Floating-Point Number Representation

numeric

Definition

Floating-Point Number Representation

Floating-point representation expresses a real number in the form

where is the base, the exponent, and the mantissa (significand) written with digits in base . The mantissa is normalised: its leading digit is nonzero, so for that leading digit is necessarily and need not be stored — the binary implicit leading bit. The stored layout is a bit word partitioned into a sign bit, an exponent field, and the mantissa field.

Floating-point is the standard number representation of real numbers on a computer, trading exactness for the ability to span a wide range of magnitudes with fixed storage.

Exponent Notation

A real number admits many decompositions — the pair is not determined by alone.

Non-unique decomposition

Every form is arithmetically equal; they differ only in where the radix point sits.

To make the representation unique in memory, the mantissa is normalised: exactly one nonzero digit sits before the radix point. The example then collapses to a single canonical form:

For the normalisation becomes free storage: the leading digit of must be , so it carries no information and need not be stored — the binary implicit leading bit already named in the definition.

Layout

The first bit stores the sign of the represented number:

• means positive, and
• means negative.

The exponent field stores the scale factor . The mantissa field stores the significant digits as a fixed-point number with exactly one digit before the radix point; for normalised binary numbers, that leading digit is implicit.

IEEE 754

Definition

IEEE 754

IEEE 754 is the dominant standard for floating-point number representation on computers, fixing and a family of precisions via the parameter set

Each format allocates a bit word into a one-bit sign, an exponent field of bits, and a mantissa field of bits (the leading of a normalised mantissa is implicit, so only the fractional bits are stored). The exponent is encoded with an excess (bias) , so the stored bits represent .

Link to original

Rounding

Rounding is important in numeric since computers can only store a certain number of bits, meaning a limited precision. There are multiple methods of rounding for floating point numbers.

Truncate

Truncating means removing a part of or the whole fractional part from a number.

Example: Truncating to 2 decimal digits is .

Directed Rounding

Directed rounding refers to rounding an number in a specific direction:

1. Round Towards Zero: Always rounds the number closer to zero.
2. Round Away from Zero: Always rounds the number farther from zero.

Examples:

• Rounding Towards Zero:
• Round Away from Zero:

Round to Nearest

Rounding to the nearest refers to rounding a number to the closest value at a specified level of precision, such as the nearest integer, tenth, hundredth, or any other place value.

Precision of Rounding