Computer Arithmetic

Computers do arithmetic with finite representations. That constraint shapes every result. An integer type covers a bounded set of whole numbers. A floating-point type covers a finite, unevenly spaced set of real-number approximations.

You need this mental model whenever values can become large, small, fractional, or sensitive to rounding. It explains why an integer can wrap, why decimal 0.1 is usually approximate in binary floating point, and why two algebraically equivalent expressions can produce different machine results.

Start with representation

A numeral such as 1011 is a pattern. Its value depends on the radix and interpretation. In base two, each position has a power-of-two weight. Unsigned interpretation gives all bits nonnegative weights. Two's-complement interpretation gives the highest bit a negative weight and provides the usual signed range.

Width matters. An unsigned value with n bits ranges from zero through 2^n - 1. A two's-complement signed value with n bits ranges from -2^(n-1) through 2^(n-1) - 1. The same bits can therefore denote different values under different types.