Millions and billions: why big numbers matter

Alpha Centauri Bb
Alpha Centauri Bb: an incredibly long way away.

We have a new neighbour. A rocky planet, barely larger than Earth, has been found orbiting Alpha Centauri B. The double-star Alpha Centauri system is our closest stellar companion, at a negligible 4.3 light years from our Sun.

That’s about 25 trillion miles, or 40 trillion kilometres. Voyager, the man-made object which has travelled furthest from the Earth, has gone about 18 billion kilometres. That sounds like a long way. But if Voyager were the Mayflower, and Alpha Centauri were Cape Cod, then the probe would currently be about half-way out of Plymouth harbour, still a mile or so from the breakwater. I lived in Plymouth when I was a toddler: if I were to look out of my old bedroom window now, 35 years after the Voyagerflower set off, I’d still be able to see it crawling at sub-glacial speeds past Drake Island. It wouldn’t reach its destination until the year 77682AD, or thereabouts. (Also, Voyager’s heading in a totally different direction to Alpha Centauri, but let’s not worry about that.)

We just can’t think in terms of big numbers. For example, US debt is $16 trillion. That’s a lot. PBS costs America $445 million a year. That’s also a lot. But the difference between the two “that’s a lots” is incomprehensible: if the US debt were the voyage of the Titanic (perhaps a better metaphor than the Mayflower, in this case), cutting PBS altogether would get it less than 170 yards away from Southampton dock.

These vast numbers are an incredibly recent invention, in human terms. The words “billion” and “trillion” were only invented in the late 17th century; some tribes are reported not to have any concept of numbers above five at all, although that’s highly controversial. In a legend about the life of the Buddha, recounted in Daniel Tammet’s book Thinking in Numbers, a great mathematician asked the young Siddhartha to multiply numbers, starting with the number koti, about 10 million. He does so, and does so again, until he reaches tallaksana – one followed by 53 zeroes – and still continues, until he reaches a number “greater than the grains of sand in one billion Ganges”. The mathematician, Arjuna, says “One with such knowledge of numbers is incomparable!”; only the truly enlightened, souls on their last earthly incarnation, can count so high.

Modern mathematicians might be pleased to know that they are so close to nirvana. But it’s clearly the case that for most of us, once we get into the millions and billions and trillions (and quadrillions, quintillions and googolplexes), that we simply hear “big numbers”. We overestimate the probability of incredibly unlikely things, like lottery wins and terror attacks, because although we can do the maths with our conscious brain, our instincts can’t make the distinction between possible and plausible odds.

You can see why this is. For a savannah-prowling early human, there’s not a great deal of use for multiplication tables: you might never see more than 100 of your species in one place at the same time, and while there might be millions of starlings, or billions of flies, a simple “lots” probably provides enough information in both cases. It would only have been around the rise of the first civilisations, as trading became more complex – about 4500BC – that even the concept of a “thousand” would have been of much help. We have not evolved to understand big numbers.

This matters. When the banks were bailed out, we were worrying about multi-billion-pound bailouts and multi-million-pound bonuses as though they were in some way comparable. The same is going on all over the news; millions, billions, trillions. As XKCD put it: “The difference between a million and a billion is the difference between me having a sip of wine and 30 seconds with your daughter, and a bottle of gin and a night with her.”

And 0.002 dollars will NEVER equal 0.002 cents.

• And, yes, I typed the first “trillion” as “million” by accident. See what I mean? Tricky, this stuff.


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