GATE CS 2008 Q6 — Number Representations (Fixed and Floating Point)

To find the values of the radix $r$ that satisfy the equation $\sqrt{121_r} = 11_r$, we convert both sides of the equation to decimal (base 10):

1. Convert $121_r$ to base 10: $$121_r = 1 \cdot r^2 + 2 \cdot r^1 + 1 \cdot r^0 = r^2 + 2r + 1 = (r+1)^2$$
2. Convert $11_r$ to base 10: $$11_r = 1 \cdot r^1 + 1 \cdot r^0 = r + 1$$

Substituting these back into the original equation gives:

Since the radix $r$ is always a positive integer, $r + 1 > 0$, which means the equation $\sqrt{(r+1)^2} = r + 1$ is an identity and holds true for any valid radix $r$.

Constraint on the radix $r$:
For a number to be validly represented in base $r$, all digits must be strictly less than $r$. The digits present in $121_r$ are $1$ and $2$. Therefore, the radix must satisfy: $$r > 2$$

Thus, the equation is satisfied for any value $r > 2$.