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Calculus/Base to Base Conversions

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Question
When i was going through some of the previous written answers about base to base conversions, I saw something that talked about fractional bases.  I'm not sure that I understand how A fractional base can exist or function.  If you could explain it to me with a few practice problems or something of that nature, I would appreciate it greatly.  I do understand whole numbered base to base conversions.  The way I learned base to base conversions stated that the digits you use in your final number have to be at least one less than the number of the base itself.  For example, in base 10, we only use digits up to 9, and for base 9, we use digits up to 8, etc.  How can this exist for a base that is less than one?  How can this exist for a fraction in general?  Once again, if you could help me with these questions I would be very grateful.

Answer
As an example, take 0.5 in base 10 and convert it base 2.
I write  numbers with a B followed by the base if the base is not 10.
Examples: 11B2 = 2+1 = 3; 46B7 = 4*7 + 6 = 34; 53B6 = 5*6 + 3 = 33;
282B5 = ... hand on ... 5^3 = 125, 5^2 = 25, 5^1 = 5, 5^0 = 1 ... 2112B5.
To check, just take each digit, multiply by the base, and add the next.
For example, 2*5 = 10, 10+1 = 11, 11*5 = 55, 55+1 = 56, 56*5 = 280, 280 + 2 = 282.

To change 0.5 to base 2, first convert it to a fraction, so 0.5 = 1/2.
Since it is just 2^-1, it would be 0.1 in base 2.

To check this out, note 1/2 + 1/2 = 1.  If 1/2 is 0.1B2, 1/2 + 1/2 = 0.1B2 + 0.1B2.
This is 0.2B2, but 2's always slide over to a 1 in the next spot in base 2, so it is 1B2.

If we have 1/5, that is 0.2.  In base 2, 0.1B2 = 0.5, 0.01B2 = 0.25, 0.001B2 = 0.125,
so that means we have 0.2 = 0.125 + 0.075 = 0.001B2 + 0.075.
0.0001B2 = 0.0625, so we have 0.001B2 + 0.075 = 0.001B2 + 0.0625 + 0.0125.
0.001B2 + 0.0001B2 + 0.0125 = 0.0011B2 + 0.0125.  Now 0.00001B2 = 0.03125, so we have
0.2 = 0.00110B2 + 0.0125 { another 0 at the end in base 2}.  Dividing by 2 again gives
0.000001B2 = 0.015625.  This gives us 0.01100B2 + 0.0125 { another 0 at the end in base 2 }.
As can be seen, most digits are difficult to convert.

If we converted 1/5 to a fraction in base 5, it would be 0.1B5, for it is 5^-1.
If we converted 1/25 to a fraction in base 5, it would be 0.01B5, for 1/25 = 1/5^2.

If we converted 1/27 to a fraction in base 10, it would be a mess.
Using base 3, however, 1/27 = 0.001B3 since 1/27 = 1/3^3.

31/64 converted to base 4 would be ... first, 31 base 4 is 133B4.
Since we are dividing this by 64, note that 64 = 4*4*4, so slide the decimal 3 places.
This gives 0.133B4 = 31/64.  If we took 1.33B4, that would be 1 + 3/4 + 3/16 =
1 + 0.75 + 0.1875 (since 1/4 = 0.25 and 1/16 = 0.0625) = 1.9375.  

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