How many billions in a trillion?
You are currently participating in a mighty event - when it is over the world will have moved two thousand kilometres through the solar system. Mankind will have expended about thirty million megajoules just staying alive, and added about four billion grams of carbon dioxide to the atmosphere in the process. About fourteen hectares of forest will have been cut down and over a hundred people will have died. The Earth will be hit by lightning six thousand times and there's not a thing you can do to stop it. That's because these hundreds, thousands, millions and billions of things happen in a minute, every minute. Yep, that mighty event and the numerous consequences occurred while you were downloading and reading the first paragraph of this article.
Two thousand kilometres may seem like a long way, but it's just a minute of the year long trip the Earth takes around the sun. Thirty million megajoules is 30000000000000 (30 quadrillion) joules, but there are over 6000000000 (6 billion) people on the earth, and it takes about 4500 joules to power your body for one minute. Four billion grams of carbon dioxide weighs more than 24 Boeing 747s, yet that's how much the human race naturally breaths out every minute of the day.
Confounded yet? You should be - only by understanding the inherent difficulty in comprehending large numbers can you begin to appreciate the pause we should all take when consuming media articles that contain them. As Xavier Rizos wrote in his post, "Science, non-science and non-sense", comfortably dealing with science is important because science is "a highly cultural and political area and thereby essential". And in a world of trillion dollar debts and billion ton pollution problems, having the numerical literacy to interpret, compare, estimate and appreciate the numbers is essential to comprehension.
There was a time when reports of million dollar corporate bonuses and billion dollar profits would shock the casual reader. Now however, after months of reports of trillion dollar deficits the casual reader is numb. How do we actually appreciate what it means to owe a trillion dollars?
One way to get a "feel" for a trillion dollars is to physically represent it. Unfortunately, even if you collected one representative token every second on your way to building your physical display of one trillion, it would take you over 30 thousand years to create your work. Fortunately, we can model the same process, like so. Getting a feel for a trillion? Seriously, take a minute to read the linked article - physical appreciation is an excellent tool for appreciating orders of magnitude, and that's exactly the skill that was being nurtured when you played with counting blocks in primary school. Stacking objects in collections of 10 or 100 or 1000 is a great way to understand the difference between a billion, 10 billion, 100 billion and a trillion. A trillion is a huge number.
But physical realisation only goes so far when you're getting into the billions and trillions. A trillion anything, physically represented, is certainly impressive, but it's still far enough beyond reality that comparisons and intelligent reckoning is difficult. Another excellent tool we have for appreciating large numbers is by making comparisons - rates and ratios. Take a look at this video to get an idea of what one trillion dollars could buy you:
Now physical realisation and rates and ratios are fine when someone has gone to the trouble of presenting them, but what are we, the casual readers of newspapers, supposed to do? As a mathematician, I'm lazy, and will always look for the simplest answer (it's an Ockham's razor thing). The time honoured tool for comprehending large numbers is the laziest method we have: orders of magnitude. Despite "orders of magnitude" being used in popular literature to mean just about anything, it has a precise meaning in the sciences. Quite simply, an order of magnitude is a power of ten. If something is one order of magnitude larger than something else, it is 10 times as big. If it is two orders of magnitude larger, it is 10 squared or 100 times as big. Three orders is 10 cubed or 1000 times. It gets even easier when you drop the "squared" and "cubed" language and note that the order of magnitude correlates to the number of zeros after the 1. Two orders of magnitude is 100 times. Six orders of magnitude is 1000000 times.
And the laziness doesn't stop there - orders of magnitude are so useful that all those zeroes become tiresome. That's why scientists shorten 1000000 to 1e6. The 'e' (or 'E') stands for exponent, but that's by the by - all you need to know is that the '6' tells you how many zeros there are. That way orders of magnitude can be read straight off the number: 1e9 is three orders of magnitude bigger than 1e6.
There's one final piece of the puzzle that will link all this orders of magnitude business to the numbers you see in newspaper articles. The numbers and unit prefixes we use just so happen to correspond to a very simple pattern of orders of magnitude:
Once you have that relationship down pat, it's a snap to see an article about the BrisConnections fiasco and realise that the $4.8 billion Airport Link project costs 4.8 with 9 orders of magnitude and is therefore 3 orders of magnitude (or a thousand times) larger than the $4.5 million (4.5 with 6 orders of magnitude) cost worn by project manager Thiess John Holland.
Or you could read about energy efficiency in Toronto schools and realise that the annual energy saving of 100 million megajoules has an order of magnitude of 14 (two zeros, plus six for million, plus six for mega) and that the $4.8 billion spent so far has cost about $4.80 per 100 kilojoules saved each year (4.8e9 dollars for 1e14 joules equals 4.8 dollars per 1e5 joules).
Not only is learning a level of numeracy a shortcut to lazy comprehension, numerical appreciation is a vital tool for anyone that wants to consume the news of today with more than robotic capability to respond.