If You Add All The Negative Integers With Positive Integers, Would You Get Zero?

The sum of all integers is a divergent series with no ordinary value. Pairing each positive integer with its negative gives zero, but that is just one ordering, and regrouping the same terms can send the total to infinity.

We know that there are infinitely many positive integers and infinitely many negative integers. If we take a finite, equal count of each, adding them together does give zero, since every positive integer cancels its negative partner. This makes both intuitive and mathematical sense.

However, is it the same for an infinite number of these integers? Let’s check out this brain teaser in more detail!

What Is A Series In Mathematics?

The addition of all the infinite integers (both positive and negative) is neither an easy nor straightforward task, so we need to begin by understanding the mathematical process behind it. To understand this process, we must first become familiar with the ideas of series and sequences in mathematics.

Let’s go in a sequence (no pun intended) and start with sequences first. So, what is a sequence in real life? In our day-to-day activities, we often arrange things sequentially. In the kitchen, the spices and herbs we use most often are kept forward, while those we rarely use are kept in the back.

When a teacher takes test papers back from her students, they arrange the papers in the order of the roll numbers or names. There are many such examples of sequences in our daily lives and the most important thing about them is a pattern. All the sequences follow a pattern, so accordingly, the members of a sequence find their place.

The same is true in math. A sequence is a collection of numbers that all follow a certain rule or order. For example, we can create a rule as a_i = i², where the value of i changes to give us the i^th element of the sequence. Following this rule, the sequence would have the form 1, 4, 9, 16, 25, …and so on.

Now, a series is just the summation of the elements of a sequence. For example, the series of the sequence mentioned above would be

Σ_i=1⁵ i².

This signifies the sum of all the elements in the sequence of i² until 5.

Integers and their addition also form sequences, and their series is used to find the answer to the titular question of this article.

What Are Convergence And Divergence?

We learned what a series is in mathematics: the addition of numbers that follow a specific rule. Now, there can be sequences that go on forever, meaning that there are infinite elements in them. The series of sequences with infinite elements are known as infinite series.

If, in the sequence mentioned in the previous section, instead of stopping at 5, we had gone on until infinity, then the infinite series would have looked something like this Σ_i=1^∞ i².

There are finite series with a single finite answer, such as the one that stopped at 5 (its answer being 55), but what about infinite series? Do they have a finite answer?

The answer is both yes and no.

The answer to an infinite series (a summation of infinite numbers) is taken by first finding a finite sum, and then applying the limit of infinity to it. What does that mean? The entire summation can be divided into n number of smaller summations. Then we apply a limit to these partial summations (applying a limit helps us find the answer using predefined formulae, without actually adding the infinite sums) to get the answer to the entire series.

Instead of actually adding all the infinite elements, we simply find the formula for the partial sum and then use the limit that tends to infinity on it and gets the answer using the predefined formulae of limits.

There are some series whose summation gets very close to a certain number. We can think of it this way: as we add more and more elements to the series (taking the number of terms toward infinity), the running total gets closer and closer to one particular number.

It seems as if the series is converging to a single number. The answer to the series of such infinite numbers is that particular number; this property is known as convergence.

Now, there are other infinite series whose summation never gets closer to any particular number, but continues increasing. In fact, it gets so big that the summation of all the infinite elements turns out to be positive or negative infinity. Such infinite series that do not converge at any particular number are divergent, and the property is known as divergence.

So What Is The Answer To The Series Of Integers?

As we have seen, integers follow a certain rule and therefore form a sequence. Since we want to find the summation of all the integers (both positive and negative), this means we want to find the answer to the series of the integer sequence.

But this is where the problem gets tricky. The summation of all integers (positive and negative) is a divergent series: as you keep adding terms, the running total never settles toward a single number, so the series has no ordinary sum at all. That is the honest answer. The reason it feels like the answer should be zero is that we instinctively pair each positive integer with its negative partner. But that pairing is just one particular way of grouping the terms, and for a divergent series the way you group and order the terms changes what you get.

Let me show you two groupings to make the point clear. If we line up each integer with its own negative, the partial sums look like this:

lim_n→∞ Σ_i=0ⁿ (i − i)

Every bracket is i − i = 0, so this particular arrangement converges to zero. But we have quietly chosen the order; we have not actually summed the integers in their natural order.

Pair the terms differently, for instance grouping each integer with the negative of the previous one:

lim_n→∞ [ Σ_i=0ⁿ (i − (i − 1)) ]

Now every bracket equals 1, so the partial sums march off to infinity and the series diverges, with no finite answer. Two regroupings of the same terms, two different outcomes, which is exactly why the series has no well-defined sum.

This is not a loophole; it is a deep feature of infinite series. The Riemann rearrangement theorem shows that even a series that does converge, but only conditionally (its positive and negative parts each blow up to infinity), can be reordered to add up to any number you like, or to diverge entirely. A blunt divergent series like the integers is even less constrained, so “the positives and negatives cancel to zero” describes one arrangement, not a true total.

Conclusion

So, would adding all the negative integers to all the positive integers give you zero? Strictly speaking, no. The sum of all integers is a divergent series with no ordinary value, so there is nothing to equal zero in the first place. Zero is simply what you get from one tempting way of pairing the terms, and a different pairing sends the same terms off to infinity.

There are more mathematically rigorous ways to handle stubborn series like this, but the honest takeaway is the same: the appealing answer of zero comes from the order in which we add, not from the integers themselves.