How many zeros do we need to add to get a nonzero value?

A student (kid) of mine asked this question to me. I am not sure what to make of it or how do I answer it.

How many zeros do I need to add to get a non-zero value?


Could someone help? If I were to explain it slightly mathematically, what approach I can take?

• I couldn’t guess exactly what they meant, but try this: hold an empty cup and act like you’re pouring its (empty) contents into an empty bowl. Repeat this several times in front of them, and ask how many times you need to do this before the bowl is no longer empty. Sep 11, 2021 at 5:33
• Answering is probably not the point at all. Once you know what they have in mind when using words like "zero" and "adding", you can provide an explanation, but not before that. And then, of course, the explanation will probably be something along the lines of "adding zeros gives zero", like @NickC suggested. Sep 11, 2021 at 7:07
• Actually it's Not Zero OR Non-Zero, given just to make it clear. No there is no division involved. Sep 11, 2021 at 12:23
• In Calculus / Analysis "adding" uncountably many 0s can be nonzero, but adding countably many 0s is 0. Sep 13, 2021 at 22:47
• I will leave an answer Sep 15, 2021 at 1:10

Suppose that you want to know that $$\underbrace{0+\cdots+ 0}_{N} = 0$$. We know that $$0+0=0$$ and also have that $$\underbrace{0+\cdots+ 0}_{n+1} = (\underbrace{0+\cdots+ 0}_{n}) +0$$ by associativity. Then proceed by mathematical induction, reducing $$(\underbrace{0+\cdots+ 0}_{n}) +0$$ to $$0+0=0$$.

explaining by contrapositive

Adding up any number of zeros gives zero; so, it is impossible to obtain a nonzero value by adding up just zeros.

Let $$c$$ be a nonzero value.

Assume that $$n\in\mathbb Z_0^+$$ zeros are required to additively obtain $$c,$$ i.e., $$\underbrace{0+0+\ldots+0}=c.\\\quad\;n\text{ times}$$

Then $$n\times 0=c;$$ so $$c=0.$$

Hence, $$c$$ is both zero and nonzero—which is absurd; our assumption must have been false; i.e., no countable number of zeros can sum to a nonzero value.

Addendum (expansion of my first comment below)

Consider a target-archery competition in which a contestant has scored $$7$$ points. Then the probability that they have hit the circular target is $$1$$ yet, given any point on it, the probability that they have hit that point is exactly $$0.$$ Here, the sum of uncountably many zeros turns out to be nonzero. (Measures only need to be countably additive.)

• @Ashish Shukla: Reply by a smart aleck kid -- What about Zeno's millet seed paradox (see also this search). Or what about the fact each point has zero length, but when you put enough of them together to form a line segment, then their lengths add up to a positive value? Sep 11, 2021 at 17:40
• @Dave Or the fact that although the probability of hitting a dartboard may be nonzero, the probability of hitting its centroid is zero? Sep 11, 2021 at 19:36
• Better is that for each point on the dartboard, the probability of hitting that point is zero. However, the probability of hitting some point on the dartboard is $1$ (assuming the dart always hits the dartboard). Of course, this is essentially the same as points adding up to a positive length (here we have zero-area points adding up to a positive-area value), but for the students one would be telling this to, I think it will seem sufficiently different to also use. Sep 11, 2021 at 19:50
• @Sue VanHattum: sometimes infinity times 0 is not 0 --- This assumes the student knows that repeated addition is multiplication. See, for instance, the sentence beginning with "My impression is that half of US university students" in this recently posted answer. Nonetheless, I suspect any student asking this kind of question is in the other half (even at a much younger age). Sep 11, 2021 at 22:52
• I would ask the student to clarify what s/he means instead of asking us. I wouldn't presume to know what the student is thinking. Sep 12, 2021 at 17:33

You don't say what age the student is or what their mathematical background is. Your question is also phrased as a mathematics question, not an education question. Both of these things make your question, in its present form, not very well suited to this site.

If the student is in high school and hasn't had calculus, then a reasonable thing to do here would be to explain that there are various number systems that extend the reals to include infinite quantities. Examples:

https://en.wikipedia.org/wiki/Extended_real_number_line

https://en.wikipedia.org/wiki/Hyperreal_number

https://en.wikipedia.org/wiki/Surreal_number

The student's question can only be answered in such an extended number system, not in the real number system, since the reals don't include infinity. The student's question is equivalent to asking 1/0, and actually none of these systems make that a well-defined thing. But, e.g., the hyperreals do allow you to divide 1 by an infinitely small number and get an infinite result.

If the student has had calculus, then you can tell them that they're basically reinventing indeterminate forms.

• Hi, Are you a Teacher? See neither age nor prior Mathematical knowledge should be required for a kid to understand Zero, and BELIEVE ME IT ISN'T. I explain them Zero and then I ask, if 1 is One of SOMETHING, THEN ZERO IS NOTHING OF ____________. and invariably kids say EVERYTHING. It's not important that they know "The Definition", the important point is THAT THEY SAW ZERO. THEY COMPLETED MY SENTENCE!!! So "question must satisfy certain parameters", "they should know this" is a perfect example of "By the Book" teaching. A kid was trying to know Zero, I couldn't answer, hence asked? Sep 20, 2021 at 4:27
• If the student has had calculus, then you can tell them that they're basically reinventing indeterminate forms. --- To me indeterminate forms are formulations of rates of growth, not some kind of generalized number, although I suppose one could classify things like Hausdorff gaps as a type of number. Also, calculus is not needed to understand indeterminate forms (only precalculus limit ideas at a non-rigorous level), although students almost never work with them, or even see them, until calculus. Sep 20, 2021 at 13:53