James' World 2

Click the link below the picture

.

Here’s a math problem that everybody can solve: What is 1 − 1? 0. So far so good. If we then add a 1, the sum grows, but if we subtract yet another 1, we’re back at 0. Let’s say, we keep doing this forever:

1 – 1 + 1 – 1 + 1 – 1 + …

What is the resulting sum? The question seems simple, silly even, but it puzzled some of the greatest mathematicians of the 18th century. Paradoxes surround the problem because multiple seemingly sound arguments about the sum reach radically different conclusions. The first person to deeply investigate it thought it explained how God created the universe. Its resolution in modern terms illustrates that mathematics is a more human enterprise than sometimes appreciated.

Take a guess at what you think the infinite sum equals. I’ll give you multiple choices:

A. 0
B. 1
C. ½
D. It does not equal anything

The argument for 0 comes naturally if we include suggestive parentheses:

(1 – 1) + (1 – 1) + (1 – 1) + …

Recall that in mathematics, the order of operations dictates that we evaluate those inside parentheses before evaluating those outside. Each (1 − 1) cancels to 0, so the above works out to 0 + 0 + 0 +…, which clearly amounts to nothing.

Yet a slight shift of the brackets yields a different result. If we set aside the first 1, then the second and third terms also cancel, and the fourth and fifth cancel:

1 + (–1 + 1) + (–1 + 1) + (–1 + 1) + …

Again, all the parentheticals add up to 0, but we have this extra positive 1 at the beginning, which suggests that the whole expression sums to 1.

Italian monk and mathematician Luigi Guido Grandi first investigated the series (the sum of infinitely many numbers) in 1703. Grandi, whom this particular series is now named after, observed that by merely shifting around parentheses he could make the series sum to 0 or 1. According to math historian Giorgio Bagni, this arithmetic inconsistency held theological significance for Grandi, who believed it showed that creation out of nothing was “perfectly plausible.

The series summing to both 0 and 1 seems paradoxical, but surely option C (½) is no less troubling. How could a sum of infinitely many integers ever yield a fraction? Yet ultimately, Grandi and many prominent 18th-­century mathematicians after him thought the answer was ½. Grandi argued for this with a parable: Imagine that two brothers inherit a single gem from their father, and each keeps it in their own museum on alternating years. If this tradition of passing the gem back and forth carried on with their descendants, then the two families would each have ½ ownership over the gem.

As proofs go, I wouldn’t recommend putting the gem story on your next math test. German mathematician Gottfried Wilhelm Leibniz agreed with Grandi’s conclusion, but he tried to support it with probabilistic reasoning. Leibniz argued that if you stop summing the series at a random point, then your sum up to that point will be either 0 or 1 with equal probability, so it makes sense to average them to ½. He thought the result was correct but acknowledged that his argument was more “metaphysical than mathematical.”

.

.

.

Click the link below for the article:

.

__________________________________________