Math 326, March 10 - Changing Bases

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We build the base change model from 4.1.  Numbers as we normally use them are expressed in base 10, which makes counting easier.  However it is sometimes useful to express numbers in different bases.  Base 2, Binary, is particularly useful for computers as they effectively have only two fingers with which to count.  The number 2308 is written so in base 10 since
    2308 = 2*10³ + 2*10² + 0*10¹ + 8*10⁰
We could also write this number in other bases, such as base 6, where it would be 14404 since
    2308 = 1*6⁴ + 4*6³ + 4*6² + 0*6¹ + 4*6⁰
To keep the bases straight we'll use subscripts, for example 2308₁₀=14404₆.

We can also express numbers in bases higher than 10 by using letters from the alphabet to represent "digits" with value 10-35.  We could go even higher provided we come up with different symbols.

Chapter 4.1 details how to build models which convert from base 10 to some new base between 2 and 35, and also from some odd base back to base 10.

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Additional Questions

• Double Switch - Combine the two models to allow conversion from base n to base m, with neither the starting nor ending base fixed as 10.
• A Little Codebreaking - As long as we have a program which can convert letters into new letters and numbers, why not use it as a code?  Since it's based on relatively straight forward mathematical conversion, I'm sure the NSA would have no trouble breaking it, but no matter.  Try decoding these phrases:

                "14644445011  546420013"
                "1AAGE  2F772  2548"
                "12N40  2C28AFL  1QCB0  1I2E  2NNPHR"
                "2CR709S  2270PO  16MF9"