How Floating-Point Numbers Are Represented

Introduction to Binary Representation00:00:00

Computers store integers using binary, where each bit represents a power of 2. However, representing fractional numbers requires different methods.

Fixed-Point Representation00:00:50

Fixed-point representation allocates a set number of bits to the integer and fractional parts. While simple, it lacks dynamic range and precision for certain values.

Floating-Point Representation00:02:37

Modern computers use the floating-point approach, similar to scientific notation. It consists of a sign bit, an exponent, and a mantissa (significand), allowing for a wider range of values.

The IEEE 754 Standard00:04:16

The IEEE 754 standard defines how floating-point numbers are stored, primarily using single-precision (32-bit) and double-precision (64-bit) formats.

Deep Dive into Single-Precision00:05:09

In 32-bit format, the exponent uses a bias of 127 to handle negative powers of 2. The mantissa assumes a leading 1, effectively increasing precision.

Handling Special Cases00:07:07

The standard uses extreme exponent values (0 and 255) to represent zero, infinity, 'not a number' (NaN), and subnormal numbers.

Double-Precision Format00:09:02

Double-precision uses 64 bits, providing 11 bits for the exponent and 52 bits for the mantissa, resulting in greater range and higher precision.