How big is a million? Do you know? and do you really care? And did you know there are two "billions"?

The word "millione" was coined in Italy from *mille *(Latin, "thousand") to mean "great thousand" and this has given us our word "million." Around 1484, N. Chuquet coined the words *billion, trillion, . . ., nonillion, *which also appeared in print in a 1520 book by Emile de la Roche.

These arithmeticians used "illion" after the prefixes

b, tr, quadr, quint, sext, sept, oct and non

to denote the

2nd, 3rd, 4th, 5th, 6th, 7th, 8th and 9th powers of a **million**.

So a billion was (1 million)^{2},
a trillion was (1 million)^{3} and so on.

Then around the middle of the 17th century, some other French arithmeticians used the same words instead for the
3rd, 4th, 5th, 6th 7th, 8th, 9th and 10th powers of a **thousand**.

which was not very logical, because if you're going to use powers of a thousand, then the "bi-(something)" should be 1000^{2} and not 1000^{3}!

Although condemned by the greatest lexicographers as "erroneous" (Littre) and "an entire perversion of the original nomenclature of Chuquet and de la Roche" (Murray), this newer usage is now standard in the US, and its use is spreading with the use of computers (an industry dominated by the US). The older logical system survives in Britain (but is not now used by the Government, and most of the daily papers...) and is still standard in continental countries.

Nothing stands still. The symbol for a million in Ancient Egypt was a pictogram of a man holding up his hands in surprise; and before this century who needed anything bigger than a million? Numbers and corporations get bigger, and inflation needs larger numbers to express what may well be smaller values... Here are some of the number-prefixes recommended by the Conference Generale des Poids et Mesures in 1991:

unit x N | unit/N | the number N |
---|---|---|

deca (da) | deci | 10 |

hecto (h) | centi | 100 |

kilo (k) | milli | 1,000 |

mega (M) | micro | 1,000,000 |

giga (G) | nano | 10^{9 } |

tera (T) | pico | 10^{12} |

peta (P) | femto | 10^{15} |

exe (E) | atto | 10^{18} |

zetta (Z) | zepto | 10^{21} |

yotta (Y) | yocto | 10^{24 } |

Is there a formula for the "-illions"? It depends on which system you're using, British or American. The n'th "illion" is 10^{6n} (British) or 10^{3n+3} (American).

Here's a table of English words for some of the powers of ten.

name | power: UK | power: US |
---|---|---|

ten | 1 | 1 |

hundred | 2 | 2 |

thousand | 3 | 3 |

myriad | 4 | 4 |

lac or lakh* | 5 | 5 |

million | 6 | 6 |

crore* | 7 | 7 |

milliard | 9 | - |

billion | 12 | 9 |

trillion | 18 | 12 |

quadrillion | 24 | 15 |

decillion | 60 | 33 |

vigintillion | 120 | 63 |

centillion | 600 | 303 |

* adopted by the British in India.

Not many people realise there are two systems. The number names are not well-known, and rarely taught in schools; probably few teachers know the difference, not having been taught the names themselves.

I think we should stick to the British series as being more logical in construction. The prefix "bi" means "two"; it is illogical to call 10^{9} a billion when 9 is not divisible by two! A quadrillion, where "quad" means "four" is equally peculiar when rated as 10^{15}, when 15 does not divide by 4... I could go on, but will not labour the point myself - here follow some of Tom Pendlebury's comments on the two systems, reprinted and slightly rewritten, from Dozenal Review no *32.

Would you say that two and two make five which really means six?

Crazy! Isn't it? Yet that is exactly the sort of thing that people are fostering when they use the word "billion" to mean a THOUSAND million.

Many do not realise that there are TWO different systems of using the words: billion, trillion, quadrillion, quintillion, etc. One is mathematically quite sound, the other, perfectly ludricous.

The sound system is called the Anglo-German, but it is also common to all eastern and northern European countries. Quite straightforward: Billion=million million. Trillion = million million million. Quadrillion=million million million million, that is a billion billion, just as two and two make four, and so on. Since a million has six noughts, a quadrillion has four times six, that is twenty-four. A sextillion has thirty-six noughts; for its square root we halve "sext-" giving "tri-" and the answer trillion (18 noughts); its cube root (three into "sext-" is "bi-") is a billion (12 noughts).

The other system is the American, and according to Webster's New Collegiate Dictionary (an American publication) "..was modeled on the French system but more recently the French system has been changed to correspond to the German and British systems."

It goes up not by millions but thousands. The calamity is, it steps off on the wrong foot, for it is the THOUSAND MILLION, that is, the thousand thousand thousand (the **third** power) which is called a **Bi**llion (9 noughts). How can you rate a "third power" as "2" ("bi")?

- A thousand "billion" is called a "trillion" (12 noughts). This is the
**fourth**power of a thousand, or second power of a million. (The Anglo-German billion) - A thousand "trillion" is called a "quadrillion" (15 noughts), the
**fifth**power. - A thousand "quadrillion" is called a "quintillion" (18 noughts), the
**sixth**power, The Anglo-German trillion). And so on!

In the American system
all the **even **prefixes "bi-, quadr-, sext- oct-" etc are used for** odd** powers and all the **odd **prefixes "tri-, quint- sept-" etc for the **even **powers! This leads to a very queer arithmetic such as: A billion billion is not a quadrillion but a quintillion, yet that quintillion does not mean the fifth but the sixth power of a thousand! As though "two and two" were "five" which really means six. Similarly, their "**tri**llion **tri**llion" is a "**sept**illion" meaning the **eighth** power of a thousand.

So it goes on: The square root of their "nonillion" is a "quadrillion", but the square root of their "octillion" is a "trillion" multiplied by the square root of a thousand! A "septillion" divided by a "billion" is not a "quintillion" but a "quadrillion" (7 - 2 = 4???).

The very purpose of these words is quite simply: to name the higher numbers; that is, to tell us how many noughts to put. A quintillion in the Anglo-German system has 5 (quint) x 6 noughts, i.e. 30 noughts. The American quintillion has eighteen noughts - and you can't get eighteen from five; you have to take "quint-" meaning five and multiply it by three (the number of noughts in a thousand), and then add another three to get the "correct" answer, eighteen noughts! Large numbers are difficult enough to comprehend without all this senseless rigmarole. We should really try to persuade people to use just the one system - the logical Anglo-German system, but for the moment, to avoid ambiguity, the Anglo-German billion (American "trillion") should be called "million million".

In the Anglo-German system the thousand million may alternatively be called a "milliard". This word means the same in all languages, and is found also in American dictionaries. This should be used instead of "billion", so that the billion can go back to being 10^{12}.....