By Heath Raftery

You are currently participating in a mighty event.

When it is over the world will have moved 2,000 kilometres through the solar system. Mankind will have expended around 30 million megajoules just staying alive and added about four billion grams of carbon dioxide to the atmosphere in the process. Approximately 14 hectares of forest will have been cut down and more than 100 people will have died.

The Earth will be hit by lightning 6,000 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 five lines 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 30,000,000,000,000 (30 quadrillion) joules, but there are over 60,000,000,00 (six billion) people on the earth, and it takes about 4,500 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.

Confounded yet? You should be. The inherent difficult in comprehending large numbers should ideally mean we pause 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 large 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 token every second on your way to building your physical display of one trillion, it would take you over 30,000 years to create your work. Fortunately, we can model the same process, like so. Getting a feel for a trillion? Take a minute to look at this representation – it’s a good one!

Physical appreciation is an excellent tool for appreciating orders of magnitude – and is exactly the skill nurtured when you played with counting blocks in primary school. Stacking objects in collections of 10 or 100 or 1,000 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 billions and trillions. A trillion of 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.

Here’s what one trillion dollars could buy you…

Physical realisation, rates and ratios are fine but it requires someone going to the trouble of presenting them. So what are we, as casual readers of newspapers, supposed to do without that help? 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 1,000 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 10,000,00 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 10,000,00 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 the $4.8 billion Airport Link project costs 4.8 with nine orders of magnitude and is therefore three orders of magnitude (or a thousand times) larger than the $4.5 million (4.5 with six orders of magnitude) cost worn by project manager Thiess John Holland.

Or you could read about energy efficiency in Toronto schools and realise 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.

There’s still a few maths-related events you can head to as part of National Science Week.

Wednesday 26th August: From certainty to fallibility: an epic tale from the history of mathematics

Friday 28th August: Mathematics and Sex with Clio Cresswell

Exhibition until Sunday 30 August: Sydney Hyperbolic Crochet Coral Reef exhibition

Tags: maths, numbers

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