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Writing Numbers in Bases Greater Than 10

Date: 04/05/2001 at 18:31:26
From: Cindy
Subject: Recording numbers in bases larger than 10

I am trying to understand how number bases above 10 are recorded.  
Examples given use only low numbers and do not answer my question.  
What would 4 x 13 [base 10] look like in base 42?  Do all bases above 
ten use the same method?  

I appreciate any information you can send me and any additional Web 
sites that I may check. Thank you.

Date: 04/05/2001 at 22:49:26
From: Doctor Peterson
Subject: Re: Recording numbers in bases larger than 10

Hi, Cindy.

Actually, the concept of base doesn't require any particular way to 
represent the digits. You could write numbers in base 10 using, say, 
Chinese characters, or any random symbols you choose, and you can use 
any symbols you want for base 42 as well. We tend to use a simple 
method for bases from 11 to 36 or so: use 0 - 9, then letters A - Z 
(possibly dropping I and O). The only bases above ten I'm familiar 
with in actual use by humans are 16 (hexadecimal, used in computers, 
which uses 0-9 and A-F), and 12 (used in schools, sometimes with 0-9 
and T and E for ten and eleven instead of A and B). But it's really up 
to you to define the symbols you use if you write numbers in an 
unfamiliar base.

There are special uses of other bases in the computer world; for 
example, e-mail programs often send attachments in "base 64" form, 
where they just chose 64 printable characters in which to encode 
everything, without having to make it human-readable at all. And the 
ancient Babylonians used a base 60 system: they wrote numbers less 
than 60 in a decimal form (like "<<<||||" for 34), and starting at 60 
they used those numbers as "digits," as in "| <<<|||" for 60+34 = 94.

Let's take that example for base 42. I'll just use [0] - [41] as the 
42 digits in the system; then your sample problem becomes

    [4] * [13] = [1] [10]

since 4*13 = 52 = 1*42 + 10.

You may find this page on the Babylonian system from the MacTutor Math 
History archives interesting:   

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Number Theory

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