What do you get when you add up every number from one to infinity? Obviously, if you stop this series at any point before infinity, it will just produce a very large number. But in fact, if you calculate the infinite series all at once as a set, you’ll get an unusually specific, non-infinite answer: -1/12.
The commutative, associative, and distributive properties that you remember from algebra class all apply to sets, but sets also enjoy additional special properties that numbers don’t. These properties make it possible to extrapolate sets of infinite series into natural numbers.
These natural numbers are not approximations of the sum of the infinite series but the average of all the series’ terms. At first glance, this might look like cheating, but it’s not. The most important thing to realize about sums of infinite series is that they correspond to reality, even if they don’t necessarily participate in reality the way rocks, trees, and human beings do.
- The result of the infinite sum (1 + 2 + 3 + 4…) is used in string theory.00:37
- The result of the infinite sum (1 – 1 + 1 – 1 + 1…) is 1/2.01:54
- It’s hard to conceive of an intuitive reason for why the infinite sum (1 + 2 + 3 + 4…) produces its strange result without writing out the proof.06:38
In fact, all numbers, sets, and mathematical principles are just abstract concepts. This should make the fact that the sum of all natural numbers is -1/12 much easier to wrap your mind around — you can’t exactly take out a calculator and add every number forever, but you can describe a system that does exactly that.
Infinite Sets Don’t Occur in Nature
Unsurprisingly, infinite sums are hard to work with. The universe doesn’t actually produce infinite energy in a laboratory, but the amount of energy the Casimir Effect produces is always in accordance with this summation. As a result, when physicists and mathematicians need to describe the impact of the Casimir Effect on other systems, they replace the sum of all natural numbers with -1/12 and, astonishingly, get accurate, observable results.