Naming Big Numbers

October 6, 2009 at 3:08 pm | Posted in Computers, History, Humor, Science | Leave a comment

Really large numbers are hard to get our head around. If you think about it, you can probably visualize a room with a few hundred people in it. Or remember a scene with a few thousand people. But can you visualize a million? What about a sextillion?

If we talk in straight “powers of 10” as a standard order of magnitude, we can visualize the exponential scale more easily, as this presentation illustrates.

However, the way we name our numbers does not go up in powers of 10. And what those names mean depend on where you live and what you do. Before I explain, I’ll outline the naming convention.

We have tens, hundreds, thousands and millions (French for big thousand).
After that, each iteration adds a (French) prefix: Billion (bi, 2), Trillion, Quadrillion, Quintillion, Sextilion, Septilion, Octillion, Nonillion, and then Decillion.

From there, the pattern repeats: Undecillion, Duodecillian, Tredecillion, and so on to Vigintillion. The next series comes to Centillion. But note the variations in the prefix over the first series.

One problem is in what numbers those names represent. Like the metric system, Europe and Spanish speaking countries use a logical progression. Billion means 2 or double million, so a billion is not a 1,000 million but a million million. They call a 1,000 million a Milliard. The effect is to name each round of 6 additional numbers with the next million name. This is called the Long Scale.

The American and scientific community use the names in a progression for each thousand: i.e.: each additional 3 place settings get the next name, called the Short Scale.

Thus a Trillion in the US is a Billion in Europe. They get progressively further apart until by the European Decillion, the US has raced up to Novemdecillion.

Here’s a simple chart comparing the progression by name.
This section of Wikipedia breaks it down even further.
This site lists all the derivations. Aside from some toungue twisters, there’s a few amusing ones like seducentillion.

I find the European system not only more logical but easier to visualize or intuit. But even there, the naming conventions are not entirely consistent. And the habit of US naming is deeply embedded in our culture.

But the bigger issue is that there is not a relationship between the name and the exponential expression that’s used in math. Tredecillion will tell you how many 0’s (depending on the naming convention you use) but not what power it is, unless you’ve memorized it or use a formulae to convert. It’s like going from Imperial to Metric even though it’s supposed to be representing the same scale.

And of course, because the names can mean very different things, they fail to communicate consistently. Scientists thus stay with numbers as they easily relate to themselves unlike the naming conventions. They simply read the notation, as in “ten to the thirty-third“. As a result, even names like Quintillion are rarely used in practice. One of the reasons you’re not familiar with names over a trillion.

Wikipedia mentions a formula that can used to calculate the power in short scale from the name. The number represented, +1, x 3. For example Nonillion = 9+1 = 10 * 3 = 30. Nonillion = 10^30th. You can reverse the formula to get back but would have to remember the correct name for 9. Or you can us a tool like this at Mathcats. This will handle other numerals midway to produce the other names in series just as you might write a cheque for Five Thousand Three Hundred and Thirty Five. If you insert a bunch of random numbers, you get a very long name with all of the series represented.

Zillion
Another name you hear occasionally, Zillion simply means any really large number. More than can be conceived. While we could call this an algebraic variable, it’s really just slang, like Humongous.

Googol
No, not the search engine. The name a 9-year-old came up with for 10^100th. More 0’s than can fit across this page. He also suggested Googolplex which is 1 followed by as many 0’s as you can write until you get tired. It was then defined as 10^googol or 10^10^100. This isn’t just 10x more but 10 scales larger, exponentially. Very large but still finite. And would be very hard indeed to write out.

Gesundheit
or Skewe’s Number: 10^10^10^34 was considered the upper bound in a mathematical proof. But much larger numbers have since been used.

SI Prefixes
Another naming convention from the French language, SI is also known as the metric prefix. It’s designed to reduce the number of zero’s like scientific notation but uses symbols for Latin prefixes – kilo (k), Mega (M), Giga (G), Tera (T), Peta(P), Exa (E), Zetta (Z), and Yotta (Y). There is a similar scale into the very small: centi, milli, micro, nano, pico, femto, etc.

Think of it as extended Metric, although some prefixes are used with non-Metric notations as well. We see this in weight, distance, computer sizes, electricity and so on. They’re not supposed to be used with time or angles, although astronomers do.

This is where computer MB (million) and GB (billion) hard drives get their naming from. You’ll know your computer has lots of storage when it has Yottabyte drives. The Short Scale equivalent is Septillion bytes, with 8 sets of 000.

Computer naming introduces 2 errors to the standard though:
1) Technically, you’re supposed to write the Kilo symbol with a small k, as in kB, not KB. K is supposed to mean Kelvin.

2)A “kilobyte” is supposed to be 1,000 bytes but it’s often actually 1024 bytes, an exponential value of 2. 2 originates from the on and off states of computer data. As hard drive sizes and network transmission rates have climbed, the discrepancy has gotten larger and larger.

Curiously, hard drive makers are actually underselling the size of the drive when they say it’s 1 TB (trillion). They’re talking less about data space and more about how it will display in a computer. (see below) And some gear does the reverse, implying  larger than it is. 11 years ago, the IEC adopted an “i” to denote the powers of 2 used in computers. Thus 1,000 bytes = 1 kB and 1,024 bytes is 1 KiB. I’ve not actually seen this in use yet, but if drive makers realize they can use this to mean more, it may catch on. Would also help a lot if the OS makers supported this though.

Windows, for example, displays a drives space in bytes and GB, dividing the bytes by 1024 to get GB and thus yielding a curiously smaller number. For example, 25.3 billion bytes becomes 23.6GB, a difference of 1.7 billion! It should actually display as 23.6 GiB or 25.3 GB.

Above k, the symbols are all upper case. Seems it would be more consistent if they used all caps for the ascending scale, however, the scale historically starts with Kilo, even though it’s not in the middle at 1.

Unlike the scales above, the SI series is not to be used in combination, as in gigakilo, meaning it doesn’t use positioning and new names will have to be added to go above Yotta. Some variations are also seen like micro for micrometer and angstrom for nanometer.

Some suggest adapting the SI prefixes to remove the international vagueness of what billion and trillion mean. But it’s not large or flexible enough a scale. And in the same way as above, the scale names don’t have a relationship with the numeric values.

-yllion
This is one persons proposal to standardize naming but it also changes the notation, using commas after every 4 numbers. Rather than being a simple naming standardization it’s another system and so unlikely to be adopted. The page also has links to many other traditional number systems.

Origins
And finally, a touch of history. Our numeric system came to Europe from the Arabs who in turn got it from invading Persia. Persia derived theirs from India. Here you can see how some of the Sanskrit names evolved through Arabia and into Greek and Latin. This includes the ideas of zero and place values. Position allows expression of all numbers with just 10 symbols. The discussion above was just on how to name the results.
David