Add up every positive whole number-one plus two plus three plus four etc -and eventually you get infinity.

And yet, if you do some careful mathematical trickery, a very different number appears: minus one twelfth.

Maybe you saw this strange result do the rounds of the internet a few years ago.

Maybe you also correctly surmised that it's just some mathematical sleight of hand.

But maybe you didn't know that exactly this way of extracting finite numbers from infinities is at the heart of some famous results in physics.

Some of them are even equal to minus a twelfth.

When you add an infinite list of numbers together, sometimes the result settles down to a finite value.

For example, a half plus a quarter plus an eighth and so on equals one.

As long as we're careful about ensuring convergence, then an infinite sum can equal a finite number.

But many infinite series do not converge-their result really is infinity.

And it is pretty obvious that the sum of the natural numbers is such a divergent series.

1+2+3+ etc equals infinity.

So why were people going around saying that it equals -1/12?

It takes a bit of mathematical hocus pocus, The OG viral video on this is on Numberphile, and their "proof" does some rather loose arithmetic with a few different infinite series.

And by loose, I mean not mathematically valid.

It's linked below, as well as my favorite debunking video.

Now, to be fair to Numberphile, that video was trying to give an intuitive description of a much more rigorous argument based on the Riemann zeta function.

This is a famous function-subject of the famous Riemann hypothesis, which as of this recording hasn't been solved by AI yet.

It can be written as sum-over-n-from-one-to-infinity of 1/n^s.

As long as s is greater than 1, the function converges.

In fact, the infinite series itself only defines the zeta function for values of s greater than one.

But a creative mathematician can rewrite the same function in other forms that agree perfectly with that series wherever it converges, and those new expressions continue smoothly into values where the original sum diverges.

That process is analytic continuation.

At s=-1, this analytically continued zeta function has the value minus one twelfth.

3blue1brown has an amazing video on this stuff, also linked below.

None of this means that 1+2+3+.. et cetera equals -1/12.

It doesn't.

That fraction is the zeta-regularized part of a genuinely divergent sum.

But this also isn't just a cheap maths trick.

This result really does show up in physics.

And in physics it's actually been measured.

With the Casimir effect.

So, in 1948 Hendrik Casimir figured out a way to delete part of the quantum vacuum.

As you may have heard, empty space isn't really empty.

Quantum field theory describes it as being filled with quantum fields in their vacuum state.

Each field supports an enormous range of possible vibrational modes, and even when none of those modes contains a real particle, each still retains an irreducible amount of quantum fluctuation at every allowed wavelength.

OK, now take a pair of conducting plates and put them extremely close together.

The plates exclude a bunch of modes in the quantum electromagnetic field in particular.

Only EM waves whose nodes line up with the plates are allowed-and that means those for which an integer number of half-wavelengths fit exactly in the gap width.

It's just like a guitar string-its sound is the combination of those notes whose half-wavelengths fit between the fixed ends of the string.

That's a 1-D analogy, but even with the Casimir effect, one dimension gives the right idea.

We can also think in terms of frequency instead of wavelength.

There's a smallest allowed frequency and then only frequencies that are an integer multiple of this are allowed, but all such frequencies are allowed.

The energy of each mode is proportional to its frequency, so if you want the total energy in allowed EM modes between Casimir plates you get some base energy times 1+2+3 etc.

That sum should now sound very familiar.

It sounds like it predicts that there's infinite energy in the vacuum between our Casimir plates.

And that's after excluding certain modes-which means there should be infinity ... plus?

energy outside the plates.

This "prediction" of infinite energy in the quantum vacuum is not something we can take literally.

To compare the vacuum within the plates to the vacuum without them, we introduce a cutoff that temporarily stops us from adding arbitrarily high-frequency modes.

For real metal plates this even has a physical motivation: at sufficiently high frequencies the metal stops behaving like a perfect reflector, so those modes barely notice that the plates are there.

But cutoff or no, as long as this background energy is the same between one location and another, it feels like there's nothing there.

We only detect differences in energy.

Which is why those excluded modes between the Casimir plates are important.

Both the inside and outside of the plates appear to add up to infinite or very large energy, but after regulating them in the same way, the vacuum energy with the plates is slightly lower than the reference vacuum outside them.

There's a process for comparing infinities to find their finite difference.

The trick, as I hinted, is to separate the varying finite part of the sum from a shared infinite part.

That latter part can be expressed as a shared high-energy cutoff.

This process of introducing a cutoff to deal with divergent terms is one way to do what we call regularization.

Zeta regularization, which involves the analytic continuation I mentioned, is another way.

So, in the Casimir effect, we figure out what divergent part is shared by the interior and exterior of the plates and subtract that.

We end up with a Casimir energy of pi-hbar-c/d * ½ * -1/12.

The first factor sets the energy scale of the fundamental mode, where (d) is the distance between the plates, ½ comes from the zero point energy of the quantum oscillator.

And then there's the -1/12.

And yes, it's back.

And it's back for the same reason-we plucked a finite part of an infinity.

Here the minus sign really matters because it tells us that energy gets even lower as the plates move closer together.

