# How to find the radix ( BASE )?

Whats the method to find the base of any given number,

eg. (121) r = (144) 8

here r and 8 is a base

So how do i find the value of r?

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By definition $$144_8=1\cdot8^2+4\cdot8^1+4\cdot8^0=64+32+4=100\;,$$ and $$121_r=1\cdot r^2+2\cdot r^1+1\cdot r^0=r^2+2r+1\;.$$ To find $r$, just solve the quadratic equation $$r^2+2r+1=100$$ by whatever method you find most convenient; the slickest method is probably to notice that $r^2+2r+1=(r+1)^2$, so $(r+1)^2=100=10^2$.
 So value of r here would be i guess, 9 ? (r+1)^2= 100 -> r+1 = 10 -> r = 10-1 -> r =9 – Kishan Thobhani Oct 20 '12 at 12:22 @Kishan: That’s right. And you can now check: $$121_9=1\cdot9^2+2\cdot9^1+1\cdot9^0=81+18+1=100\;.$$ – Brian M. Scott Oct 20 '12 at 12:27