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What's it all About?The S I PrefixesComputer PrefixesI E C PrefixesMillion etc.Archimedes |

In ordinary language prefixes have an important to play in modifying the meanings of words. For example, consider these, where the (prefix) offered allows us to see two words at once. (un)desirable, (anti)clockwise, (in)variable, (im)possible, (re)new, and even (pre)fix itself. It is also possible to run two prefixes into one like | In some cases, more than one prefix is possible for the word which follows, especially in cases where a number is implied by the prefix. In each of these, any one (but only one) of the prefixes offered could be attached. (uni)(bi)(tri)cycle. (penta)(deca)(dodeca)gon (tetra)(hexa)(octa)hedron And it is even possible to combine them as in that well-known musical expression |

The Bureau International des Poids et Mesures (BIPM) was set up in 1875 with the principal aim of establishing and publishing standards of physical measurement. Its headquarters is near Paris. Over the years since then its responsibilities have increased, together with the number of different committees involved. However, its main drive has been to produce a rational and unified system of basic measures. And this was launched in 1960 as the Système International d'Unités or SISince then various modifications have been made but most are of little concern to the ordinary everyday user. | There are rules and guides for working with the SI and the main ones are to be found in the Dictionary of Units under Note, unlike ordinary language, SI prefixes may NOT be doubled. |

Prefix | Multiplier | ||||

yotta | Y | = | 10^{24} | = | 1 000 000 000 000 000 000 000 000 |

zetta | Z | = | 10^{21} | = | 1 000 000 000 000 000 000 000 |

exa | E | = | 10^{18} | = | 1 000 000 000 000 000 000 |

peta | P | = | 10^{15} | = | 1 000 000 000 000 000 |

tera | T | = | 10^{12} | = | 1 000 000 000 000 |

giga | G | = | 10^{9} | = | 1 000 000 000 |

mega | M | = | 10^{6} | = | 1 000 000 |

kilo | k | = | 10^{3} | = | 1 000 |

hecto | h | = | 10^{2} | = | 100 |

*deca | da | = | 10^{1} | = | 10 |

10^{0} | = | 1 | |||

deci | d | = | 10^{-1} | = | 0.1 |

centi | c | = | 10^{-2} | = | 0.01 |

milli | m | = | 10^{-3} | = | 0.001 |

micro | µ | = | 10^{-6} | = | 0.000 001 |

nano | n | = | 10^{-9} | = | 0.000 000 001 |

pico | p | = | 10^{-12} | = | 0.000 000 000 001 |

femto | f | = | 10^{-15} | = | 0.000 000 000 000 001 |

atto | a | = | 10^{-18} | = | 0.000 000 000 000 000 001 |

zepto | z | = | 10^{-21} | = | 0.000 000 000 000 000 000 001 |

yocto | y | = | 10^{-24} | = | 0.000 000 000 000 000 000 000 001 |

(What standards?)

For example, in the SI system 'kilo' means 1000 whereas a computer reference to 'kilo' as in 'kilobytes' meant 1024 bytes.

This was because the computer prefixes were based on a binary scale, and each one is a power of 2 so comparative values were

SI values | Computer values |
||||

kilo = | (10³)^{1} = | 1 000 | (2^{10})^{1} = | 1 024 | |

mega = | (10³)² = | 1 000 000 | (2^{10})² = | 1 048 576 | |

giga = | (10³)^{3} = | 1 000 000 000 | (2^{10})^{3} = | 1 073 741 824 | |

tera = | (10³)^{4} = | 1 000 000 000 000 | (2^{10})^{4} = | 1 099 511 627 776 | |

and so on . . . . | and so on . . . . | ||||

The patterns in the way the values grow is clear. The differences in value from the SI system are not enormous but they do get bigger as you move 'up' the scale. Here they are shown as percentages (to 3 significant figures). A computer 'kilo' is . . . 2 .4% bigger than an SI 'kilo'.A computer 'mega' is . . 4 .86% bigger than an SI 'mega'.A computer 'giga' is . . . 7 .37% bigger than an SI 'giga'.A computer 'tera' is . . . 9 .95% bigger than an SI 'tera'. Etc. |

To avoid all ambiguity and any confusion, in 1998, the International Electrotechnical Commission [IEC] approved a new series of prefixes to be used beside the SI set. First of all if, say, 'kilo' is intended then that prefix or 'k' is used. However, if the computer value is meant then the prefix ' kibi' is used.This is formed by using the first two characters of kilo & binary respectively.This rule extends to yield mebi for megabinarygibi for gigabinarytebi for terabinaryand so on . . .
A fuller account of this, together with suggestions on pronounciation, can be found at the US National Institute of Standards and Technology [NIST]or by going directly to this document here |

