Changing Bases Made Simple: A Guide

[cdhotfire]

Is there an easy way to change bases?

 

[Tom McCurdy]

yeah find a program to do it, or write your own... lol what bases are you talking about .. there are tricks for some.

 

[cdhotfire]

i mean like any bases, what tricks are there, i like tricks.

 

[HallsofIvy]

Since you can easily do arithmetic in base 10, it is relatively easy to change from base 10 to any other base.

Example: write 116 in base 3.

3 divides into 116 38 times with remainder 2: 116= 3(38)+ 2
3 divides into 38 12 times with remainder 2: 38= 3(12)+ 2 so
116= 3(3(12)+ 2)+ 2
3 divides into 12 4 times with remainder 0: 12= 3(4) so
116= 3(3(3(4))+ 2)+ 2
3 divides into 4 once with remainder 1:
116= 3(3(3(3+ 1))+ 2)+ 2= 1*3⁴+1*3³+ 0*3²+ 2*3+ 2 = 11022₃.

That is, the last quotient, followed by the remainders.


Similarly, to convert 389 to base 7: 7 divides into 389 55 times with remainder 4
7 divides into 55 7 times with remainder 6
7 divides into 7 1 time with remainder 0.
Thus 389= 1064₇.

IF you could easily do arithmetic in, say base 7, changing from base 7 to base 10 would be just as easy. Unfortunately I'm not that good in other bases so the simplest way of changing from base 7 to base 10 is just to use the definition of "base" notation.
1064₇= 1*7³+ 0*7²+ 6*7+ 4= 1*343+ 0*49+ 6*7+ 4= 343+ 42+ 4= 389₁₀.

 

[cdhotfire]

that really helps alot, thank you.