The system therefore lowers its energy by shrinking the gap-and that means an attractive force.

The resulting force has been successfully measured, and the Casimir effect has been verified, and it's exactly what we expect from the weird maths.

The real three-dimensional electromagnetic calculation is more complicated because the modes can also carry momentum parallel to the plates and because the electromagnetic field has multiple polarizations.

But the same basic logic works, though the -1/12 is instead +1/120.

But yeah, the infinite sum really does sometimes come out to -1/12 or something, but only in the context of regularization and renormalization, which I'll come back to.

But first it's time for some string theory-which may or may not have anything to do with nature.

But our -1/12 shows up here too, and actually gives us one of string theory's weirdest properties.

So, string theory replaces point particles with tiny filaments vibrating across multiple compact dimensions.

Just like a guitar string, a quantum string can vibrate in different modes, and those modes determine the properties of the emergent particles.

In the guitar string and Casimir, the allowed modes come from the fact that their ends are fixed in place.

Quantum strings also have a boundary condition, but in the case of closed strings, which we'll focus on, it's because the strings are loops.

Every oscillation has to match up with itself when traced around the loop.

But the result is the same.

Allowed frequency and energy modes are every integer multiple of the base mode.

Which means we once again have a divergent sum.

In the case of the quantum string, regularization involves introducing a cutoff.

However, this cutoff feels less ad hoc than in regular QFT.

The thing we want to calculate based on these stringy oscillations is the overall energy spectrum of the string.

What are the possible global oscillations?

The shortest-wavelength modes may correspond to extreme energies.

But at those tiny scales, such a fluctuation only sees an infinitesimal patch of the total string-it doesn't know about the global periodicity.

Those high-frequency contributions are at both local and constant all around the string, and don't contribute to the string's energy spectrum that we're trying to calculate.

We can ignore them in the same way that we can regularize away the high-frequency modes between the Casimir plates when those tiny fluctuations have no idea the plates are even there.

So again we have a cutoff for our infinite sum and lets us figure out the size of the finite component of the string's vibrations.

A couple of things for perfect precision.

First, I've been talking about oscillations around a 1-D loop.

Really, we think of the string's loop as sweeping out a 2-D surface over time that we call a worldsheet.

Second, all of this stuff-the string, its worldsheet-are embedded within some larger number of spatial dimensions.

Could be 3 like our universe, could be more.

We'll say it's D dimensions for now and see what happens.

Once we account for the redundancies in how we describe the surface of the worldsheet, the genuinely physical vibrations of the string are transverse to the string's direction of motion.

That leaves D-2 independent transverse directions in which the string can oscillate.

In ordinary four-dimensional spacetime that would leave two physical transverse directions, but more if D is larger.

And every one of those directions has its own infinite tower of modes.

So the total zero-point contribution is proportional to the energy is this: \frac{D-2}{2} (1+2+3+4+\ldots).

Now use the same regulation trick as we used for the Casimir effect to get rid of the inconsequential high energy modes.

In the quantized string this shows up as a shift in the string's energy spectrum-the so-called normal-ordering constant.

We again use the Riemann zeta function to replace the infinite sum with -1/12, giving us: - \frac{D-2}{24}.

And this is where the -1/12 has some crazy consequences.

Now I have to get a bit hand-wavey here, so please forgive me.

If we bring in relativity and force the quantum theory on the worldsheet to preserve Lorentz symmetry, the D-2/24 part has to equal one.

This is the only relationship consistent with our non-negotiable symmetries.

But this is only possible if D equals 26.

And D is the number of dimensions.

So yeah, if you've heard that string theory requires lots of hidden dimensions, it's because that - 1/12 is forcing string theory's hand.

D here is the famous critical dimension of bosonic string theory.

Now, bosonic string theory itself is not a realistic theory of nature.

Among other problems, it doesn't naturally have fermions, which are needed for actual matter.

Modern string theories do include fermionic vibrations on the worldsheet, and these partially cancel the bosonic vibrations, reducing the number of dimensions needed.

But the basic idea survives: regulate the infinite quantum fluctuations, keep track of their finite quantum contribution, and demand that the symmetries of the theory remain intact.

Now for superstrings that consistency condition gives ten spacetime dimensions rather than twenty-six.

But the same minus-one-twelfth-style regularization is still hiding inside the calculation.

So this brings us back to the original mathematical trick.

In the physics examples, regularization gives us a controlled way to handle an infinite mode sum, and renormalization tells us which divergent parts are universal or unobservable and can be subtracted away.

Zeta regularization-the mathematical procedure that got us our 1+2+3..=-1/12 - is related to this.

But instead of introducing a cutoff, it uses analytic continuation to assign a finite value to the divergent series.

And, in fact, for the one-dimensional Casimir example and the simplest bosonic-string calculation, we can literally use this as the Riemann zeta function to regularize the same divergent sum.

Minus one twelfth is not what you get by adding all the positive integers.

It's what can be left after the universal infinity has been stripped away, and knowing this lets us tame the infinite quantum fluctuations of empty spacetime.