Base Conversions: Trial and Error or Formula?

Date: 11/26/2002 at 03:51:18
From: Bgpremasudha
Subject: Number system

Find base x of 211(x)=152(8)

I.e, find base x of a given number that is equivalent to a number in 
any base.

Answer I found by trial and error: convert 152 in base 8 to a base 10 
number and try by trial and error to find x by dividing the base 10 
number successively.

Date: 11/26/2002 at 04:47:15
From: Doctor Mike
Subject: Re: Number system

Hello,

Yes, first certainly you should find out what octal 152 is equal to in 
base ten. Call that T.  I'm not clear on what you mean by the next 
step. Perhaps you mean to use trial and error with some different 
values of x, to see if 211 base x is equal to the base ten number T 
that you found.

Here is another way that does NOT use trial and error. You know that 
the digits in 211 (base x) stand for powers of x. The 3 different 
places in this number stand for (from right to left) 1=x^0, x=x^1, and 
x*x=x^2, where ^ means to raise to a power. Is that clear? So, one way 
of writing what 211 base x means (in the base ten system) is .......
  
    2*x*x + x + 1
  
The way we want it to come out is for that to equal T (base 10).
   
    2*x*x + x + 1 = T
  
If you then subtract T from both sides, you get the equivalent,
   
    2*x*x + x + (1-T) = 0
  
which is a quadratic equation in standard form, with zero on the 
right. You can solve this for x with the Quadratic Formula. Be sure to 
check your value for x. Note: the Q.F. will give you two solution 
values for x. Only the positive one makes sense.
  
Which way do you like better?  Trial and error, or Not?  

- Doctor Mike, The Math Forum
  http://mathforum.org/dr.math/