Numbers, letters and exponential growth

Mixing slightly arbitrary abstract systems can be fun. The two systems that nearly everyone has to deal with are our systems for sound (the alphabet) and for quantity and order (numbers). So taking each letter in turn, what is the lowest number with that letter?

Just three letters are left out: j, k and z. Of course z does turn up in the number system in Zero, but curiously all three of these turn up a lot in the list of fake large numbers in English. Think of “kajillion” or “zillion”.

It is interesting how long we have to wait for several common letters, especially c. We can visualise this by plotting the smallest number for each letter. As the numbers get so large we use a logarithmic scale so the smaller numbers don’t become invisible.

In fact the logarithmic scale makes a lot of sense, after all we have more letters available every time we have used one. We can look at how many letters are needed for all the numbers up to 31:

A further reason for a logarithmic scale is that, as the numbers get larger the number of words that are needed also declines. We start at a quick pace, all numbers up to twenty have a new word. Yet after twenty we just have twenty-one, twenty-two and so on. No new words until thirty, then forty and so on in tens up to hundred. From there we do not need a new word until a thousand and then a million. We can plot the largest number expressible given the number of distinct words:

With standard practice this will continue in a straight line (on the exponential plot) until we run out of words. They do go a long way though.  There is a small question that arrises here though. That is the size of a billion. This has been used for both 1,000,000,000 (a thousand million) and 1,000,000,000,000 (a million million). In each system a trillion has a corresponding meaning. In the first 1,000,000,000,000 (a million million) in the second 1,000,000,000,000,000,000 (a million million million). The same for quadrillion, quintillion and so on. The first system is sometimes called the short count and the second the long count. In the image above I used the first, more common usage. Yet the second version has a distinct advantage if we associate a million with 1, billion with 2, trillion with 3 and so on we can then consider what happens when we multiply. A million times a million gives a billion, just as 1+1 = 2, a billion times a trillion will give a quintillion, found easily by 2+3=5. This ability to add to work out a multiplication relates directly to the effects of using logarithms. It is not easy to come up with an simple way to find the answer when multiplying large numbers in the short count.

So, suppose we use the standard number rules and allow 35 words (that is all the regular numbers and the -illions up to heptillion), what is the largest number we can count up to? In the short count it is nine hundred and ninety nine heptillion, nine hundred and ninety nine sextillion, nine hundred and ninety nine quintillion, nine hundred and ninety nine quadrillion, nine hundred and ninety nine trillion, nine hundred and ninety nine billion, nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine or:

Which is pretty large, while also show why we might prefer a numeral system over words for these big numbers. The long count does even better with nine hundred and ninety nine thousand nine hundred and ninety nine heptillion, nine hundred and ninety nine thousand nine hundred and ninety nine sextillion, nine hundred and ninety nine thousand nine hundred and ninety nine quintillion, nine hundred and ninety nine thousand nine hundred and ninety nine quadrillion, nine hundred and ninety nine thousand nine hundred and ninety nine trillion, nine hundred and ninety nine thousand nine hundred and ninety nine billion, nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine or:

Which is a lot bigger. We can however make an even more powerful system without really bending the rules that much. We already have smaller numbers multiplying larger ones, for example a hundred thousand. What happens if we allow each power of ten number after a million to be multiplied by all smaller ones, without repetition. With this system we can consider a hundred thousand million before we need a billion in the long count. Note that to make this fit I am arbitarily denying the use of ten. In this system therefore a trillion is not a million billion, but a billion billion, a quadrillion is a trillion trillion and so on. With the same 35 words we can now get to:

nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine trillion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine quadrillion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine trillion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine quintillion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine trillion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine quadrillion nine hundred and ninety nine thousand nine hundred and ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine billion nine 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or:

This massive growth comes as we are now using the exponential of an exponential. Why not try to come up with even more efficient methods that need even fewer words? The binary system is worth investigating, especially if you consider why it might be efficient and how this links to its use in computers. You could also venture into the delightful game of constructing larger and still more massive numbers.