As Obama proposes a $1,000,000,000,000 rescue plan, Marcus du Sautoy offers a handy guide to the mind-bogglingly big numbers being bandied about. [via guardian]

The global recession has brought us a slew of numbers so large, with so many noughts attached, that it's getting difficult to put them into any kind of perspective. The Bank of England recently announced it was injecting up to £150 billion of new money into the British economy, an unimaginable amount - yet now we hear Barack Obama is proposing to splash out a further $1trn (one trillion dollars) to rescue Wall Street's floundering institutions. And even that's not as much as Britain's national debt has been recalculated at - £1.5trn - following the classification of Lloyds and the Royal Bank of Scotland as public corporations.

Millions, billions, trillions - names most of us are familiar with, even if we can't specify the number of zeros. In January, Zimbabwe printed a dollar note with a number containing 11 zeros, only to further deflate its currency a month later. And it still doesn't match the Hungarian National Bank in 1946, which came up with the highest denomination banknote ever issued: a 100 quintillion (20 zeros) peng note.

To make any sense of what's going on (and how bad things really are), you need a feeling for quite how big these numbers are. So here's a brief guide, from zero right up to the biggest of them all.

## 0 or zero

A relative newcomer on the mathematical scene, zero wasn't recognised as a number in its own right until the Indians started exploring its properties in the seventh century AD (they are also responsible for the other nine symbols we use for recording numbers, known as the Arabic-Hindu system). Zero was introduced to Europe by the Italian mathematician Fibonacci in the 12th century - and the authorities were so suspicious of it that in 1299 the government of Florence banned its use.

The Indians' invention of the number zero is directly related to their fascination with large numbers. The Sanskrit saga Lalitavistara gives an account of Gautama Buddha, who is asked at one point to name all of the numbers up to those with 421 zeros. A time-consuming task.

## 10

The base-10 system we use today is a direct result of the fact we count on our 10 fingers (the Simpsons, presumably, are working in base eight). Other cultures were not so hooked on powers of 10: the ancient Babylonians collected things in powers of 60, and we see hangovers of their system of numbers in the modern world. The fact that there are 60 minutes in an hour and 360 degrees in a circle is a relic of the Babylonians' choice of base 60. The effect of putting zeros on the end of a Babylonian number is therefore even more devastating than on our modern decimal notation.

## 1,000,000 (one million)

We really start to see the power of the Arabic-Hindu system coming into its own as we hit the big numbers. The Romans had to keep on cooking up new letters every time their numbers got bigger - C for 100, D for 500, M for 1,000 - because they didn't have zeros to add on to the end. To give a sense of how big a million is, 1m seconds is just over 11½ days and if you laid 1m pound coins end to end, they would stretch for 14 miles.

## 1,000,000,000 (one billion)

In the UK, this number used to be called, simply, 1,000 million, while a billion was reserved for a million million (a number with 12 zeros). But pressure to standardise our numbers with the US drove Harold Wilson to announce in 1974 that any government mention of a billion would from then on mean a number with nine zeros.

If you really want someone to blame for the confusion over billions, however, it's the French. Throughout history, they have flip-flopped between different definitions, wreaking havoc on the names of numbers. In 1480, they proposed that a billion have 12 zeros, which is what the British adopted. Then, in the middle of the 17th century, they knocked three zeros off, so a billion became a number with nine zeros. The young United States inherited this new definition. Then in 1948, the French reverted back to the old system.

## 1,000,000,000,000 (one trillion)

To help Obama put the full scale of his rescue plan into perspective, one trillion seconds would take you back 31,709 years to the time of the hunter-gatherers. If you lined up the 1.5tn pound coins that were reported to have been wiped off the global markets on one single black Friday, they would get you from here to Mars.

## 1,000,000,000,000,000 (one quadrillion)

Mathematicians write this number as 10^{15}: the superscript tells you how many zeros there are after the one. Given that we are already wiping trillions off the markets, this is the next order of magnitude that's surely soon going to start appearing on the scene.

The Americans and British call this number a quadrillion, although the European name is a billiard. The world's derivative market has a notional value of nearly half a quadrillion dollars - that's 10 times the value of the world's output, which is why it is regarded as a ticking timebomb by some analysts. Line up a quadrillion pound coins, and they will take you outside our solar system.

## 10^{100} ... (one googol)

This numerical name was coined in 1938 by a nine-year-old boy, Milton Sirotta, who was asked by his mathematical uncle to think of a name for a number with one followed by 100 zeros. If that's not mindboggling enough, a "googolplex" is a number with a googol number of zeros. As surely everyone knows, a misspelt version of this number is now the name of a rather well-known internet search engine. It was also the answer to the million pound question given by Who Wants To Be A Millionaire? cheat Major Charles Ingram.

## 316470269330 ... 66697152511

This is the largest prime number that has been discovered (with the aid of a large computer) to date. It has nearly 13m digits and was only found in August of last year. Printing the full number would require a G2 page about 30 miles long, and it would take more than two months to read aloud all the digits. It earned the discoverer a prize of $100,000 for the first prime number to break the 10m-digit barrier. The next prize on offer is $150,000, for a prime number with more than 100m digits. Thanks to the ancient Greek mathematician Euclid, we know that there are prime numbers out there with as many digits as we want.

## A zillion

Ask a child to name a really big number and they will often go for a zillion. This name does not correspond to any particular number, but has gone into the lexicon to mean a number of indefinitely large magnitude, coined by the American writer Damon Runyon, the author of Guys and Dolls.

## Infinity

The smart kids will go for infinity as the largest number imaginable. Until the end of the 19th century, the concept represented the unknowable - but amazingly, in 1874, a mathematician called Georg Cantor revealed that there are many sorts of infinity, some larger than others. He also showed how one can make sense of adding and multiplying infinities. He paid for his investigation, however, spending much of his life in a German mental asylum in Halle.

So, in the great scheme of the mathematical universe, the numbers being bandied about over the last few weeks are pretty small beer. However bad it gets, mathematicians will always be ready with a name and notation to tackle the next onslaught of bad economic figures.

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