Is there an equation to express short scale (US) number names vs long scale (EU)?

[darkpraxis]

For example, long scale number names are as follows (French/Spanish naming standard):
10⁶ = million
10¹² = billion
10²⁴ = trillion

You can determine the name of the number by expressing it as 10⁶ⁿ where n equals n-lion. And in reverse, understanding what a trillion is in long scale is simple because tri- equals 3, and 10⁶⁽³⁾ equals 10²⁴.

But when we look at short scale (US/English) number names:
10⁶ million
10⁹ = billion
10¹² = trillion
10¹⁵ = quad
10¹⁸ = quint, etc.

...what simple equation could we use to express this naming convention, where n is the only variable and equals n-lion?

 

[Dr.Peterson]

Equations don't work on words. The translation is done by a dictionary ...

But if you're thinking of translating a number n, corresponding to 1="mi", 2="bi", 3="tri", to a power of ten, p, then observe that increasing n by 1 increases p by 3, so you want a linear function with slope 3 through the points (n, p) = (1, 6), (2, 9), (3, 12). What is that equation? It's pretty simple.

 

[darkpraxis]

3n+3=p !

Therefore,
10³⁽¹⁾⁺³ = 1 million, mi=1
10³⁽²⁾⁺³ = 1 billion, bi=2
10³⁽³⁾⁺³ = 1 trillion, tri=3 etc.

Thank you. I’m obviously rusty at this stuff sinI wasn’t thinking of the other side of the equation as a moving target/variable of p. Although you kind of gave the answer away when you said “n by 1 increases p by 3... linear function with slope of 3...”

n*3 for the increase in p, and +3 for the linear slope.

Thanks again, except now I have to admit defeat that long scale number counting is superior to short scale, at least when it comes to simplicity of the expressed equation.

 

[Dr.Peterson]

3n+3=p !

Therefore,
10³⁽¹⁾⁺³ = 1 million, mi=1
10³⁽²⁾⁺³ = 1 billion, bi=2
10³⁽³⁾⁺³ = 1 trillion, tri=3 etc.

Thank you. I’m obviously rusty at this stuff sinI wasn’t thinking of the other side of the equation as a moving target/variable of p. Although you kind of gave the answer away when you said “n by 1 increases p by 3... linear function with slope of 3...”

n*3 for the increase in p, and +3 for the linear slope.

Thanks again, except now I have to admit defeat that long scale number counting is superior to short scale, at least when it comes to simplicity of the expressed equation.

The way I think of it is 3(n+1), which is not so bad; so given, say trillion, I think "tri = 3, add 1 to get 4, triple that and it's 10^12."

If you're interested, here is a mini-debate between me and a British colleague (20 years ago) about the history and relative merits of the two systems. In particular, I said, "There is logic behind the usage; in this system, billion doesn't mean "million squared" but "second -illion", counting by thousands. It's hardly different from deciding whether to index arrays starting at 0 or 1. You just have to choose where to start and how big a step to take, and the numbers follow a logical progression."