HOW BIG?

We were talking about big numbers. She was saying how it's wacky how we talk about millions, billions, and trillions almost interchangeably. I was agreeing and noting how if you ask folks to point to the place on a linear plot from zero to a billion where they think a million would land, they commonly point to a spot somewhere in the middle of the line, about half way between a billion and zero, and rather far from where you would actually find: it right at the start next to the zero.

It just seems to be true that most of us don't have good intuition for big numbers. Why would we? Even very small numbers are challenging to manage. Don't most folks, unless they're really trying, have a tough time storing and recalling more than eight digits from their short-term memory. Eight does seem like a number residing at a very human scale.

From there, we talked about our favourite large number stories and visualizations. Mine were the Sand Reckoner, 52 factorial, and -1/12.

ARCHI' THE RECKONER

Somewhere along your mathematics journey in school most of us learned about how thousands of years ago Archimedes, the Ancient Greek mathematician, set out to determine how many sand grains you could pack into the universe. To get there, we're told he had to invent new language. At the time of his thought experiment the Greek numbering system topped out at 10,000, called myriad (or μυριάς). No, I'm not sure why 10,000+1 was inconceivable.

Archi' eventually realized he could talk about a myriad myriads, 100,000,000 (or 10⁸). He also figured he could multiply such large numbers. (Doing so he also found and explained the law of exponents. 10⁸x10⁸=10¹⁶) And he went on naming numbers as he scaled up to the obscenely large, eventually capping at orders of the myriad-myriadth period. He took 10⁸ to the power of 10⁸ to the power of 10⁸... *Gak!* As I understand, to write the number in words we would say it was a one followed by eighty quadrillion zeroes. That seems like plenty. Archimedes felt so too.

With this system in place he went on to grok the scale of the universe (two thousand years ago.) His starting assumption? Well, he went with the universe as described by his colleague Aristarchus, who and understood that the stars and Sun were stationary and the Earth revolved around the Sun. Observing no shift in the stars relative to other stars, stellar parallax, the ancient Greeks reasoned that the stars must be at a tremendous distance from the Earth. From there he had to make a bunch of assumptions, such as the perimeter of the Earth, the scale of the Sun, the universe being spherical, and more.

He landed on a diameter for all there is being not more than 10¹⁴ stadia. In modern units we would say this is a distance of two light years. He had to sort out how many grains of sand in a stadia and from there did a little geometry to land upon the upper limit for the number of grains of sand you could pack into his universe-sphere. The number was 10⁶³. We have a name for that: one vigintillion... That's Archimedes, the reckoner of sand.

FIFTY-TWO!

Now, how to count to:

80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000

In scientific notation this number approximates to 8.0658 × 10⁶⁷ (an 8 followed by 67 zeros, for a total of 68 digits). Folks like using this particular number to illustrate very large numbers as it's the number of possible combinations you can make out of a deck of cards. Also known as 52 factorial (also written as 52!). Though obscenely large, you can write it out as above and, like a million or a billion or a trillion, this makes it seem approachable. To shatter that illusion and bring you back to reality, folks have worked out way to try and conceive of eight to the power of sixty-seven:

• Stand on the equator and start a stopwatch.

• Watch the stopwatch tick away, tallying the seconds that have elapsed until one billion years pass. (Even with all those seconds, you have not even begun counting seconds.)

• Once a billion years have elapsed take one step forward. Then repeat: wait a billion years, take a step.

• After you've walked around the entire perimeter of the Earth and returned back to where you started, remove one drop of water from the Pacific Ocean.

• Repeat the process until you've emptied the Pacific Ocean of all its water.

• Once completely dry, refill the ocean and place a single sheet of paper on the ground.

• Then repeat this process of circling the globe, taking a drop of water, emptying the ocean and refilling it, and placing sheets of paper until your stack of paper reaches the Sun.

• When you eventually reach the Sun you will notice that if you stack paper to the Sun one thousand more times you still have the vast majority of your counting still ahead of you.

And that is why each time you shuffle a deck of cards the sequence of those cards is truly unique. Love that.

NEGATIVE ONE-TWELFTH

This is my favourite number. I came across the number, like most who have, in the Numberphile video from 2014 featuring physicists Ed Copeland and Tony Padilla. If you want to watch, check out their revisit from 2024, here (where Tony also mentions his new paper on the matter, that "introduces a new ultra-violet regularisation scheme for loop integrals in quantum field theory" inspired by "the method of smoothed asymptotics developed by Terence Tao.")

One intuits that if you add all the natural numbers, 1+2+3+4+5+6..., to infinity (known in math as a "divergent infinite series") that whatever that sum is it will be pretty darn big. In fact, nothing is more obvious than that adding ever-larger positive numbers for the rest of time and beyond results in something boundlessly large, if that could even make sense.

It seems perfectly obvious that the above reckonings of sand or all possible sortings of all possible decks of cards would be just the smallest fraction of this much more gargantuan total.

The Terence Tao, tells us "NO!" As infinities like this spring up all the time in the worlds of math and physics, you're either left hiding under your bed trying to think happy thoughts or needing to find a work-around.

Copeland, Padilla, Tao and their ilk us (mere mortals) that using fancy-pants techniques like Ramanujan summation, zeta function regularization, or analytic continuation of the Riemann zeta function, mathematicians and physicists are able to do real work when they assign what should be an infinite value the finite and truly radical value -1/12.

This number, negative one-twelfth, isn't the literal sum of all natural numbers (which would be nonsense in standard addition) of course, but — with rigorous definitions and careful caveats — this regularized value makes such divergent quantities finite and useful in the deepest of math and theoretical physics.

So next time someone asks you for your favourite number or just a big one do as I do and go with -1/12.