Million billion trillion - making sense of large numbers

Part of a series of hubs aimed at demystifying tricky topics in everyday mathematics

Ever wondered where that famous search engine (shan't mention its name here but it begins with G) got its name? It actually comes from the word googol, which is ten multiplied by ten 100 times, or to put it another way, a one with 100 zeros after it. Not surprisingly perhaps, googol isn't a term in common use - even the size of my country's national debt has some way to go before reaching that magnitude.

But even if we're not (yet) measuring things in googols, large numbers do occasionally cause confusion. Wondering how you would write 1.2 million as a number with zeros after it? Want to know how to divide two million into forty billion? Want to remember how many megabytes make a gigabyte? Read on - this could be the hub for you.

Below this paragraph is a handy table showing some large numbers in words, figures and standard form, which is a shorthand way of writing numbers. A googol in standard form would be written as 10¹⁰⁰, which you would say out loud as "ten to the power of one hundred". Similarly, 100 is ten to the power of two (10²), i.e. 10 x 10. One thousand is ten to the power of three (10³), i.e. 10 x 10 x 10. And so on.

In Britain, we used to refer to 1,000,000,000 as a milliard rather than a billion (to us, a billion was a million million instead of a thousand million), but now we've adopted the American usage. So this, at least, is one instance where the two nations are no longer divided by a common language.

How do I deal with large numbers when they're written as decimals?

You might have noticed that articles in the newspapers often refer to sums of money like this (e.g. "£1.2 billion" or "$21.08 million"). If you've wondered how these numbers would look written out in full with loads of zeros, you can make use of one of my favourite maths "helpers", the place value table:

With these two particular numbers, I know that the "1" in these numbers corresponds to "1 billion" and "1 million" respectively. I can then put the other numbers in the appropriate columns, and fill the gaps down to the units column with zeros, giving £1,200,000,000 and $21,080,000 respectively. (In case you were wondering, the numbers said out loud and in non-decimal format would be "one billion two hundred million pounds" and "twenty-one million eighty thousand dollars".) Once you've used a place value table in this way a few times, you'll be able to visualise the numbers in your head and you won't need it any more!


OK, so what about all those gigabytes then?

When I first used a computer back in the late Eighties, you were doing well if your machine even had a hard drive, never mind one with over 150 gigabytes (Gb) of memory such as is common today. But just how much memory is 150 Gb?

1 gigabyte (Gb) = one billion (1,000,000,000) bytes (a byte is the basic unit of computer memory). Similarly:

1 kilobyte (kb) = 1,000 bytes
1 megabyte (Mb) = one million (1,000,000) bytes.


To put it in context, a typical text-only MS Word file is about 20 kb.

Suppose you want to know how many 20 kb files can fit on a 1 Gb memory stick? This is where it helps if you know what these quantities look like as actual figures. Then you can turn them into a top-heavy fraction and cancel all those zeros! Thus:

20 kb = 20,000 bytes, and 1 Gb = 1,000,000,000 bytes

1 Gb divided by 20 kb can be written in fraction form as

1,000,000,000
    20,000

Cancel four zeros from both numbers and you end up with

100,000 = 50,000
    2

(Yes, it's really true. You can fit 50,000 20 kb files onto a 1 Gb memory stick.)

You can use this cancelling method any time you're dividing whole numbers that both have zeros on the end.

How to multiply numbers with (lots of) zeros on the end

Suppose we were doing the above calculation in the other direction, i.e.

50,000 x 20,000 = ?

There is no need to panic when confronted with something like this - it's quite easy to do manually, on a piece of paper. (Which is just as well, because many calculators - the basic ones at any rate - can't cope with it.) You can do it in three stages:

1. Multiply the 5 and the 2 to make 10
2. Total up the number of zeros in the two starting numbers (there are four in each, so that makes eight)
3. Stick eight zeros onto the end of the 10, to give your 1,000,000,000. (If you're adding commas or spaces or whatever you use to separate the groups of three zeros, make sure you go from right to left!)

Try another? How about 1,200 x 3,000,000

1. Multiply 12 x 3 to give 36
2. Total up the number of zeros (there are eight again)
3. Stick your eight zeros onto the end of the 36, to give 3,600,000,000 (3.6 billion or 3 billion 600 million).

The same principle applies to smaller numbers as well - 30 x 250 is much easier to do when you know it's a question of multiplying 3 x 25 and then adding two zeros!

© Empress Felicity February 2010