Is a million times a million a billion or is it a trillion? For my entire childhood, it was a billion, only for me to grow up and realise, as an adult, that it had better be a trillion!

## Growing Up With the Long Scale

As a child, I stumbled upon an old dictionary lying around our house and that was where I discovered the names of large numbers. The dictionary used the long scale system, which is based on powers of a million. According to the long scale system,

- a million times a million is a billion,
- a million times a million times a million is a trillion,
- a million multiplied by itself four times is a quadrillion,
- multiply it five times and we get a quintillion
- and so on.

These long scale names made a lot of sense to me. Each name told us exactly how many times we multiplied a million by itself. The little prefix at the start, like bi-, tri-, quadri-, quinti-, etc. matched the number of times a million appeared in the multiplication: twice for a billion, three times for a trillion and so on. If I came across a number like quintillion, I immediately knew it meant a million multiplied by itself five times, so a quintillion had to be \( (10^6)^5 = 10^{30} \). It was simple and intuitive.

Now imagine my disappointment when I left home for university, got access to computers and the World Wide Web and discovered that the names I had learnt were off by several orders of magnitude compared with what everyone else was using. The long scale names I had grown up with were irrelevant in this new world I was stepping into. It was the short scale that had taken over most of the technology world, especially in computing. In the short scale, a million times a million is no longer a billion. Instead, somehow, it is a trillion! Similarly, a million multiplied three times is no longer a trillion. It is a quintillion! What on earth was going on?

Naturally, I had to go through a period of adjustment, letting go of my old intuition for the long scale names and getting used to the short scale ones. At the same time, I found a way to trick my mind into making sense of the short scale. Before I get into the mental gymnastics that made the short scale click for me, let us first take a quick look at the long scale and short scale names side by side.

## Same Names, Different Numbers

The long scale and short scale are simply two different ways of naming large numbers. To see how they differ, here is a table showing what each name corresponds to in the two systems.

Number | Long Scale | Short Scale
--- | --- | ---
\( 10^6 \) | Million | Million
\( 10^9 \) | Milliard | Billion
\( 10^{12} \) | Billion | Trillion
\( 10^{15} \) | Billiard | Quadrillion
\( 10^{18} \) | Trillion | Quintillion
\( 10^{21} \) | Trilliard | Sextillion
\( 10^{24} \) | Quadrillion | Septillion
\( 10^{27} \) | Quadrilliard | Octillion
\( 10^{30} \) | Quintillion | Nonillion
\( 10^{33} \) | Quintilliard | Decillion

As the table shows, the two systems agree on the value of a million, but diverge immediately after. From that point on, the same names represent increasingly different magnitudes.

## Making Sense of the Short Scale

I used to quite like the long scale names because the prefix in each name is determined by how many times we multiply a million by itself. Each prefix reflects the value of \( n \) in the number \( 10^{6n} \), where \( n \) is a positive integer. For example, if \( n = 3 \), we get \( 10^{18} \), which in the long scale is called a trillion. But in the short scale, the same number is called a quintillion. How can we make sense of it?

One way to interpret the short scale is to count how many additional thousands are multiplied onto one thousand. A thousand is, of course, just one thousand. Multiply it by another thousand and we get a million. Multiply that by another thousand and we get a billion. One more multiplication gives us a trillion and so on. I do not find this quite as neat as the long scale but it works.

In effect, the prefix in the short scale corresponds to \( n \) of the integer \( 1000 \times 1000^n \), where \( n \) is a positive integer. Stated more succinctly, it represents \( n \) of \( 10^{3(n + 1)} \). So for example, if \( n = 5 \), then \[ 10^{3(n + 1)} = 10^{18}, \] which is called a quintillion in the short scale. The quinti- prefix still represents five but now it is tied to the number of additional thousands we multiplied to the first thousand. It is not quite the elegant pattern I grew up with but this is the convention that dominates today. In the world of technology, computing and finance, as well as in most English-speaking contexts, the short scale has prevailed.

## Conclusion

Today, of course, I use only the short scale, because that is the convention adopted by the part of the world relevant to me. I no longer use the long scale. But it remains in memory, simply because I spent a large part of my early life with it. It still lives in some corner of my heart, partly out of nostalgia and partly due to its elegance.