GATE CS 2014 Set 1 Q8 — Number Representations (Fixed and Floating Point)

Let the base (or radix) of the number system be $r$. Since the digits $0, 1, 2,$ and $3$ are used in the equation, the base must satisfy $r > 3$.

The given equation is:

We can rewrite this as:

Converting both sides to their base-10 algebraic representations:

• $312_r = 3r^2 + 1r + 2$
• $20_r = 2r + 0 = 2r$
• $13.1_r = 1r^1 + 3r^0 + 1r^{-1} = r + 3 + \frac{1}{r}$

Substituting these into the equation:

Expanding the right-hand side:

Subtracting $2r^2 + 6r + 2$ from both sides gives:

Since the radix $r$ must be a positive integer greater than $3$, we discard $r = 0$.

Thus, the base of the number system is 5.