By G. H. Hardy

ISBN-10: 1149378085

ISBN-13: 9781149378083

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**Extra resources for The general theory of Dirichlet's series**

**Example text**

8 multiplied by a number formed from 1 followed by nine zeros. This method of representing numbers is known as scientific notation. We can also represent small numbers with this notation. 01 (1/100) as 10-2 • Hence, the number lO- n is a decimal with allocated n places to the right of the DISCOVERY OF THE NUMBER SEQUENCE 35 decimal point. Therefore we can show very small numbers with a corresponding large n. On a practical scale we have almost reached the limit of numbers used by the general population.

On a practical scale we have almost reached the limit of numbers used by the general population. 5 x 1012), where one trillion is one thousand billion. How long would it take to count up to our national debt if we counted at the rate of 72 per minute? Approximately 145,000 years. Larger numbers are primarily used by scientists. We have actually given names to numbers beyond one trillion, but there is so little use for them that they are generally unfamiliar to most of US. 14 102 103 106 109 1012 1015 1018 1021 1024 1027 1030 1033 1036 1039 1042 1045 1048 1051 1054 1057 1060 1063 = 100 = 1,000 = 1,000,000 = 1,000,000,000 = 1,000,000,000,000 = = = = = = = = = = = = hundred thousand million billion trillion quadrillion quintillion sextillion septillion octillion nonillion decillion undecillion duo decillion tredecillion quattuordecillion quindecillion sexdecillion septendecillion octo decillion novemdecillion vigintillion Other than the observation that some of the names are amusing, their usefulness is questionable since we don't encounter numbers 36 MATHEMATICAL MYSTERIES over a trillion frequently enough to have need of a name for them.

Even though the number, by our standards, is large, compared to all the numbers that exist that are larger, it is still exceedingly small. In fact, it is so small that the chances of choosing such a small number are almost nonexistent. A randomly chosen number must be much larger. But how large? This is where we begin to get into trouble, for no matter what number we come up with as an example, we immediately see there exists only a finite number of numbers that are smaller, and an infinity of numbers that are bigger.

### The general theory of Dirichlet's series by G. H. Hardy

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