The word million (=1 000 000) appeared in the 1300's. Then, in 1484, Nicolas Chuquet, a French mathematician published a system for naming larger numbers.It was based on the million. The next number would be million × million. That is two millions multiplied together. The Latin prefix for two is 'bi' so it would be bi-millions or billion.Next tri-millions or trillion (million×million×million)Then quad-millions or quadrillion and so on. | Nothing much happened about this for a few centuries and then it was slowly taken up. Britain used Chuquet's system, but France modified it to make each step increase a thousand times rather than a million times. Later, the USA adopted this French system. And so the seeds of confusion were sown, which was not helped by the fact that, later still, France changed to the original Chuquet system! |

In the second half of the 1900's with much more happening internationally rather than just nationally, the modified Chuquet system came to be more generally used. This was particularly important for the billion (= a thousand million) which was being bandied about more and more. And let's be pragmatic, you get to be a billionaire more quickly with that one! Here we will refer to the modified Chuquet system as the Modern system. The difference between the two is shown in the table below. |

Chuquet | Modern | |||||

million | = | million^{1} | = | 1000^{2} | = | million |

thousand million | = | 1000^{3} | = | billion | ||

billion | = | million^{2} | = | 1000^{4} | = | trillion |

thousand billion | = | 1000^{5} | = | quadrillion | ||

trillion | = | million^{3} | = | 1000^{6} | = | quintillion |

thousand trillion | = | 1000^{7} | = | sextillion | ||

quadrillion | = | million^{4} | = | 1000^{8} | = | septillion |

A more extensive list of the Modern names,

and their definitions, is given below.

The 'Modern' Naming System | ||||||||||

1000^{1} | = | thousand | = | 10^{3} | 1000^{11} | = | decillion | = | 10^{33} | |

1000^{2} | = | million | = | 10^{6} | 1000^{12} | = | undecillion | = | 10^{36} | |

1000^{3} | = | billion | = | 10^{9} | 1000^{13} | = | duodecillion | = | 10^{39} | |

1000^{4} | = | trillion | = | 10^{12} | 1000^{14} | = | tredecillion | = | 10^{42} | |

1000^{5} | = | quadrillion | = | 10^{15} | 1000^{15} | = | quattuordecillion | = | 10^{45} | |

1000^{6} | = | quintillion | = | 10^{18} | 1000^{16} | = | quindecillion | = | 10^{48} | |

1000^{7} | = | sextillion | = | 10^{21} | 1000^{17} | = | sexdecillion | = | 10^{51} | |

1000^{8} | = | septillion | = | 10^{24} | 1000^{18} | = | septendecillion | = | 10^{54} | |

1000^{9} | = | octillion | = | 10^{27} | 1000^{19} | = | octodecillion | = | 10^{57} | |

1000^{10} | = | nonillion | = | 10^{30} | 1000^{20} | = | novemdecillion | = | 10^{60} | |

1000^{21} | = | vigintillion | = | 10^{63} | ||||||

(and who needs them anyway?) until we get to 1000 ^{101} = centillion = 10^{303}and beyond! |

The table is (at first sight) very well-behaved. The powers of 1000 go up in 1's and the powers of 10 go up in 3's, so that the latter is always three times the size of the former. However, if we study the names of the numbers we can see an anomaly. Matching their prefixes (bi=2, tri=3 quad=4 etc.) against the power of the 1000's on their left shows that they are always 1 behind. The reason for this is apparent from the previous table. It can be resolved by re-writing in the manner suggested on the right. |
and so on . . . |

Archimedes (287-212 BC) is generally credited as being one of the greatest mathematicians of all time. Among his many works that we know of, is one called The Sand Reckoner. Its title is derived from his attempt to estimate how many grains of sand there might be in the entire universe or, more correctly, to set an upper limit on how many there might be. It fact it is a treatise on how we can go on generating a naming system for bigger and bigger numbers. | Briefly, using a modern notation, it went like this. A myriad (=10,000) was their largest number.Take a myriad (he said) and multiply it by itself. Call this X. ^{8})^{1} . . X^{2} . . X^{3} . . X^{4} . . and so onup to X ^{X} and call this P (=10^{800000000})We use P since Archimedes referred to all numbers now possible at this point as belonging to the first period. Up to P² was the second period and so on up to P ^{X}And there he stopped. (What about P ^{P}?) |

^{X} simply write a 1 followed by80,000,000,000,000,000 noughts that is 80 quadrillion noughts! Finally In very recent times we have had two large numbers added to our language. They were devised by Edward Kasner (who defined them) in collaboration with his 9 year old nephew (who named them). First there is the googol = 10^{100}and then there is the googolplex = 10^{googol} |

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