means” no tens; 2 “means” two left over. 102 is not the same as one hundred two, but rather, it’s the most standard conventional way of representing the concept “one hundred two”.

Further than that, 0 by itself can sometimes be argued to be a placeholder. Consider this problem: \[4x – 2 = 3x\]

My students don’t have a particular problem with moving things across the equal sign, but this sort of problem gives them a surprising amount of difficulty. Many of them want to move the \(3x\) to the left side: \[x – 2 = \]

Then they’re stuck. They’ll conclude that \(x = ^-2\) because that’s all they have left. They didn’t learn, consistently, to put in a zero when everything has been removed from one side. They have a variable and a number and an operator and an equal sign, and they use those pieces to build a valid equation, even if it doesn’t happen to have the same truth value.

Point being, in this case, the 0 is as much of a “place-holder” as it is in 102: It’s “only” there because we can at any point in mathematics declare that we have zero of anything we don’t happen to have any of. I have 0 unicorns, 0 griffins, and 0 ghosts living in my house. I could add more to this list, but I’d only actually seriously tell you what I have zero of if it might lead to confusion for me to leave it out entirely (hence, 0102 is well-formed in cases where I have to submit a four-digit number for some purpose).

That doesn’t mean it’s not a value, though. It’s a perfectly true statement: 0 means we have zero of whatever units things are being measured in. There’s none of them, which isn’t the same as the null set.

Wandering into the realm of negative integers, we add an additional layer. Negative-one is a value, just as one is a value; -1 is a representation of that value. As I’ve done upstream, I will occasionally write negative values with a superscripted negative sign to reinforce that 1 represents a single value, as opposed to being a unary function on a value. \(-1 = ^-1\), but the first is the unary opposite operator operating on the value one (just as -(-1) represents two applications of the unary opposite operator on the value one, yielding one), while the latter is the specific value negative one (just as -(-1) represents two applications of the unary opposite operator on the value one, yielding one).

I bring this up because, just as 0 represents an intrinsic part of the conventional representation of a number, so does the negative sign. It’s fine, in other words, to write -1, using the contrast in the previous paragraph, because the value of a properly parsed -1 is the same as the value of a properly parsed 1, that is, “negative one”.

This can be expanded to real numbers, as well. But students also struggle with fractions and irrationals and such. Most students, in my experience, are more comfortable with 1.48 than they are with \(\frac{\sqrt{2}\pi}{3}\), even when the latter is the correct value, because the latter doesn’t look like a number.

Granted, 1.48 is much easier to work with, but I think another reason for this is that we don’t sufficiently reinforce that numeric representations are not the same as their underlying numbers. We do reinforce this throughout the lower grades: Here are twelve apples in a grid. How can we write this? We can write this as 12. We can write it as 4 + 4 + 4, or as 3 + 3 + 3 + 3, or as 4 × 3, or as 3 × 4… these are all ways to represent our twelve apples. Over time, though, 12 emerges as “the best” way to write twelve, which leads to the further implication that 12 “is” twelve. When it’s not, not really.

When we present 0 as “just a place-holder” in 102, for instance, I think we diminish the significance of that distinction between representation and value. And while that might have a short-term advantage, I wonder about its negative long-term effect on number sense.

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