Powers of ten & big numbers

Around 8 billion people live on Earth, your phone stores 128 gigabytes, and governments calculate in trillions. Numbers like these show up in the news all the time, but who can really picture them? Powers of ten give you a tool that suddenly makes huge numbers manageable.

Every step means: times ten

Instead of 10 times 10 times 10, we write 10 to the power of 3, so 10^3 = 1000. The small exponent tells you how often 10 appears as a factor, and conveniently also how many zeros the number has: 10^3 has three zeros. Each step up means times ten: 100 becomes 1000, 1000 becomes 10,000. This builds a ladder that takes you from 1 to the biggest numbers in the universe in just a few steps.

Million, billion, trillion

The big names sit on fixed rungs of the ladder: a million is 10^6, a 1 with six zeros. A billion is 10^9, a thousand million. A trillion is 10^12, a thousand billion. Watch out when switching between English and German: the German Milliarde is the English billion, and the German Billion is the English trillion, which causes plenty of confusion. To compare big numbers, look at the exponent first: 3 times 10^9 is bigger than 8 times 10^8, because a whole step of ten outweighs any factor in front. This only holds, though, as long as both numbers are written properly in scientific notation, that is with a factor between 1 and 10 in front of the power of ten.

Big numbers in everyday life

With the ladder in your head, you can make sense of news and spec sheets. The world population is about 8 times 10^9 people. A gigabyte is roughly 10^9 bytes, a terabyte roughly 10^12 bytes, a thousand times as much. And if a country owes 2 trillion euros, that is 2 times 10^12 euros. Spread across the 8 times 10^9 people on Earth, that comes to 250 euros each. Dividing works in two steps: you divide the factors in front (2 divided by 8 gives 0.25) and subtract the exponents (12 minus 9 gives 3). Together that makes 0.25 times 10^3 = 250.