Is the Sum of All Real Numbers Between -0.5 and 0.5 Equal to Zero?

The question of whether summing all real numbers between -0.5 and 0.5 results in zero seems straightforward at first glance. After all, there appears to be a nice symmetry - for every positive number in this interval, there is a corresponding negative number of equal magnitude. Surely these should all cancel out to zero?

However, the mathematical reality is more nuanced. The key issue lies in how we define the sum of infinitely many numbers in the first place. In standard analysis, there are two main notions of infinite sums - absolute convergence and conditional convergence.

For a sum to be absolutely convergent, the sum of the absolute values of the terms must be finite. It’s not hard to see this fails for the sum in question. Even restricting to a subinterval like [0, 0.5], the sum 0.5 + 0.25 + 0.125 + … already diverges to infinity. So the full sum from -0.5 to 0.5 is not absolutely convergent.

What about conditional convergence? Here the requirement is weaker - we just need the sum itself to converge to a finite value, even if the absolute values blow up. The classic example is the alternating harmonic series 1 - 1/2 + 1/3 - 1/4 + … which conditionally converges to ln(2).

However, conditional convergence comes with a major caveat. The terms can be rearranged to make the sum converge to any desired value! This is because the positive and negative parts are each divergent series, and these can be interleaved in different ways. So even if we could define a conditionally convergent sum for all numbers from -0.5 to 0.5 (which itself is doubtful), the particular value it converges to would be essentially arbitrary.

In fact, a fundamental theorem states that if a sum is conditionally convergent, then the set of terms must be countable. But the interval [-0.5, 0.5] is uncountable, immediately implying the sum cannot be conditionally convergent either.

So in the standard framework of analysis, the proposed sum is simply undefined and has no meaningful value assigned to it, let alone the specific value of zero. This is an instructive example of how naive intuitions about infinite processes can go awry, and why we need rigorous definitions and theorems to reason about them correctly.

That said, it may be possible to construct alternative definitions of infinite sums that allow assigning values to sums like this one. For example, one could look at sums over arbitrary index sets using ideas from topology like nets and filters. But any such definition would need close examination to avoid paradoxes and align with key theorems of analysis. The standard notions of absolute and conditional convergence have proven their worth and can’t be discarded lightly.

In summary, while the idea that the sum of numbers from -0.5 to 0.5 “should” equal zero might seem appealing, it doesn’t hold up to mathematical scrutiny with the standard tools at hand. Infinite sums and series are subtle creatures, and extending them to uncountable settings like the real numbers requires great care. This question provides a great opportunity to explore the boundaries of the infinite and see why precision is so important.