TEZPUR 2025 — Computer PYQ

The correct answer is (b) Rounding errors.

Floating-point numbers are used to represent real numbers in computers. Because computers have finite memory, they can only store a limited number of bits for the significand (fractional part) and the exponent. This leads to the fundamental limitation of representing infinite real numbers with finite precision.

• Floating-Point Representation: Most systems follow the IEEE 754 standard. When a number cannot be represented exactly in binary form (like $0.1$), the system must round it to the nearest representable value.

• Cumulative Error: In long chains of calculations, these small rounding errors can accumulate, leading to significant inaccuracies.

Mathematically, if $x$ is the true value and $fl(x)$ is the floating-point representation, the relative error $\delta$ is:

$$\delta =\left|\frac{x-fl(x)}{x}\right|$$

When performing an arithmetic operation $\odot$ (where $\odot \in \{+,-,\times ,÷\}$), the machine result is:

$$fl(x\odot y)=(x\odot y)(1+ϵ)$$