Orders of Magnitude

Truck and Forklift by Ricecracker

Imagine you're building yourself a new house, from scratch, and you're trying to guess how many bricks you'll need. You can't be sure in advance what the exact number will be, but you can try to calculate within the right ballpark: will it be thousands? Tens of thousands? Hundreds of thousands? Millions?

Mathematicians call these figures orders of magnitude. Orders of magnitude are calculated by powers of ten: 10^2 (i.e. hundreds), 10^3 (i.e. thousands), 10^4 (i.e. tens of thousands) and so on. For example, any number between 100 and 999 has an order of magnitude 2, because any number in that range is "in the hundreds" and one hundred = 10^2. Any number between 1,000 and 9,999 has an order of magnitude 3, because it's "in the thousands" and one thousand = 10^3. Any number between 10,000 and 99,999 has an order of magnitude 4, because it's "in the tens of thousands" and ten thousand = 10^4. Obviously an order of magnitude can include a pretty massive range — 99,9999 is very much bigger than 10,000 — again, the point is just to make sure that we're in the right ballpark.

Finally you're ready to start building the house, and you realise that you need some way to get the bricks over to the building site. Your little cousin Toly says he wants to help out, and he tells you proudly that he's very strong and can bring you 7 bricks a day. Your other cousin Andrew overhears this and says he wants to help out too — unfortunately he's a bit of a weakling and can only bring you 4 bricks a day. Then your cousin Hannah shows up and tells you she'll just bring you her tractor and carry 12,000 bricks for you every day. Immediately you realise something very important: Toly and Andrew don't matter at all. If you're calculating how many days it'll take to get all the bricks you need you can just divide the total number of bricks by 12,000 and ignore Toly and Andrew's (generous) contributions completely. Hannah is bringing in more 10,000 times more bricks — that is, more than 4 orders of magnitude more bricks — than the boys combined. This is a recurring theme with orders of magnitude: if one of your inputs is several orders of magnitude bigger than the others, you can often just ignore the little ones for simplicity's sake.

Further reading: xkcd

Uri Bram writes popular non-fiction books with a conceptual approach to mathematical, scientific and analytical thinking. He is the author of Thinking Statistically and Write Harder.

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