Million Times Million

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, straightforward, 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.

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. But the long scale still lives in some corner of my heart, partly out of nostalgia and partly due to its elegance. I no longer use it. I no longer rely on it. But it remains in memory, simply because I spent a large part of my early life with it.