Science & Technology

Add Up Every Number From 1 to Infinity, And You Get ...

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.

Wait, What?

First, we'll need to define some mathematical terms. We're talking about natural numbers, which are positive whole numbers. If you throw fractions and other numbers into the mix, you'll get infinities of a different value and produce wildly different results. Second, the series is what's known as "divergent," because its terms keep growing without limit. This is simple addition, but instead of adding individual numbers together (since that would take an infinite amount of time), we're applying addition to sets of numbers — and our sets of numbers happen to already be infinite.

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.

– Numberphile

Key Facts In This Video

1. The result of the infinite sum (1 + 2 + 3 + 4...) is used in string theory. 00:37

2. The result of the infinite sum (1 - 1 + 1 - 1 + 1...) is 1/2. 01:54

3. 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

Neither numbers nor infinite sets actually occur by themselves. There is no physical thing that corresponds to the number -1/12, just like there is no physical thing that corresponds to the sum of all natural numbers. But when these concepts produce testable, observable results in physical experiments, they become part of mathematical reality. This is what makes numbers useful in the first place — for counting things and solving problems. If an equation works, it can accurately be described as true.

The sum of all natural numbers is a great example of this because physicists actually use it. For instance, it's used to represent the energy difference of a vacuum-energy force quantum physicists call the Casimir Effect. The Casimir Effect describes a force that permeates the entire universe and can be "trapped" in a way by placing two uncharged conducting plates sufficiently close to one another. A force that permeates the entire universe must be infinite — and experiments have shown that the difference in energy between the observed force (between the two plates) and the force acting on the rest of the universe corresponds to the sum of all natural numbers.

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.  